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Numerical Methods for Engineers S E VEN TH ED IT I ON

Steven C. Chapra Berger Chair in Computing and Engineering Tufts University

Raymond P. Canale Professor Emeritus of Civil Engineering University of Michigan

NUMERICAL METHODS FOR ENGINEERS, SEVENTH EDITION Published by McGraw-Hill Education, 2 Penn Plaza, New York, NY 10121. Copyright © 2015 by McGraw-Hill Education. All rights reserved. Printed in the United States of America. Previous editions © 2010, 2006, and 2002. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of McGraw-Hill Education, including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. Some ancillaries, including electronic and print components, may not be available to customers outside the United States. This book is printed on acid-free paper. 1 2 3 4 5 6 7 8 9 0 DOC/DOC 1 0 9 8 7 6 5 4 ISBN 978–0–07–339792–4 MHID 0–07–339792–x Senior Vice President, Products & Markets: Kurt L. Strand Vice President, General Manager, Products & Markets: Marty Lange Vice President, Content Production & Technology Services: Kimberly Meriwether David Executive Brand Manager: Bill Stenquist Managing Director: Thomas Timp Global Publisher: Raghothaman Srinivasan Developmental Editor: Lorraine Buczek Marketing Manager: Heather Wagner Director, Content Production: Terri Schiesl Senior Content Project Manager: Melissa M. Leick Buyer: Jennifer Pickel Cover Designer: Studio Montage, St. Louis, MO Cover Image: Peak towering above clouds: Royalty-Free/CORBIS; Skysurfers: Getty Images/Digital Vision/RF Media Project Manager: Sandra M. Schnee Compositor: Aptara®, Inc. Typeface: 10/12 Time Roman Printer: R. R. Donnelley All credits appearing on page or at the end of the book are considered to be an extension of the copyright page. Library of Congress Cataloging-in-Publication Data Chapra, Steven C. Numerical methods for engineers / Steven C. Chapra, Berger chair in computing and engineering, Tufts University, Raymond P. Canale, professor emeritus of civil engineering, University of Michigan. — Seventh edition. pages cm Includes bibliographical references and index. ISBN 978-0-07-339792-4 (alk. paper) — ISBN 0-07-339792-X (alk. paper) 1. Engineering mathematics—Data processing. 2. Numerical calculations—Data processing 3. Microcomputers—Programming. I. Canale, Raymond P. II. Title. TA345.C47 2015 518.024’62—dc23

2013041704

To Margaret and Gabriel Chapra Helen and Chester Canale

CONTENTS PREFACE

xiv

ABOUT THE AUTHORS

xvi

PART ONE MODELING, COMPUTERS, AND ERROR ANALYSIS 3

PT1.1 Motivation 3 PT1.2 Mathematical Background PT1.3 Orientation 8

5

CHAPTER 1 Mathematical Modeling and Engineering Problem Solving 1.1 A Simple Mathematical Model 11 1.2 Conservation Laws and Engineering Problems 21 CHAPTER 2 Programming and Software

18

27

2.1 Packages and Programming 27 2.2 Structured Programming 28 2.3 Modular Programming 37 2.4 Excel 39 2.5 MATLAB 43 2.6 Mathcad 47 2.7 Other Languages and Libraries 48 Problems 49 CHAPTER 3 Approximations and Round-Off Errors 3.1 Signiﬁcant Figures 56 3.2 Accuracy and Precision 3.3 Error Deﬁnitions 59 3.4 Round-Off Errors 65 Problems 79

iv

58

55

11

CONTENTS

v

CHAPTER 4 Truncation Errors and the Taylor Series

81

4.1 The Taylor Series 81 4.2 Error Propagation 97 4.3 Total Numerical Error 101 4.4 Blunders, Formulation Errors, and Data Uncertainty Problems 108 EPILOGUE: PART ONE 110 PT1.4 Trade-Offs 110 PT1.5 Important Relationships and Formulas 113 PT1.6 Advanced Methods and Additional References

106

113

PART TWO ROOTS OF EQUATIONS 117

PT2.1 Motivation 117 PT2.2 Mathematical Background PT2.3 Orientation 120 CHAPTER 5 Bracketing Methods

119

123

5.1 Graphical Methods 123 5.2 The Bisection Method 127 5.3 The False-Position Method 135 5.4 Incremental Searches and Determining Initial Guesses Problems 142 CHAPTER 6 Open Methods

145

6.1 Simple Fixed-Point Iteration 146 6.2 The Newton-Raphson Method 151 6.3 The Secant Method 157 6.4 Brent’s Method 162 6.5 Multiple Roots 166 6.6 Systems of Nonlinear Equations 169 Problems 173 CHAPTER 7 Roots of Polynomials

176

7.1 Polynomials in Engineering and Science 7.2 Computing with Polynomials 179 7.3 Conventional Methods 182

176

141

vi

CONTENTS 7.4 Müller’s Method 183 7.5 Bairstow’s Method 187 7.6 Other Methods 192 7.7 Root Location with Software Packages Problems 202 CHAPTER 8 Case Studies: Roots of Equations

192

204

8.1 Ideal and Nonideal Gas Laws (Chemical/Bio Engineering) 204 8.2 Greenhouse Gases and Rainwater (Civil/Environmental Engineering) 8.3 Design of an Electric Circuit (Electrical Engineering) 209 8.4 Pipe Friction (Mechanical/Aerospace Engineering) 212 Problems 215 EPILOGUE: PART TWO 226 PT2.4 Trade-Offs 226 PT2.5 Important Relationships and Formulas 227 PT2.6 Advanced Methods and Additional References

PART THREE LINEAR ALGEBRAIC EQUATIONS 231

PT3.1 Motivation 231 PT3.2 Mathematical Background PT3.3 Orientation 241 CHAPTER 9 Gauss Elimination

233

245

9.1 Solving Small Numbers of Equations 245 9.2 Naive Gauss Elimination 252 9.3 Pitfalls of Elimination Methods 258 9.4 Techniques for Improving Solutions 264 9.5 Complex Systems 271 9.6 Nonlinear Systems of Equations 271 9.7 Gauss-Jordan 273 9.8 Summary 275 Problems 275 CHAPTER 10 LU Decomposition and Matrix Inversion 10.1 LU Decomposition 278 10.2 The Matrix Inverse 287 10.3 Error Analysis and System Condition Problems 297

291

278

227

207

CONTENTS

vii

CHAPTER 11 Special Matrices and Gauss-Seidel

300

11.1 Special Matrices 300 11.2 Gauss-Seidel 304 11.3 Linear Algebraic Equations with Software Packages Problems 316 CHAPTER 12 Case Studies: Linear Algebraic Equations

311

319

12.1 Steady-State Analysis of a System of Reactors (Chemical/Bio Engineering) 319 12.2 Analysis of a Statically Determinate Truss (Civil/Environmental Engineering) 322 12.3 Currents and Voltages in Resistor Circuits (Electrical Engineering) 326 12.4 Spring-Mass Systems (Mechanical/Aerospace Engineering) 328 Problems 331 EPILOGUE: PART THREE 341 PT3.4 Trade-Offs 341 PT3.5 Important Relationships and Formulas 342 PT3.6 Advanced Methods and Additional References

342

PART FOUR OPTIMIZATION

345

PT4.1 Motivation 345 PT4.2 Mathematical Background PT4.3 Orientation 351

350

CHAPTER 13 One-Dimensional Unconstrained Optimization

355

13.1 Golden-Section Search 356 13.2 Parabolic Interpolation 363 13.3 Newton’s Method 365 13.4 Brent’s Method 366 Problems 368 CHAPTER 14 Multidimensional Unconstrained Optimization 14.1 Direct Methods 371 14.2 Gradient Methods 375 Problems 388

370

viii

CONTENTS CHAPTER 15 Constrained Optimization

390

15.1 Linear Programming 390 15.2 Nonlinear Constrained Optimization 401 15.3 Optimization with Software Packages 402 Problems 413 CHAPTER 16 Case Studies: Optimization

416

16.1 Least-Cost Design of a Tank (Chemical/Bio Engineering) 416 16.2 Least-Cost Treatment of Wastewater (Civil/Environmental Engineering) 421 16.3 Maximum Power Transfer for a Circuit (Electrical Engineering) 425 16.4 Equilibrium and Minimum Potential Energy (Mechanical/Aerospace Engineering) 429 Problems 431 EPILOGUE: PART FOUR 438 PT4.4 Trade-Offs 438 PT4.5 Additional References 439

PART FIVE CURVE FITTING 441

PT5.1 Motivation 441 PT5.2 Mathematical Background PT5.3 Orientation 452 CHAPTER 17 Least-Squares Regression

443

456

17.1 Linear Regression 456 17.2 Polynomial Regression 472 17.3 Multiple Linear Regression 476 17.4 General Linear Least Squares 479 17.5 Nonlinear Regression 483 Problems 487 CHAPTER 18 Interpolation

490

18.1 Newton’s Divided-Difference Interpolating Polynomials 18.2 Lagrange Interpolating Polynomials 502 18.3 Coefﬁcients of an Interpolating Polynomial 507 18.4 Inverse Interpolation 507 18.5 Additional Comments 508 18.6 Spline Interpolation 511 18.7 Multidimensional Interpolation 521 Problems 524

491

CONTENTS CHAPTER 19 Fourier Approximation

ix

526

19.1 Curve Fitting with Sinusoidal Functions 527 19.2 Continuous Fourier Series 533 19.3 Frequency and Time Domains 536 19.4 Fourier Integral and Transform 540 19.5 Discrete Fourier Transform (DFT) 542 19.6 Fast Fourier Transform (FFT) 544 19.7 The Power Spectrum 551 19.8 Curve Fitting with Software Packages 552 Problems 561 CHAPTER 20 Case Studies: Curve Fitting

563

20.1 Linear Regression and Population Models (Chemical/Bio Engineering) 563 20.2 Use of Splines to Estimate Heat Transfer (Civil/Environmental Engineering) 567 20.3 Fourier Analysis (Electrical Engineering) 569 20.4 Analysis of Experimental Data (Mechanical/Aerospace Engineering) 570 Problems 572 EPILOGUE: PART FIVE 582 PT5.4 Trade-Offs 582 PT5.5 Important Relationships and Formulas 583 PT5.6 Advanced Methods and Additional References

PART SIX NUMERICAL DIFFERENTIATION AND INTEGRATION 587

PT6.1 Motivation 587 PT6.2 Mathematical Background PT6.3 Orientation 599

597

CHAPTER 21 Newton-Cotes Integration Formulas 21.1 The Trapezoidal Rule 605 21.2 Simpson’s Rules 615 21.3 Integration with Unequal Segments 21.4 Open Integration Formulas 627 21.5 Multiple Integrals 627 Problems 629

603

624

584

x

CONTENTS CHAPTER 22 Integration of Equations

633

22.1 Newton-Cotes Algorithms for Equations 22.2 Romberg Integration 634 22.3 Adaptive Quadrature 640 22.4 Gauss Quadrature 642 22.5 Improper Integrals 650 Problems 653 CHAPTER 23 Numerical Differentiation

633

655

23.1 High-Accuracy Differentiation Formulas 655 23.2 Richardson Extrapolation 658 23.3 Derivatives of Unequally Spaced Data 660 23.4 Derivatives and Integrals for Data with Errors 661 23.5 Partial Derivatives 662 23.6 Numerical Integration/Differentiation with Software Packages Problems 670 CHAPTER 24 Case Studies: Numerical Integration and Differentiation

663

673

24.1 Integration to Determine the Total Quantity of Heat (Chemical/Bio Engineering) 673 24.2 Effective Force on the Mast of a Racing Sailboat (Civil/Environmental Engineering) 675 24.3 Root-Mean-Square Current by Numerical Integration (Electrical Engineering) 677 24.4 Numerical Integration to Compute Work (Mechanical/Aerospace Engineering) 680 Problems 684 EPILOGUE: PART SIX 694 PT6.4 Trade-Offs 694 PT6.5 Important Relationships and Formulas 695 PT6.6 Advanced Methods and Additional References

PART SEVEN ORDINARY DIFFERENTIAL EQUATIONS 699

PT7.1 Motivation 699 PT7.2 Mathematical Background PT7.3 Orientation 705

703

695

CONTENTS CHAPTER 25 Runge-Kutta Methods

xi

709

25.1 Euler’s Method 710 25.2 Improvements of Euler’s Method 721 25.3 Runge-Kutta Methods 729 25.4 Systems of Equations 739 25.5 Adaptive Runge-Kutta Methods 744 Problems 752 CHAPTER 26 Stiffness and Multistep Methods 26.1 Stiffness 755 26.2 Multistep Methods Problems 779

755

759

CHAPTER 27 Boundary-Value and Eigenvalue Problems

781

27.1 General Methods for Boundary-Value Problems 782 27.2 Eigenvalue Problems 789 27.3 Odes and Eigenvalues with Software Packages 801 Problems 808 CHAPTER 28 Case Studies: Ordinary Differential Equations

811

28.1 Using ODEs to Analyze the Transient Response of a Reactor (Chemical/Bio Engineering) 811 28.2 Predator-Prey Models and Chaos (Civil/Environmental Engineering) 818 28.3 Simulating Transient Current for an Electric Circuit (Electrical Engineering) 822 28.4 The Swinging Pendulum (Mechanical/Aerospace Engineering) 827 Problems 831 EPILOGUE: PART SEVEN 841 PT7.4 Trade-Offs 841 PT7.5 Important Relationships and Formulas 842 PT7.6 Advanced Methods and Additional References

PART EIGHT PARTIAL DIFFERENTIAL EQUATIONS 845

PT8.1 Motivation 845 PT8.2 Orientation 848

842

xii

CONTENTS CHAPTER 29 Finite Difference: Elliptic Equations

852

29.1 The Laplace Equation 852 29.2 Solution Technique 854 29.3 Boundary Conditions 860 29.4 The Control-Volume Approach 866 29.5 Software to Solve Elliptic Equations 869 Problems 870 CHAPTER 30 Finite Difference: Parabolic Equations

873

30.1 The Heat-Conduction Equation 873 30.2 Explicit Methods 874 30.3 A Simple Implicit Method 878 30.4 The Crank-Nicolson Method 882 30.5 Parabolic Equations in Two Spatial Dimensions Problems 888 CHAPTER 31 Finite-Element Method

885

890

31.1 The General Approach 891 31.2 Finite-Element Application in One Dimension 31.3 Two-Dimensional Problems 904 31.4 Solving PDEs with Software Packages 908 Problems 912

895

CHAPTER 32 Case Studies: Partial Differential Equations

915

32.1 One-Dimensional Mass Balance of a Reactor (Chemical/Bio Engineering) 915 32.2 Deﬂections of a Plate (Civil/Environmental Engineering) 919 32.3 Two-Dimensional Electrostatic Field Problems (Electrical Engineering) 921 32.4 Finite-Element Solution of a Series of Springs (Mechanical/Aerospace Engineering) 924 Problems 928 EPILOGUE: PART EIGHT 931 PT8.3 Trade-Offs 931 PT8.4 Important Relationships and Formulas 931 PT8.5 Advanced Methods and Additional References

932

CONTENTS

xiii

APPENDIX A: THE FOURIER SERIES

933

APPENDIX B: GETTING STARTED WITH MATLAB

935

APPENDIX C: GETTING STARTED WITH MATHCAD BIBLIOGRAPHY INDEX

957

954

943

PREFACE It has been over twenty years since we published the first edition of this book. Over that period, our original contention that numerical methods and computers would figure more prominently in the engineering curriculum—particularly in the early parts—has been dramatically borne out. Many universities now offer freshman, sophomore, and junior courses in both introductory computing and numerical methods. In addition, many of our colleagues are integrating computer-oriented problems into other courses at all levels of the curriculum. Thus, this new edition is still founded on the basic premise that student engineers should be provided with a strong and early introduction to numerical methods. Consequently, although we have expanded our coverage in the new edition, we have tried to maintain many of the features that made the first edition accessible to both lower- and upper-level undergraduates. These include: • Problem Orientation. Engineering students learn best when they are motivated by problems. This is particularly true for mathematics and computing. Consequently, we have approached numerical methods from a problem-solving perspective. • Student-Oriented Pedagogy. We have developed a number of features to make this book as student-friendly as possible. These include the overall organization, the use of introductions and epilogues to consolidate major topics and the extensive use of worked examples and case studies from all areas of engineering. We have also endeavored to keep our explanations straightforward and oriented practically. • Computational Tools. We empower our students by helping them utilize the standard “point-and-shoot” numerical problem-solving capabilities of packages like Excel, MATLAB, and Mathcad software. However, students are also shown how to develop simple, well-structured programs to extend the base capabilities of those environments. This knowledge carries over to standard programming languages such as Visual Basic, Fortran 90, and C/C11. We believe that the current flight from computer programming represents something of a “dumbing down” of the engineering curriculum. The bottom line is that as long as engineers are not content to be tool limited, they will have to write code. Only now they may be called “macros” or “M-files.” This book is designed to empower them to do that. Beyond these five original principles, the seventh edition has new and expanded problem sets. Most of the problems have been modified so that they yield different numerical solutions from previous editions. In addition, a variety of new problems have been included. The seventh edition also includes McGraw-Hill’s Connect® Engineering. This online homework management tool allows assignment of algorithmic problems for homework, quizzes, and tests. It connects students with the tools and resources they’ll need to achieve success. To learn more, visit www.mcgrawhillconnect.com. McGraw-Hill LearnSmart™ is also available as an integrated feature of McGraw-Hill Connect® Engineering. It is an adaptive learning system designed to help students learn faster, study more efficiently, and retain more knowledge for greater success. LearnSmart assesses xiv

PREFACE

xv

a student’s knowledge of course content through a series of adaptive questions. It pinpoints concepts the student does not understand and maps out a personalized study plan for success. Visit the following site for a demonstration. www.mhlearnsmart.com As always, our primary intent in writing this book is to provide students with a sound introduction to numerical methods. We believe that motivated students who enjoy numerical methods, computers, and mathematics will, in the end, make better engineers. If our book fosters an enthusiasm for these subjects, we will consider our efforts a success. Acknowledgments. We would like to thank our friends at McGraw-Hill. In particular, Lorraine Buczek and Bill Stenquist, who provided a positive and supportive atmosphere for creating this edition. As usual, Beatrice Sussman did a masterful job of copyediting the manuscript and Arpana Kumari of Aptara also did an outstanding job in the book’s final production phase. As in past editions, David Clough (University of Colorado), Mike Gustafson (Duke), and Jerry Stedinger (Cornell University) generously shared their insights and suggestions. Useful suggestions were also made by Bill Philpot (Cornell University), Jim Guilkey (University of Utah), Dong-Il Seo (Chungnam National University, Korea), Niall Broekhuizen (NIWA, New Zealand), and Raymundo Cordero and Karim Muci (ITESM, Mexico). The present edition has also benefited from the reviews and suggestions by the following colleagues: Betty Barr, University of Houston Jalal Behzadi, Shahid Chamran University Jordan Berg, Texas Tech University Jacob Bishop, Utah State University Estelle M. Eke, California State University, Sacramento Yazan A. Hussain, Jordan University of Science & Technology Yogesh Jaluria, Rutgers University S. Graham Kelly, The University of Akron Subha Kumpaty, Milwaukee School of Engineering Eckart Meiburg, University of California-Santa Barbara Prashant Mhaskar, McMaster University Luke Olson, University of Illinois at Urbana-Champaign Richard Pates Jr., Old Dominion University Joseph H. Pierluissi, University of Texas at El Paso Juan Perán, Universidad Nacional de Educación a Distancia (UNED) Scott A. Socolofsky, Texas A&M University It should be stressed that although we received useful advice from the aforementioned individuals, we are responsible for any inaccuracies or mistakes you may detect in this edition. Please contact Steve Chapra via e-mail if you should detect any errors in this edition. Finally, we would like to thank our family, friends, and students for their enduring patience and support. In particular, Cynthia Chapra, Danielle Husley, and Claire Canale are always there providing understanding, perspective, and love. Steven C. Chapra Medford, Massachusetts [email protected] Raymond P. Canale Lake Leelanau, Michigan

ABOUT THE AUTHORS Steve Chapra teaches in the Civil and Environmental Engineering Department at Tufts University where he holds the Louis Berger Chair in Computing and Engineering. His other books include Surface Water-Quality Modeling and Applied Numerical Methods with MATLAB. Dr. Chapra received engineering degrees from Manhattan College and the University of Michigan. Before joining the faculty at Tufts, he worked for the Environmental Protection Agency and the National Oceanic and Atmospheric Administration, and taught at Texas A&M University and the University of Colorado. His general research interests focus on surface water-quality modeling and advanced computer applications in environmental engineering. He is a Fellow of the ASCE, and has received a number of awards for his scholarly contributions, including the Rudolph Hering Medal (ASCE), and the Meriam-Wiley Distinguished Author Award (American Society for Engineering Education). He has also been recognized as the outstanding teacher among the engineering faculties at Texas A&M University, the University of Colorado, and Tufts University. Raymond P. Canale is an emeritus professor at the University of Michigan. During his over 20-year career at the university, he taught numerous courses in the area of computers, numerical methods, and environmental engineering. He also directed extensive research programs in the area of mathematical and computer modeling of aquatic ecosystems. He has authored or coauthored several books and has published over 100 scientific papers and reports. He has also designed and developed personal computer software to facilitate engineering education and the solution of engineering problems. He has been given the Meriam-Wiley Distinguished Author Award by the American Society for Engineering Education for his books and software and several awards for his technical publications. Professor Canale is now devoting his energies to applied problems, where he works with engineering firms and industry and governmental agencies as a consultant and expert witness.

xvi

Numerical Methods for Engineers

PART ONE

MODELING, COMPUTERS, AND ERROR ANALYSIS PT1.1

MOTIVATION Numerical methods are techniques by which mathematical problems are formulated so that they can be solved with arithmetic operations. Although there are many kinds of numerical methods, they have one common characteristic: they invariably involve large numbers of tedious arithmetic calculations. It is little wonder that with the development of fast, efficient digital computers, the role of numerical methods in engineering problem solving has increased dramatically in recent years. PT1.1.1 Noncomputer Methods Beyond providing increased computational firepower, the widespread availability of computers (especially personal computers) and their partnership with numerical methods has had a significant influence on the actual engineering problem-solving process. In the precomputer era there were generally three different ways in which engineers approached problem solving: 1.

2.

3.

Solutions were derived for some problems using analytical, or exact, methods. These solutions were often useful and provided excellent insight into the behavior of some systems. However, analytical solutions can be derived for only a limited class of problems. These include those that can be approximated with linear models and those that have simple geometry and low dimensionality. Consequently, analytical solutions are of limited practical value because most real problems are nonlinear and involve complex shapes and processes. Graphical solutions were used to characterize the behavior of systems. These graphical solutions usually took the form of plots or nomographs. Although graphical techniques can often be used to solve complex problems, the results are not very precise. Furthermore, graphical solutions (without the aid of computers) are extremely tedious and awkward to implement. Finally, graphical techniques are often limited to problems that can be described using three or fewer dimensions. Calculators and slide rules were used to implement numerical methods manually. Although in theory such approaches should be perfectly adequate for solving complex problems, in actuality several difficulties are encountered. Manual calculations are slow and tedious. Furthermore, consistent results are elusive because of simple blunders that arise when numerous manual tasks are performed.

During the precomputer era, significant amounts of energy were expended on the solution technique itself, rather than on problem definition and interpretation (Fig. PT1.1a). This unfortunate situation existed because so much time and drudgery were required to obtain numerical answers using precomputer techniques. 3

4

MODELING, COMPUTERS, AND ERROR ANALYSIS

FORMULATION Fundamental laws explained briefly

SOLUTION

FIGURE PT1.1 The three phases of engineering problem solving in (a) the precomputer and (b) the computer era. The sizes of the boxes indicate the level of emphasis directed toward each phase. Computers facilitate the implementation of solution techniques and thus allow more emphasis to be placed on the creative aspects of problem formulation and interpretation of results.

Elaborate and often complicated method to make problem tractable

FORMULATION In-depth exposition of relationship of problem to fundamental laws

SOLUTION Easy-to-use computer method

INTERPRETATION

INTERPRETATION

In-depth analysis limited by timeconsuming solution

Ease of calculation allows holistic thoughts and intuition to develop; system sensitivity and behavior can be studied

(a)

(b)

Today, computers and numerical methods provide an alternative for such complicated calculations. Using computer power to obtain solutions directly, you can approach these calculations without recourse to simplifying assumptions or time-intensive techniques. Although analytical solutions are still extremely valuable both for problem solving and for providing insight, numerical methods represent alternatives that greatly enlarge your capabilities to confront and solve problems. As a result, more time is available for the use of your creative skills. Thus, more emphasis can be placed on problem formulation and solution interpretation and the incorporation of total system, or “holistic,” awareness (Fig. PT1.1b). PT1.1.2 Numerical Methods and Engineering Practice Since the late 1940s the widespread availability of digital computers has led to a veritable explosion in the use and development of numerical methods. At first, this growth was somewhat limited by the cost of access to large mainframe computers, and, consequently, many engineers continued to use simple analytical approaches in a significant portion of their work. Needless to say, the recent evolution of inexpensive personal

PT1.2

MATHEMATICAL BACKGROUND

5

computers has given us ready access to powerful computational capabilities. There are several additional reasons why you should study numerical methods: 1.

2.

3.

4.

5.

PT1.2

Numerical methods are extremely powerful problem-solving tools. They are capable of handling large systems of equations, nonlinearities, and complicated geometries that are not uncommon in engineering practice and that are often impossible to solve analytically. As such, they greatly enhance your problem-solving skills. During your careers, you may often have occasion to use commercially available prepackaged, or “canned,” computer programs that involve numerical methods. The intelligent use of these programs is often predicated on knowledge of the basic theory underlying the methods. Many problems cannot be approached using canned programs. If you are conversant with numerical methods and are adept at computer programming, you can design your own programs to solve problems without having to buy or commission expensive software. Numerical methods are an efficient vehicle for learning to use computers. It is well known that an effective way to learn programming is to actually write computer programs. Because numerical methods are for the most part designed for implementation on computers, they are ideal for this purpose. Further, they are especially well-suited to illustrate the power and the limitations of computers. When you successfully implement numerical methods on a computer and then apply them to solve otherwise intractable problems, you will be provided with a dramatic demonstration of how computers can serve your professional development. At the same time, you will also learn to acknowledge and control the errors of approximation that are part and parcel of large-scale numerical calculations. Numerical methods provide a vehicle for you to reinforce your understanding of mathematics. Because one function of numerical methods is to reduce higher mathematics to basic arithmetic operations, they get at the “nuts and bolts” of some otherwise obscure topics. Enhanced understanding and insight can result from this alternative perspective.

MATHEMATICAL BACKGROUND Every part in this book requires some mathematical background. Consequently, the introductory material for each part includes a section, such as the one you are reading, on mathematical background. Because Part One itself is devoted to background material on mathematics and computers, this section does not involve a review of a specific mathematical topic. Rather, we take this opportunity to introduce you to the types of mathematical subject areas covered in this book. As summarized in Fig. PT1.2, these are 1.

2.

Roots of Equations (Fig. PT1.2a). These problems are concerned with the value of a variable or a parameter that satisfies a single nonlinear equation. These problems are especially valuable in engineering design contexts where it is often impossible to explicitly solve design equations for parameters. Systems of Linear Algebraic Equations (Fig. PT1.2b). These problems are similar in spirit to roots of equations in the sense that they are concerned with values that

6 FIGURE PT1.2 Summary of the numerical methods covered in this book.

MODELING, COMPUTERS, AND ERROR ANALYSIS

(a) Part 2: Roots of equations

f(x)

Solve f(x) = 0 for x.

Root x

(b) Part 3: Linear algebraic equations Given the a’s and the c’s, solve a11x1 + a12x2 = c1 a21x1 + a22x2 = c2

x2

Solution

for the x’s.

x1

(c) Part 4: Optimization Determine x that gives optimum f(x).

f(x)

Minimum x

(d) Part 5: Curve fitting f(x)

f(x)

Interpolation

Regression

(e) Part 6: Integration I = 兰ab f (x) dx Find the area under the curve.

x

x f(x)

I x

PT1.2 FIGURE PT1.2 (concluded)

MATHEMATICAL BACKGROUND

7

( f ) Part 7: Ordinary differential equations Given dy ⌬y ⯝ = f (t, y) dt ⌬t

y

solve for y as a function of t. yi + 1 = yi + f (ti , yi ) ⌬t

Slope = f(t i , y i )

⌬t ti

(g) Part 8: Partial differential equations

ti + 1

t

Given ⭸2u + ⭸2u = f(x, y) ⭸x2 ⭸y2

y

solve for u as a function of x and y

x

3.

4.

5.

satisfy equations. However, in contrast to satisfying a single equation, a set of values is sought that simultaneously satisfies a set of linear algebraic equations. Such equations arise in a variety of problem contexts and in all disciplines of engineering. In particular, they originate in the mathematical modeling of large systems of interconnected elements such as structures, electric circuits, and fluid networks. However, they are also encountered in other areas of numerical methods such as curve fitting and differential equations. Optimization (Fig. PT1.2c). These problems involve determining a value or values of an independent variable that correspond to a “best” or optimal value of a function. Thus, as in Fig. PT1.2c, optimization involves identifying maxima and minima. Such problems occur routinely in engineering design contexts. They also arise in a number of other numerical methods. We address both single- and multi-variable unconstrained optimization. We also describe constrained optimization with particular emphasis on linear programming. Curve Fitting (Fig. PT1.2d). You will often have occasion to fit curves to data points. The techniques developed for this purpose can be divided into two general categories: regression and interpolation. Regression is employed where there is a significant degree of error associated with the data. Experimental results are often of this kind. For these situations, the strategy is to derive a single curve that represents the general trend of the data without necessarily matching any individual points. In contrast, interpolation is used where the objective is to determine intermediate values between relatively error-free data points. Such is usually the case for tabulated information. For these situations, the strategy is to fit a curve directly through the data points and use the curve to predict the intermediate values. Integration (Fig. PT1.2e). As depicted, a physical interpretation of numerical integration is the determination of the area under a curve. Integration has many

8

MODELING, COMPUTERS, AND ERROR ANALYSIS

6.

7.

PT1.3

applications in engineering practice, ranging from the determination of the centroids of oddly shaped objects to the calculation of total quantities based on sets of discrete measurements. In addition, numerical integration formulas play an important role in the solution of differential equations. Ordinary Differential Equations (Fig. PT1.2f ). Ordinary differential equations are of great significance in engineering practice. This is because many physical laws are couched in terms of the rate of change of a quantity rather than the magnitude of the quantity itself. Examples range from population-forecasting models (rate of change of population) to the acceleration of a falling body (rate of change of velocity). Two types of problems are addressed: initial-value and boundary-value problems. In addition, the computation of eigenvalues is covered. Partial Differential Equations (Fig. PT1.2g). Partial differential equations are used to characterize engineering systems where the behavior of a physical quantity is couched in terms of its rate of change with respect to two or more independent variables. Examples include the steady-state distribution of temperature on a heated plate (two spatial dimensions) or the time-variable temperature of a heated rod (time and one spatial dimension). Two fundamentally different approaches are employed to solve partial differential equations numerically. In the present text, we will emphasize finite-difference methods that approximate the solution in a pointwise fashion (Fig. PT1.2g). However, we will also present an introduction to finite-element methods, which use a piecewise approach.

ORIENTATION Some orientation might be helpful before proceeding with our introduction to numerical methods. The following is intended as an overview of the material in Part One. In addition, some objectives have been included to focus your efforts when studying the material. PT1.3.1 Scope and Preview Figure PT1.3 is a schematic representation of the material in Part One. We have designed this diagram to provide you with a global overview of this part of the book. We believe that a sense of the “big picture” is critical to developing insight into numerical methods. When reading a text, it is often possible to become lost in technical details. Whenever you feel that you are losing the big picture, refer back to Fig. PT1.3 to reorient yourself. Every part of this book includes a similar figure. Figure PT1.3 also serves as a brief preview of the material covered in Part One. Chapter 1 is designed to orient you to numerical methods and to provide motivation by demonstrating how these techniques can be used in the engineering modeling process. Chapter 2 is an introduction and review of computer-related aspects of numerical methods and suggests the level of computer skills you should acquire to efficiently apply succeeding information. Chapters 3 and 4 deal with the important topic of error analysis, which must be understood for the effective use of numerical methods. In addition, an epilogue is included that introduces the trade-offs that have such great significance for the effective implementation of numerical methods.

PT1.3

ORIENTATION

PT 1.1 Motivation

9

PT 1.2 Mathematical background

PT 1.3 Orientation

PART 1 Modeling, Computers, and Error Analysis

PT 1.6 Advanced methods

1.1 A simple model

PT 1.5 Important formulas

CHAPTER 1 Mathematical Modeling and Engineering Problem Solving

EPILOGUE PT 1.4 Trade-offs

1.2 Conservation laws

2.1 Packages and programming

4.4 Miscellaneous errors

4.3 Total numerical error

2.2 Structured programming

CHAPTER 4 Truncation Errors and the Taylor Series

4.2 Error propagation

CHAPTER 2 Programming and Software

2.4 Excel

CHAPTER 3 Approximations and Round-Off Errors

4.1 Taylor series

2.5 MATLAB

2.7 Languages and libraries 2.6 Mathcad

3.4 Round-off errors

3.1 Significant figures 3.3 Error definitions

2.3 Modular programming

3.2 Accuracy and precision

FIGURE PT1.3 Schematic of the organization of the material in Part One: Modeling, Computers, and Error Analysis.

10

MODELING, COMPUTERS, AND ERROR ANALYSIS TABLE PT1.1 Speciﬁc study objectives for Part One. 1. Recognize the difference between analytical and numerical solutions. 2. Understand how conservation laws are employed to develop mathematical models of physical systems. 3. Deﬁne top-down and modular design. 4. Delineate the rules that underlie structured programming. 5. Be capable of composing structured and modular programs in a high-level computer language. 6. Know how to translate structured ﬂowcharts and pseudocode into code in a high-level language. 7. Start to familiarize yourself with any software packages that you will be using in conjunction with this text. 8. Recognize the distinction between truncation and round-off errors. 9. Understand the concepts of signiﬁcant ﬁgures, accuracy, and precision. 10. Recognize the difference between true relative error et, approximate relative error ea, and acceptable error es, and understand how ea and es are used to terminate an iterative computation. 11. Understand how numbers are represented in digital computers and how this representation induces round-off error. In particular, know the difference between single and extended precision. 12. Recognize how computer arithmetic can introduce and amplify round-off errors in calculations. In particular, appreciate the problem of subtractive cancellation. 13. Understand how the Taylor series and its remainder are employed to represent continuous functions. 14. Know the relationship between ﬁnite divided differences and derivatives. 15. Be able to analyze how errors are propagated through functional relationships. 16. Be familiar with the concepts of stability and condition. 17. Familiarize yourself with the trade-offs outlined in the Epilogue of Part One.

PT1.3.2 Goals and Objectives Study Objectives. Upon completing Part One, you should be adequately prepared to embark on your studies of numerical methods. In general, you should have gained a fundamental understanding of the importance of computers and the role of approximations and errors in the implementation and development of numerical methods. In addition to these general goals, you should have mastered each of the specific study objectives listed in Table PT1.1. Computer Objectives. Upon completing Part One, you should have mastered sufficient computer skills to develop your own software for the numerical methods in this text. You should be able to develop well-structured and reliable computer programs on the basis of pseudocode, flowcharts, or other forms of algorithms. You should have developed the capability to document your programs so that they may be effectively employed by users. Finally, in addition to your own programs, you may be using software packages along with this book. Packages like Excel, Mathcad, or The MathWorks, Inc. MATLAB® program are examples of such software. You should become familiar with these packages, so that you will be comfortable using them to solve numerical problems later in the text.

1

CHAPTER

Mathematical Modeling and Engineering Problem Solving Knowledge and understanding are prerequisites for the effective implementation of any tool. No matter how impressive your tool chest, you will be hard-pressed to repair a car if you do not understand how it works. This is particularly true when using computers to solve engineering problems. Although they have great potential utility, computers are practically useless without a fundamental understanding of how engineering systems work. This understanding is initially gained by empirical means—that is, by observation and experiment. However, while such empirically derived information is essential, it is only half the story. Over years and years of observation and experiment, engineers and scientists have noticed that certain aspects of their empirical studies occur repeatedly. Such general behavior can then be expressed as fundamental laws that essentially embody the cumulative wisdom of past experience. Thus, most engineering problem solving employs the two-pronged approach of empiricism and theoretical analysis (Fig. 1.1). It must be stressed that the two prongs are closely coupled. As new measurements are taken, the generalizations may be modified or new ones developed. Similarly, the generalizations can have a strong influence on the experiments and observations. In particular, generalizations can serve as organizing principles that can be employed to synthesize observations and experimental results into a coherent and comprehensive framework from which conclusions can be drawn. From an engineering problem-solving perspective, such a framework is most useful when it is expressed in the form of a mathematical model. The primary objective of this chapter is to introduce you to mathematical modeling and its role in engineering problem solving. We will also illustrate how numerical methods figure in the process.

1.1

A SIMPLE MATHEMATICAL MODEL A mathematical model can be broadly defined as a formulation or equation that expresses the essential features of a physical system or process in mathematical terms. In a very general sense, it can be represented as a functional relationship of the form forcing independent Dependent , parameters, b 5f a variables functions variable

(1.1) 11

12

MATHEMATICAL MODELING AND ENGINEERING PROBLEM SOLVING

Problem definition

THEORY

Mathematical model

DATA

Problem-solving tools: computers, statistics, numerical methods, graphics, etc.

Numeric or graphic results Societal interfaces: scheduling, optimization, communication, public interaction, etc.

Implementation

FIGURE 1.1 The engineering problemsolving process.

where the dependent variable is a characteristic that usually reflects the behavior or state of the system; the independent variables are usually dimensions, such as time and space, along which the system’s behavior is being determined; the parameters are reflective of the system’s properties or composition; and the forcing functions are external influences acting upon the system. The actual mathematical expression of Eq. (1.1) can range from a simple algebraic relationship to large complicated sets of differential equations. For example, on the basis of his observations, Newton formulated his second law of motion, which states that the time rate of change of momentum of a body is equal to the resultant force acting on it. The mathematical expression, or model, of the second law is the wellknown equation F 5 ma

(1.2)

where F 5 net force acting on the body (N, or kg m/s2), m 5 mass of the object (kg), and a 5 its acceleration (m/s2).

1.1 FU

A SIMPLE MATHEMATICAL MODEL

13

The second law can be recast in the format of Eq. (1.1) by merely dividing both sides by m to give a5

F m

(1.3)

where a 5 the dependent variable reflecting the system’s behavior, F 5 the forcing function, and m 5 a parameter representing a property of the system. Note that for this simple case there is no independent variable because we are not yet predicting how acceleration varies in time or space. Equation (1.3) has several characteristics that are typical of mathematical models of the physical world: 1. 2. FD

FIGURE 1.2 Schematic diagram of the forces acting on a falling parachutist. FD is the downward force due to gravity. FU is the upward force due to air resistance.

3.

It describes a natural process or system in mathematical terms. It represents an idealization and simplification of reality. That is, the model ignores negligible details of the natural process and focuses on its essential manifestations. Thus, the second law does not include the effects of relativity that are of minimal importance when applied to objects and forces that interact on or about the earth’s surface at velocities and on scales visible to humans. Finally, it yields reproducible results and, consequently, can be used for predictive purposes. For example, if the force on an object and the mass of an object are known, Eq. (1.3) can be used to compute acceleration.

Because of its simple algebraic form, the solution of Eq. (1.2) can be obtained easily. However, other mathematical models of physical phenomena may be much more complex, and either cannot be solved exactly or require more sophisticated mathematical techniques than simple algebra for their solution. To illustrate a more complex model of this kind, Newton’s second law can be used to determine the terminal velocity of a freefalling body near the earth’s surface. Our falling body will be a parachutist (Fig. 1.2). A model for this case can be derived by expressing the acceleration as the time rate of change of the velocity (dy兾dt) and substituting it into Eq. (1.3) to yield dy F 5 m dt

(1.4)

where y is velocity (m/s) and t is time (s). Thus, the mass multiplied by the rate of change of the velocity is equal to the net force acting on the body. If the net force is positive, the object will accelerate. If it is negative, the object will decelerate. If the net force is zero, the object’s velocity will remain at a constant level. Next, we will express the net force in terms of measurable variables and parameters. For a body falling within the vicinity of the earth (Fig. 1.2), the net force is composed of two opposing forces: the downward pull of gravity FD and the upward force of air resistance FU: F 5 FD 1 FU

(1.5)

If the downward force is assigned a positive sign, the second law can be used to formulate the force due to gravity, as FD 5 mg

(1.6)

where g 5 the gravitational constant, or the acceleration due to gravity, which is approximately equal to 9.81 m/s2.

14

MATHEMATICAL MODELING AND ENGINEERING PROBLEM SOLVING

Air resistance can be formulated in a variety of ways. A simple approach is to assume that it is linearly proportional to velocity1 and acts in an upward direction, as in FU 5 2cy

(1.7)

where c 5 a proportionality constant called the drag coefficient (kg/s). Thus, the greater the fall velocity, the greater the upward force due to air resistance. The parameter c accounts for properties of the falling object, such as shape or surface roughness, that affect air resistance. For the present case, c might be a function of the type of jumpsuit or the orientation used by the parachutist during free-fall. The net force is the difference between the downward and upward force. Therefore, Eqs. (1.4) through (1.7) can be combined to yield mg 2 cy dy 5 m dt

(1.8)

or simplifying the right side, dy c 5g2 y m dt

(1.9)

Equation (1.9) is a model that relates the acceleration of a falling object to the forces acting on it. It is a differential equation because it is written in terms of the differential rate of change (dy兾dt) of the variable that we are interested in predicting. However, in contrast to the solution of Newton’s second law in Eq. (1.3), the exact solution of Eq. (1.9) for the velocity of the falling parachutist cannot be obtained using simple algebraic manipulation. Rather, more advanced techniques, such as those of calculus, must be applied to obtain an exact or analytical solution. For example, if the parachutist is initially at rest (y 5 0 at t 5 0), calculus can be used to solve Eq. (1.9) for y(t) 5

gm ( 1 2 e 2 (cym)t ) c

(1.10)

Note that Eq. (1.10) is cast in the general form of Eq. (1.1), where y(t) 5 the dependent variable, t 5 the independent variable, c and m 5 parameters, and g 5 the forcing function. EXAMPLE 1.1

Analytical Solution to the Falling Parachutist Problem Problem Statement. A parachutist of mass 68.1 kg jumps out of a stationary hot air balloon. Use Eq. (1.10) to compute velocity prior to opening the chute. The drag coefficient is equal to 12.5 kg/s. Solution. Inserting the parameters into Eq. (1.10) yields y(t) 5

9.81(68.1) ( 1 2 e 2 (12.5y68.1)t ) 5 53.44 ( 1 2 e 20.18355t ) 12.5

which can be used to compute 1

In fact, the relationship is actually nonlinear and might better be represented by a power relationship such as FU 5 2cy2. We will explore how such nonlinearities affect the model in problems at the end of this chapter.

1.1

A SIMPLE MATHEMATICAL MODEL t, s

15

v, m/s

0 2 4 6 8 10 12 `

0.00 16.42 27.80 35.68 41.14 44.92 47.54 53.44

According to the model, the parachutist accelerates rapidly (Fig. 1.3). A velocity of 44.92 m/s is attained after 10 s. Note also that after a sufficiently long time, a constant velocity, called the terminal velocity, of 53.44 m/s is reached. This velocity is constant because, eventually, the force of gravity will be in balance with the air resistance. Thus, the net force is zero and acceleration has ceased.

Equation (1.10) is called an analytical, or exact, solution because it exactly satisfies the original differential equation. Unfortunately, there are many mathematical models that cannot be solved exactly. In many of these cases, the only alternative is to develop a numerical solution that approximates the exact solution. As mentioned previously, numerical methods are those in which the mathematical problem is reformulated so it can be solved by arithmetic operations. This can be illustrated

FIGURE 1.3 The analytical solution to the falling parachutist problem as computed in Example 1.1. Velocity increases with time and asymptotically approaches a terminal velocity.

Terminal velocity

v, m/s

40

20

0

0

4

8 t, s

12

16

MATHEMATICAL MODELING AND ENGINEERING PROBLEM SOLVING

v(ti +1) True slope dv/dt v Approximate slope v v(ti +1) – v(ti ) = t –t i +1 i t

v(ti )

FIGURE 1.4 The use of a ﬁnite difference to approximate the ﬁrst derivative of v with respect to t.

ti +1

ti

t

t

for Newton’s second law by realizing that the time rate of change of velocity can be approximated by (Fig. 1.4): y(ti11 ) 2 y(ti ) dy ¢y > 5 dt ¢t ti11 2 ti

(1.11)

where Dy and Dt 5 differences in velocity and time, respectively, computed over finite intervals, y(ti) 5 velocity at an initial time ti, and y(ti+1) 5 velocity at some later time ti+1. Note that dy/dt > ¢yy¢t is approximate because Dt is finite. Remember from calculus that dy ¢y 5 lim dt ¢tS0 ¢t Equation (1.11) represents the reverse process. Equation (1.11) is called a finite divided difference approximation of the derivative at time ti. It can be substituted into Eq. (1.9) to give y(ti11 ) 2 y(ti ) c 5 g 2 y(ti ) m ti11 2 ti This equation can then be rearranged to yield y(ti11 ) 5 y(ti ) 1 c g 2

c y(t ) d (ti11 2 ti ) m i

(1.12)

Notice that the term in brackets is the right-hand side of the differential equation itself [Eq. (1.9)]. That is, it provides a means to compute the rate of change or slope of y. Thus, the differential equation has been transformed into an equation that can be used to determine the velocity algebraically at ti11 using the slope and previous values of

1.1

A SIMPLE MATHEMATICAL MODEL

17

y and t. If you are given an initial value for velocity at some time ti, you can easily compute velocity at a later time ti11. This new value of velocity at ti11 can in turn be employed to extend the computation to velocity at ti12 and so on. Thus, at any time along the way, New value 5 old value 1 slope 3 step size Note that this approach is formally called Euler’s method. EXAMPLE 1.2

Numerical Solution to the Falling Parachutist Problem Problem Statement. Perform the same computation as in Example 1.1 but use Eq. (1.12) to compute the velocity. Employ a step size of 2 s for the calculation. Solution. At the start of the computation (ti 5 0), the velocity of the parachutist is zero. Using this information and the parameter values from Example 1.1, Eq. (1.12) can be used to compute velocity at ti11 5 2 s: y 5 0 1 c 9.81 2

12.5 (0) d 2 5 19.62 m/s 68.1

For the next interval (from t 5 2 to 4 s), the computation is repeated, with the result y 5 19.62 1 c 9.81 2

12.5 (19.62)d 2 5 32.04 m/s 68.1

The calculation is continued in a similar fashion to obtain additional values: t, s

v, m/s

0 2 4 6 8 10 12 `

0.00 19.62 32.04 39.90 44.87 48.02 50.01 53.44

The results are plotted in Fig. 1.5 along with the exact solution. It can be seen that the numerical method captures the essential features of the exact solution. However, because we have employed straight-line segments to approximate a continuously curving function, there is some discrepancy between the two results. One way to minimize such discrepancies is to use a smaller step size. For example, applying Eq. (1.12) at l-s intervals results in a smaller error, as the straight-line segments track closer to the true solution. Using hand calculations, the effort associated with using smaller and smaller step sizes would make such numerical solutions impractical. However, with the aid of the computer, large numbers of calculations can be performed easily. Thus, you can accurately model the velocity of the falling parachutist without having to solve the differential equation exactly. As in the previous example, a computational price must be paid for a more accurate numerical result. Each halving of the step size to attain more accuracy leads to a doubling

18

MATHEMATICAL MODELING AND ENGINEERING PROBLEM SOLVING

Terminal velocity Approximate, numerical solution

v, m/s

40

Exact, analytical solution 20

FIGURE 1.5 Comparison of the numerical and analytical solutions for the falling parachutist problem.

0

0

4

8

12

t, s

of the number of computations. Thus, we see that there is a trade-off between accuracy and computational effort. Such trade-offs figure prominently in numerical methods and constitute an important theme of this book. Consequently, we have devoted the Epilogue of Part One to an introduction to more of these trade-offs.

1.2

CONSERVATION LAWS AND ENGINEERING Aside from Newton’s second law, there are other major organizing principles in engineering. Among the most important of these are the conservation laws. Although they form the basis for a variety of complicated and powerful mathematical models, the great conservation laws of science and engineering are conceptually easy to understand. They all boil down to Change 5 increases 2 decreases

(1.13)

This is precisely the format that we employed when using Newton’s law to develop a force balance for the falling parachutist [Eq. (1.8)]. Although simple, Eq. (1.13) embodies one of the most fundamental ways in which conservation laws are used in engineering—that is, to predict changes with respect to time. We give Eq. (1.13) the special name time-variable (or transient) computation. Aside from predicting changes, another way in which conservation laws are applied is for cases where change is nonexistent. If change is zero, Eq. (1.13) becomes Change 5 0 5 increases 2 decreases or Increases 5 decreases

(1.14)

1.2

CONSERVATION LAWS AND ENGINEERING

19

Pipe 2 Flow in = 80

Pipe 1 Flow in = 100

FIGURE 1.6 A ﬂow balance for steady-state incompressible ﬂuid ﬂow at the junction of pipes.

Pipe 4 Flow out = ?

Pipe 3 Flow out = 120

Thus, if no change occurs, the increases and decreases must be in balance. This case, which is also given a special name—the steady-state computation—has many applications in engineering. For example, for steady-state incompressible fluid flow in pipes, the flow into a junction must be balanced by flow going out, as in Flow in 5 flow out For the junction in Fig. 1.6, the balance can be used to compute that the flow out of the fourth pipe must be 60. For the falling parachutist, steady-state conditions would correspond to the case where the net force was zero, or [Eq. (1.8) with dy兾dt 5 0] mg 5 cy

(1.15)

Thus, at steady state, the downward and upward forces are in balance, and Eq. (1.15) can be solved for the terminal velocity y5

mg c

Although Eqs. (1.13) and (1.14) might appear trivially simple, they embody the two fundamental ways that conservation laws are employed in engineering. As such, they will form an important part of our efforts in subsequent chapters to illustrate the connection between numerical methods and engineering. Our primary vehicles for making this connection are the engineering applications that appear at the end of each part of this book. Table 1.1 summarizes some of the simple engineering models and associated conservation laws that will form the basis for many of these engineering applications. Most of the chemical engineering applications will focus on mass balances for reactors. The mass balance is derived from the conservation of mass. It specifies that the change of mass of a chemical in the reactor depends on the amount of mass flowing in minus the mass flowing out. Both the civil and mechanical engineering applications will focus on models developed from the conservation of momentum. For civil engineering, force balances are utilized to analyze structures such as the simple truss in Table 1.1. The same principles are employed for the mechanical engineering applications to analyze the transient up-and-down motion or vibrations of an automobile.

20

MATHEMATICAL MODELING AND ENGINEERING PROBLEM SOLVING

TABLE 1.1 Devices and types of balances that are commonly used in the four major areas of engineering. For each case, the conservation law upon which the balance is based is speciﬁed. Field

Device

Chemical engineering

Organizing Principle

Mathematical Expression

Conservation of mass

Mass balance: Input

Reactors

Output

Over a unit of time period mass = inputs – outputs Civil engineering

Conservation of momentum

Structure

Force balance: + FV – FH

+ FH – FV

At each node horizontal forces (FH ) = 0 vertical forces (FV ) = 0 Mechanical engineering

Machine

Conservation of momentum

Force balance:

Upward force x=0 Downward force

2 m d x2 = downward force – upward force dt

Electrical engineering

Conservation of charge Current balance: + –

For each node current (i ) = 0

+ i1

– i3 + i2

Circuit i1R1

Conservation of energy Voltage balance:

i2R2 i3R3

Around each loop emf’s – voltage drops for resistors = 0 – iR = 0

PROBLEMS

21

TABLE 1.2 Some practical issues that will be explored in the engineering applications at the end of each part of this book. 1. Nonlinear versus linear. Much of classical engineering depends on linearization to permit analytical solutions. Although this is often appropriate, expanded insight can often be gained if nonlinear problems are examined. 2. Large versus small systems. Without a computer, it is often not feasible to examine systems with over three interacting components. With computers and numerical methods, more realistic multicomponent systems can be examined. 3. Nonideal versus ideal. Idealized laws abound in engineering. Often there are nonidealized alternatives that are more realistic but more computationally demanding. Approximate numerical approaches can facilitate the application of these nonideal relationships. 4. Sensitivity analysis. Because they are so involved, many manual calculations require a great deal of time and effort for successful implementation. This sometimes discourages the analyst from implementing the multiple computations that are necessary to examine how a system responds under different conditions. Such sensitivity analyses are facilitated when numerical methods allow the computer to assume the computational burden. 5. Design. It is often a straightforward proposition to determine the performance of a system as a function of its parameters. It is usually more difﬁcult to solve the inverse problem—that is, determining the parameters when the required performance is speciﬁed. Numerical methods and computers often permit this task to be implemented in an efﬁcient manner.

Finally, the electrical engineering applications employ both current and energy balances to model electric circuits. The current balance, which results from the conservation of charge, is similar in spirit to the flow balance depicted in Fig. 1.6. Just as flow must balance at the junction of pipes, electric current must balance at the junction of electric wires. The energy balance specifies that the changes of voltage around any loop of the circuit must add up to zero. The engineering applications are designed to illustrate how numerical methods are actually employed in the engineering problem-solving process. As such, they will permit us to explore practical issues (Table 1.2) that arise in real-world applications. Making these connections between mathematical techniques such as numerical methods and engineering practice is a critical step in tapping their true potential. Careful examination of the engineering applications will help you to take this step.

PROBLEMS 1.1 Use calculus to solve Eq. (1.9) for the case where the initial velocity, y(0) is nonzero. 1.2 Repeat Example 1.2. Compute the velocity to t 5 8 s, with a step size of (a) 1 and (b) 0.5 s. Can you make any statement regarding the errors of the calculation based on the results? 1.3 Rather than the linear relationship of Eq. (1.7), you might choose to model the upward force on the parachutist as a secondorder relationship, FU 5 2c¿y2 where c9 5 a bulk second-order drag coefficient (kg/m). (a) Using calculus, obtain the closed-form solution for the case where the jumper is initially at rest (y 5 0 at t 5 0).

(b) Repeat the numerical calculation in Example 1.2 with the same initial condition and parameter values, but with second-order drag. Use a value of 0.22 kg/m for c9. 1.4 For the free-falling parachutist with linear drag, assume a first jumper is 70 kg and has a drag coefficient of 12 kg/s. If a second jumper has a drag coefficient of 15 kg/s and a mass of 80 kg, how long will it take him to reach the same velocity the first jumper reached in 9 s? 1.5 Compute the velocity of a free-falling parachutist using Euler’s method for the case where m 5 80 kg and c 5 10 kg/s. Perform the calculation from t 5 0 to 20 s with a step size of 1 s. Use an initial condition that the parachutist has an upward velocity of 20 m/s at t 5 0. At t 5 10 s, assume that the chute is instantaneously deployed so that the drag coefficient jumps to 60 kg/s.

22

MATHEMATICAL MODELING AND ENGINEERING PROBLEM SOLVING

1.6 The following information is available for a bank account: Date

Deposits

Withdrawals

220.13

327.26

216.80

378.51

450.35

106.80

127.31

350.61

5/1

Interest

Balance 1522.33

6/1 7/1 8/1

(b) Plot the solution on a semilog graph (i.e., ln c versus t) and determine the slope. Interpret your results. 1.8 A group of 35 students attend a class in a room that measures 11 m by 8 m by 3 m. Each student takes up about 0.075 m3 and gives out about 80 W of heat (1 W 5 1 J/s). Calculate the air temperature rise during the first 20 minutes of the class if the room is completely sealed and insulated. Assume the heat capacity, Cy, for air is 0.718 kJ/(kg K). Assume air is an ideal gas at 208C and 101.325 kPa. Note that the heat absorbed by the air Q is related to the mass of the air m, the heat capacity, and the change in temperature by the following relationship:

9/1

Q5m Note that the money earns interest which is computed as

#

T2

CydT 5 mCy (T2 2 T1 )

T1

The mass of air can be obtained from the ideal gas law:

Interest 5 i Bi where i 5 the interest rate expressed as a fraction per month, and Bi the initial balance at the beginning of the month. (a) Use the conservation of cash to compute the balance on 6/1, 7/1, 8/1, and 9/1 if the interest rate is 1% per month (i 5 0.01/month). Show each step in the computation. (b) Write a differential equation for the cash balance in the form dB 5 f (D(t), W(t), i) dt where t 5 time (months), D(t) 5 deposits as a function of time ($/month), W(t) 5 withdrawals as a function of time ($/month). For this case, assume that interest is compounded continuously; that is, interest 5 iB. (c) Use Euler’s method with a time step of 0.5 month to simulate the balance. Assume that the deposits and withdrawals are applied uniformly over the month. (d) Develop a plot of balance versus time for (a) and (c). 1.7 The amount of a uniformly distributed radioactive contaminant contained in a closed reactor is measured by its concentration c (becquerel/liter or Bq/L). The contaminant decreases at a decay rate proportional to its concentration—that is,

PV 5

m RT MwT

where P is the gas pressure, V is the volume of the gas, Mwt is the molecular weight of the gas (for air, 28.97 kg/kmol), and R is the ideal gas constant [8.314 kPa m3/(kmol K)]. 1.9 A storage tank contains a liquid at depth y, where y 5 0 when the tank is half full. Liquid is withdrawn at a constant flow rate Q to meet demands. The contents are resupplied at a sinusoidal rate 3Q sin2(t). y

0

FIGURE P1.9

decay rate 5 2kc where k is a constant with units of day21. Therefore, according to Eq. (1.13), a mass balance for the reactor can be written as dc dt

5

Equation (1.13) can be written for this system as d(Ay) dt

2kc

change decrease a b5a b in mass by decay (a) Use Euler’s method to solve this equation from t 5 0 to 1 d with k 5 0.175d21. Employ a step size of Dt 5 0.1. The concentration at t 5 0 is 100 Bq/L.

a

5 3Q sin 2 (t) 2

Q

change in b 5 (inflow) 2 (outflow) volume

or, since the surface area A is constant dy Q Q 5 3 sin 2 (t) 2 dt A A

PROBLEMS

23

Use Euler’s method to solve for the depth y from t 5 0 to 10 d with a step size of 0.5 d. The parameter values are A 5 1250 m2 and Q 5 450 m3/d. Assume that the initial condition is y 5 0. 1.10 For the same storage tank described in Prob. 1.9, suppose that the outflow is not constant but rather depends on the depth. For this case, the differential equation for depth can be written as dy a(1 1 y) 1.5 Q 5 3 sin 2 (t) 2 dt A A Use Euler’s method to solve for the depth y from t 5 0 to 10 d with a step size of 0.5 d. The parameter values are A 5 1250 m2, Q 5 450 m3/d, and a 5 150. Assume that the initial condition is y 5 0. 1.11 Apply the conservation of volume (see Prob. 1.9) to simulate the level of liquid in a conical storage tank (Fig. P1.11). The liquid flows in at a sinusoidal rate of Qin 5 3 sin2(t) and flows out according to Qout 5 3(y 2 yout ) 1.5

y . yout

Qout 5 0

y # yout

where flow has units of m3/d and y 5 the elevation of the water surface above the bottom of the tank (m). Use Euler’s method to solve for the depth y from t 5 0 to 10 d with a step size of 0.5 d. The parameter values are rtop 5 2.5 m, ytop 5 4 m, and yout 5 1 m. Assume that the level is initially below the outlet pipe with y(0) 5 0.8 m.

rtop

Qin

ytop

y yout

s

Qout 1

0

FIGURE P1.11

1.12 In our example of the free-falling parachutist, we assumed that the acceleration due to gravity was a constant value. Although this is a decent approximation when we are examining falling objects near the surface of the earth, the gravitational force decreases as we move above sea level. A more general representation based on Newton’s inverse square law of gravitational attraction can be written as g(x) 5 g(0)

R2 (R 1 x) 2

where g(x) 5 gravitational acceleration at altitude x (in m) measured upward from the earth’s surface (m/s2), g(0) 5 gravitational acceleration at the earth’s surface (> 9.81 m/s2), and R 5 the earth’s radius (> 6.37 3 106 m). (a) In a fashion similar to the derivation of Eq. (1.9) use a force balance to derive a differential equation for velocity as a function of time that utilizes this more complete representation of gravitation. However, for this derivation, assume that upward velocity is positive. (b) For the case where drag is negligible, use the chain rule to express the differential equation as a function of altitude rather than time. Recall that the chain rule is dy dy dx 5 dt dx dt (c) Use calculus to obtain the closed form solution where y 5 y0 at x 5 0. (d) Use Euler’s method to obtain a numerical solution from x 5 0 to 100,000 m using a step of 10,000 m where the initial velocity is 1500 m/s upward. Compare your result with the analytical solution. 1.13 Suppose that a spherical droplet of liquid evaporates at a rate that is proportional to its surface area. dV 5 2kA dt where V 5 volume (mm3), t 5 time (min), k 5 the evaporation rate (mm/min), and A 5 surface area (mm2). Use Euler’s method to compute the volume of the droplet from t 5 0 to 10 min using a step size of 0.25 min. Assume that k 5 0.08 mm/min and that the droplet initially has a radius of 2.5 mm. Assess the validity of your results by determining the radius of your final computed volume and verifying that it is consistent with the evaporation rate. 1.14 Newton’s law of cooling says that the temperature of a body changes at a rate proportional to the difference between its temperature and that of the surrounding medium (the ambient temperature), dT 5 2k(T 2 Ta ) dt where T 5 the temperature of the body (8C), t 5 time (min), k 5 the proportionality constant (per minute), and Ta 5 the ambient temperature (8C). Suppose that a cup of coffee originally has a temperature of 708C. Use Euler’s method to compute the temperature from t 5 0 to 10 min using a step size of 2 min if Ta 5 208C and k 5 0.019/min. 1.15 As depicted in Fig. P1.15, an RLC circuit consists of three elements: a resistor (R), and inductor (L) and a capacitor (C). The flow of current across each element induces a voltage drop.

24

MATHEMATICAL MODELING AND ENGINEERING PROBLEM SOLVING

Kirchhoff’s second voltage law states that the algebraic sum of these voltage drops around a closed circuit is zero, iR 1 L

Q1

Q3

Q5

q di 1 50 dt C

where i 5 current, R 5 resistance, L 5 inductance, t 5 time, q 5 charge, and C 5 capacitance. In addition, the current is related to charge as in

Q2

Q4

Q6

Q7

dq 5i dt (a) If the initial values are i(0) 5 0 and q(0) 5 1 C, use Euler’s method to solve this pair of differential equations from t 5 0 to 0.1 s using a step size of Dt 5 0.01 s. Employ the following parameters for your calculation: R 5 200 V, L 5 5 H, and C 5 10–4 F. (b) Develop a plot of i and q versus t. Resistor Inductor Capacitor q di L c iR dt

i

FIGURE P1.15

1.16 Cancer cells grow exponentially with a doubling time of 20 h when they have an unlimited nutrient supply. However, as the cells start to form a solid spherical tumor without a blood supply, growth at the center of the tumor becomes limited, and eventually cells start to die. (a) Exponential growth of cell number N can be expressed as shown, where m is the growth rate of the cells. For cancer cells, find the value of m. dN 5 mN dt (b) Write an equation that will describe the rate of change of tumor volume during exponential growth given that the diameter of an individual cell is 20 microns. (c) After a particular type of tumor exceeds 500 microns in diameter, the cells at the center of the tumor die (but continue to take up space in the tumor). Determine how long it will take for the tumor to exceed this critical size. 1.17 A fluid is pumped into the network shown in Fig. P1.17. If Q2 5 0.6, Q3 5 0.4, Q7 5 0.2, and Q8 5 0.3 m3/s, determine the other flows.

Q10

Q9

Q8

FIGURE P1.17

1.18 The velocity is equal to the rate of change of distance x (m), dx 5 y(t) dt

(P1.18)

(a) Substitute Eq. (1.10) and develop an analytical solution for distance as a function of time. Assume that x(0) 5 0. (b) Use Euler’s method to numerically integrate Eqs. (P1.18) and (1.9) in order to determine both the velocity and distance fallen as a function of time for the first 10 s of free-fall using the same parameters as in Example 1.2. (c) Develop a plot of your numerical results together with the analytical solution. 1.19 You are working as a crime-scene investigator and must predict the temperature of a homicide victim over a 5-hr period. You know that the room where the victim was found was at 108C when the body was discovered. (a) Use Newton’s law of cooling (Prob. 1.14) and Euler’s method to compute the victim’s body temperature for the 5-hr period using values of k 5 0.12/hr and Dt 5 0.5 hr. Assume that the victim’s body temperature at the time of death was 378C, and that the room temperature was at a constant value of 108C over the 5-hr period. (b) Further investigation reveals that the room temperature had actually dropped linearly from 20 to 108C over the 5-hr period. Repeat the same calculation as in (a) but incorporate this new information. (c) Compare the results from (a) and (b) by plotting them on the same graph. 1.20 Suppose that a parachutist with linear drag (m 5 70 kg, c 5 12.5 kg/s) jumps from an airplane flying at an altitude of a kilometer with a horizontal velocity of 180 m/s relative to the ground. (a) Write a system of four differential equations for x, y, yx 5 dx/dt and yy 5 dy/dt.

PROBLEMS (b) If the initial horizontal position is defined as x 5 0, use Euler’s methods with Dt 5 1 s to compute the jumper’s position over the first 10 s. (c) Develop plots of y versus t and y versus x. Use the plot to graphically estimate when and where the jumper would hit the ground if the chute failed to open. 1.21 As noted in Prob. 1.3, drag is more accurately represented as depending on the square of velocity. A more fundamental representation of the drag force, which assumes turbulent conditions (i.e., a high Reynolds number), can be formulated as

25 (b) At steady-state, use this equation to solve for the particle’s terminal velocity. (c) Employ the result of (b) to compute the particle’s terminal velocity in m/s for a spherical silt particle settling in water: d 5 10 mm, r 5 1 g/cm3, rs 5 2.65 g/cm3, and m 5 0.014 g/(cm?s). (d) Check whether flow is laminar. (e) Use Euler’s method to compute the velocity from t 5 0 to 2215 s with Dt 5 2218 s given the parameters given previously along with the initial condition: y (0) 5 0. FD FB

1 Fd 5 2 rACdyZyZ 2 where Fd 5 the drag force (N), r 5 fluid density (kg/m3), A 5 the frontal area of the object on a plane perpendicular to the direction of motion (m2), y 5 velocity (m/s), and Cd 5 a dimensionless drag coefficient. (a) Write the pair of differential equations for velocity and position (see Prob. 1.18) to describe the vertical motion of a sphere with diameter d (m) and a density of rs (kg/km3). The differential equation for velocity should be written as a function of the sphere’s diameter. (b) Use Euler’s method with a step size of Dt 5 2 s to compute the position and velocity of a sphere over the first 14 s. Employ the following parameters in your calculation: d 5 120 cm, r 5 1.3 kg/m3, rs 5 2700 kg/m3, and Cd 5 0.47. Assume that the sphere has the initial conditions: x(0) 5 100 m and y(0) 5 –40 m/s. (c) Develop a plot of your results (i.e., y and y versus t) and use it to graphically estimate when the sphere would hit the ground. (d) Compute the value for the bulk second-order drag coefficient cd9 (kg/m). Note that, as described in Prob. 1.3, the bulk secondorder drag coefficient is the term in the final differential equation for velocity that multiplies the term y ZyZ. 1.22 As depicted in Fig. P1.22, a spherical particle settling through a quiescent fluid is subject to three forces: the downward force of gravity (FG), and the upward forces of buoyancy (FB) and drag (FD). Both the gravity and buoyancy forces can be computed with Newton’s second law with the latter equal to the weight of the displaced fluid. For laminar flow, the drag force can be computed with Stokes’s law, FD 5 3pmdy where m 5 the dynamic viscosity of the fluid (N s/m2), d 5 the particle diameter (m), and y 5 the particle’s settling velocity (m/s). Note that the mass of the particle can be expressed as the product of the particle’s volume and density rs (kg/m3) and the mass of the displaced fluid can be computed as the product of the particle’s volume and the fluid’s density r (kg/m3). The volume of a sphere is pd3/6. In addition, laminar flow corresponds to the case where the dimensionless Reynolds number, Re, is less than 1, where Re 5 rdy/m. (a) Use a force balance for the particle to develop the differential equation for dy/dt as a function of d, r, rs, and m.

d

FG

FIGURE P1.22 1.23 As described in Prob. 1.22, in addition to the downward force of gravity (weight) and drag, an object falling through a fluid is also subject to a buoyancy force that is proportional to the displaced volume. For example, for a sphere with diameter d (m), the sphere’s volume is V 5 pd3/6 and its projected area is A 5 pd2/4. The buoyancy force can then be computed as Fb 5 –rVg. We neglected buoyancy in our derivation of Eq. (1.9) because it is relatively small for an object like a parachutist moving through air. However, for a more dense fluid like water, it becomes more prominent. (a) Derive a differential equation in the same fashion as Eq. (1.9), but include the buoyancy force and represent the drag force as described in Prob. 1.21. (b) Rewrite the differential equation from (a) for the special case of a sphere. (c) Use the equation developed in (b) to compute the terminal velocity (i.e., for the steady-state case). Use the following parameter values for a sphere falling through water: sphere diameter 5 1 cm, sphere density 5 2700 kg/m3, water density 5 1000 kg/m3, and Cd 5 0.47. (d) Use Euler’s method with a step size of Dt 5 0.03125 s to numerically solve for the velocity from t 5 0 to 0.25 s with an initial velocity of zero. 1.24 As depicted in Fig. P1.24, the downward deflection y (m) of a cantilever beam with a uniform load w (kg/m) can be computed as y5

w ( x4 2 4Lx3 1 6L2x2 ) 24EI

where x 5 distance (m), E 5 the modulus of elasticity 5 2 3 1011 Pa, I 5 moment of inertia 5 3.25 3 10–4 m4, w 5 10,000 N/m, and

26

MATHEMATICAL MODELING AND ENGINEERING PROBLEM SOLVING

L 5 length 5 4 m. This equation can be differentiated to yield the slope of the downward deflection as a function of x: dy w 5 ( 4x3 2 12Lx2 1 12L2x ) dx 24EI If y 5 0 at x 5 0, use this equation with Euler’s method (Dx 5 0.125 m) to compute the deflection from x 5 0 to L. Develop a plot of your results along with the analytical solution computed with the first equation. x=0

1.26 Beyond fluids, Archimedes’ principle has proven useful in geology when applied to solids on the earth’s crust. Figure P1.26 depicts one such case where a lighter conical granite mountain “floats on” a denser basalt layer at the earth’s surface. Note that the part of the cone below the surface is formally referred to as a frustum. Develop a steady-state force balance for this case in terms of the following parameters: basalt’s density (rb), granite’s density (rg), the cone’s bottom radius (r), and the height above (h1) and below (h2) the earth’s surface.

x=L

0

w h1

y

H

FIGURE P1.24 A cantilever beam.

r1

Granite

1.25 Use Archimedes’ principle to develop a steady-state force balance for a spherical ball of ice floating in seawater (Fig. P1.25). The force balance should be expressed as a third-order polynomial (cubic) in terms of height of the cap above the water line (h), the seawater’s density (rf), the ball’s density (rs), and the ball’s radius (r).

h r

FIGURE P1.25

Basalt

FIGURE P1.26

h2 r2

2

CHAPTER

Programming and Software

In Chap. 1, we used a net force to develop a mathematical model to predict the fall velocity of a parachutist. This model took the form of a differential equation, dy c 5g2 y m dt We also learned that a solution to this equation could be obtained by a simple numerical approach called Euler’s method, yi11 5 yi 1

dyi ¢t dt

Given an initial condition, this equation can be implemented repeatedly to compute the velocity as a function of time. However, to obtain good accuracy, many small steps must be taken. This would be extremely laborious and time-consuming to implement by hand. However, with the aid of the computer, such calculations can be performed easily. So our next task is to figure out how to do this. The present chapter will introduce you to how the computer is used as a tool to obtain such solutions.

2.1

PACKAGES AND PROGRAMMING Today, there are two types of software users. On one hand, there are those who take what they are given. That is, they limit themselves to the capabilities found in the software’s standard mode of operation. For example, it is a straightforward proposition to solve a system of linear equations or to generate a plot of x-y values with either Excel or MATLAB software. Because this usually involves a minimum of effort, most users tend to adopt this “vanilla” mode of operation. In addition, since the designers of these packages anticipate most typical user needs, many meaningful problems can be solved in this way. But what happens when problems arise that are beyond the standard capability of the tool? Unfortunately, throwing up your hands and saying, “Sorry boss, no can do!” is not acceptable in most engineering circles. In such cases, you have two alternatives. First, you can look for a different package and see if it is capable of solving the problem. That is one of the reasons we have chosen to cover both Excel and MATLAB in this book. As you will see, neither one is all encompassing and each has different 27

28

PROGRAMMING AND SOFTWARE

strengths. By being conversant with both, you will greatly increase the range of problems you can address. Second, you can grow and become a “power user” by learning to write Excel VBA1 macros or MATLAB M-files. And what are these? They are nothing more than computer programs that allow you to extend the capabilities of these tools. Because engineers should never be content to be tool limited, they will do whatever is necessary to solve their problems. A powerful way to do this is to learn to write programs in the Excel and MATLAB environments. Furthermore, the programming skills required for macros and M-files are the same as those needed to effectively develop programs in languages like Fortran 90 or C. The major goal of the present chapter is to show you how this can be done. However, we do assume that you have been exposed to the rudiments of computer programming. Therefore, our emphasis here is on facets of programming that directly affect its use in engineering problem solving. 2.1.1 Computer Programs Computer programs are merely a set of instructions that direct the computer to perform a certain task. Since many individuals write programs for a broad range of applications, most high-level computer languages, like Fortran 90 and C, have rich capabilities. Although some engineers might need to tap the full range of these capabilities, most merely require the ability to perform engineering-oriented numerical calculations. Looked at from this perspective, we can narrow down the complexity to a few programming topics. These are: • • • • • •

Simple information representation (constants, variables, and type declarations). Advanced information representation (data structure, arrays, and records). Mathematical formulas (assignment, priority rules, and intrinsic functions). Input/output. Logical representation (sequence, selection, and repetition). Modular programming (functions and subroutines).

Because we assume that you have had some prior exposure to programming, we will not spend time on the first four of these areas. At best, we offer them as a checklist that covers what you will need to know to implement the programs that follow. However, we will devote some time to the last two topics. We emphasize logical representation because it is the single area that most influences an algorithm’s coherence and understandability. We include modular programming because it also contributes greatly to a program’s organization. In addition, modules provide a means to archive useful algorithms in a convenient format for subsequent applications.

2.2

STRUCTURED PROGRAMMING In the early days of computers, programmers usually did not pay much attention to whether their programs were clear and easy to understand. Today, it is recognized that there are many benefits to writing organized, well-structured code. Aside from the obvious benefit of making software much easier to share, it also helps generate much more efficient 1

VBA is the acronym for Visual Basic for Applications.

2.2

STRUCTURED PROGRAMMING

29

program development. That is, well-structured algorithms are invariably easier to debug and test, resulting in programs that take a shorter time to develop, test, and update. Computer scientists have systematically studied the factors and procedures needed to develop high-quality software of this kind. In essence, structured programming is a set of rules that prescribe good style habits for the programmer. Although structured programming is flexible enough to allow considerable creativity and personal expression, its rules impose enough constraints to render the resulting codes far superior to unstructured versions. In particular, the finished product is more elegant and easier to understand. A key idea behind structured programming is that any numerical algorithm can be composed using the three fundamental control structures: sequence, selection, and repetition. By limiting ourselves to these structures, the resulting computer code will be clearer and easier to follow. In the following paragraphs, we will describe each of these structures. To keep this description generic, we will employ flowcharts and pseudocode. A flowchart is a visual or graphical representation of an algorithm. The flowchart employs a series of blocks and arrows, each of which represents a particular operation or step in the algorithm (Fig. 2.1). The arrows represent the sequence in which the operations are implemented. Not everyone involved with computer programming agrees that flowcharting is a productive endeavor. In fact, some experienced programmers do not advocate flowcharts. However, we feel that there are three good reasons for studying them. First, they are still used for expressing and communicating algorithms. Second, even if they are not employed routinely, there will be times when they will prove useful in planning, unraveling, or communicating the logic of your own or someone else’s program. Finally, and most important for our purposes, they are excellent pedagogical tools. From a

FIGURE 2.1 Symbols used in ﬂowcharts. SYMBOL

NAME

FUNCTION

Terminal

Represents the beginning or end of a program.

Flowlines

Represents the flow of logic. The humps on the horizontal arrow indicate that it passes over and does not connect with the vertical flowlines.

Process

Represents calculations or data manipulations.

Input/output

Represents inputs or outputs of data and information.

Decision

Represents a comparison, question, or decision that determines alternative paths to be followed.

Junction

Represents the confluence of flowlines.

Off-page connector

Represents a break that is continued on another page.

Count-controlled loop

Used for loops which repeat a prespecified number of iterations.

30

PROGRAMMING AND SOFTWARE

teaching perspective, they are ideal vehicles for visualizing some of the fundamental control structures employed in computer programming. An alternative approach to express an algorithm that bridges the gap between flowcharts and computer code is called pseudocode. This technique uses code-like statements in place of the graphical symbols of the flowchart. We have adopted some style conventions for the pseudocode in this book. Keywords such as IF, DO, INPUT, etc., are capitalized, whereas the conditions, processing steps, and tasks are in lowercase. Additionally, the processing steps are indented. Thus the keywords form a “sandwich” around the steps to visually define the extent of each control structure. One advantage of pseudocode is that it is easier to develop a program with it than with a flowchart. The pseudocode is also easier to modify and share with others. However, because of their graphic form, flowcharts sometimes are better suited for visualizing complex algorithms. In the present text, we will use flowcharts for pedagogical purposes. Pseudocode will be our principal vehicle for communicating algorithms related to numerical methods. 2.2.1 Logical Representation Sequence. The sequence structure expresses the trivial idea that unless you direct it otherwise, the computer code is to be implemented one instruction at a time. As in Fig. 2.2, the structure can be expressed generically as a flowchart or as pseudocode. Selection. In contrast to the step-by-step sequence structure, selection provides a means to split the program’s flow into branches based on the outcome of a logical condition. Figure 2.3 shows the two most fundamental ways for doing this. The single-alternative decision, or IF/THEN structure (Fig. 2.3a), allows for a detour in the program flow if a logical condition is true. If it is false, nothing happens and the program moves directly to the next statement following the ENDIF. The double-alternative decision, or IF/THEN/ELSE structure (Fig. 2.3b), behaves in the same manner for a true condition. However, if the condition is false, the program implements the code between the ELSE and the ENDIF.

FIGURE 2.2 (a) Flowchart and (b) pseudocode for the sequence structure.

Instruction1

Instruction2

Instruction3

Instruction1 Instruction2 Instruction3 Instruction4

Instruction4

(a) Flowchart

(b) Pseudocode

2.2

STRUCTURED PROGRAMMING

Flowchart

Condition ?

31

Pseudocode

True

True Block

IF condition THEN True block ENDIF

(a) Single-alternative structure (IF/THEN)

False

FIGURE 2.3 Flowchart and pseudocode for simple selection constructs. (a) Single-alternative selection (IF/THEN) and (b) doublealternative selection (IF/THEN/ELSE).

False Block

Condition ?

True

True Block

IF condition THEN True block ELSE False block ENDIF

(b) Double-alternative structure (IF/ THEN/ELSE)

Although the IF/THEN and the IF/THEN/ELSE constructs are sufficient to construct any numerical algorithm, two other variants are commonly used. Suppose that the ELSE clause of an IF/THEN/ELSE contains another IF/THEN. For such cases, the ELSE and the IF can be combined in the IF/THEN/ELSEIF structure shown in Fig. 2.4a. Notice how in Fig. 2.4a there is a chain or “cascade” of decisions. The first one is the IF statement, and each successive decision is an ELSEIF statement. Going down the chain, the first condition encountered that tests true will cause a branch to its corresponding code block followed by an exit of the structure. At the end of the chain of conditions, if all the conditions have tested false, an optional ELSE block can be included. The CASE structure is a variant on this type of decision making (Fig. 2.4b). Rather than testing individual conditions, the branching is based on the value of a single test expression. Depending on its value, different blocks of code will be implemented. In addition, an optional block can be implemented if the expression takes on none of the prescribed values (CASE ELSE). Repetition. Repetition provides a means to implement instructions repeatedly. The resulting constructs, called loops, come in two “flavors” distinguished by how they are terminated.

32

PROGRAMMING AND SOFTWARE

Flowchart

Pseudocode

False

False

False

Condition3 ?

Block4

Condition1 ? True

Condition2 ? True

True

Block1

Block2

Block3

IF condition1 THEN Block1 ELSEIF condition2 Block2 ELSEIF condition3 Block3 ELSE Block4 ENDIF

(a) Multialternative structure (IF/THEN/ELSEIF)

Test expression

Value1 Block1

Value2 Block2

Value3 Block3

Else Block4

SELECT CASE Test Expression CASE Value1 Block1 CASE Value2 Block2 CASE Value3 Block3 CASE ELSE Block4 END SELECT

(b) CASE structure (SELECT or SWITCH) FIGURE 2.4 Flowchart and pseudocode for supplementary selection or branching constructs. (a) Multiplealternative selection (IF/THEN/ELSEIF) and (b) CASE construct.

The first and most fundamental type is called a decision loop because it terminates based on the result of a logical condition. Figure 2.5 shows the most generic type of decision loop, the DOEXIT construct, also called a break loop. This structure repeats until a logical condition is true. It is not necessary to have two blocks in this structure. If the first block is not included, the structure is sometimes called a pretest loop because the logical test is performed before anything occurs. Alternatively, if the second block is omitted, it is

2.2

STRUCTURED PROGRAMMING

Flowchart

Pseudocode

Block1

Condition ?

33

True

DO Block1 IF condition EXIT Block2 ENDDO

False Block2

FIGURE 2.5 The DOEXIT or break loop.

Flowchart

True

i = start i > finish i = i + step ? False

FIGURE 2.6 The count-controlled or DOFOR construct.

Pseudocode

DOFOR i = start, finish, step Block ENDDO

Block

called a posttest loop. Because both blocks are included, the general case in Fig. 2.5 is sometimes called a midtest loop. It should be noted that the DOEXIT loop was introduced in Fortran 90 in an effort to simplify decision loops. This control construct is a standard part of the Excel VBA macro language but is not standard in C or MATLAB, which use the so-called WHILE structure. Because we believe that the DOEXIT is superior, we have adopted it as our decision loop structure throughout this book. In order to ensure that our algorithms are directly implemented in both MATLAB and Excel, we will show how the break loop can be simulated with the WHILE structure later in this chapter (see Sec. 2.5). The break loop in Fig. 2.5 is called a logical loop because it terminates on a logical condition. In contrast, a count-controlled or DOFOR loop (Fig. 2.6) performs a specified number of repetitions, or iterations. The count-controlled loop works as follows. The index (represented as i in Fig. 2.6) is a variable that is set at an initial value of start. The program then tests whether the

34

PROGRAMMING AND SOFTWARE

index is less than or equal to the final value, finish. If so, it executes the body of the loop, and then cycles back to the DO statement. Every time the ENDDO statement is encountered, the index is automatically increased by the step. Thus the index acts as a counter. Then, when the index is greater than the final value (finish), the computer automatically exits the loop and transfers control to the line following the ENDDO statement. Note that for nearly all computer languages, including those of Excel and MATLAB, if the step is omitted, the computer assumes it is equal to 1.2 The numerical algorithms outlined in the following pages will be developed exclusively from the structures outlined in Figs. 2.2 through 2.6. The following example illustrates the basic approach by developing an algorithm to determine the roots for the quadratic formula. EXAMPLE 2.1

Algorithm for Roots of a Quadratic Problem Statement. The roots of a quadratic equation ax2 1 bx 1 c 5 0 can be determined with the quadratic formula, 2b 6 2Zb2 2 4acZ x1 5 x2 2a

(E2.1.1)

Develop an algorithm that does the following: Step 1: Prompts the user for the coefﬁcients, a, b, and c. Step 2: Implements the quadratic formula, guarding against all eventualities (for example, avoiding division by zero and allowing for complex roots). Step 3: Displays the solution, that is, the values for x. Step 4: Allows the user the option to return to step 1 and repeat the process.

Solution. We will use a top-down approach to develop our algorithm. That is, we will successively refine the algorithm rather than trying to work out all the details the first time around. To do this, let us assume for the present that the quadratic formula is foolproof regardless of the values of the coefficients (obviously not true, but good enough for now). A structured algorithm to implement the scheme is DO INPUT a, b, c r1 5 (2b 1 SQRT(b2 2 4ac))y(2a) r2 5 (2b 2 SQRT(b2 2 4ac))y(2a) DISPLAY r1, r2 DISPLAY 'Try again? Answer yes or no' INPUT response IF response 5 'no' EXIT ENDDO 2

A negative step can be used. In such cases, the loop terminates when the index is less than the final value.

2.2

STRUCTURED PROGRAMMING

35

A DOEXIT construct is used to implement the quadratic formula repeatedly as long as the condition is false. The condition depends on the value of the character variable response. If response is equal to ‘yes’ the calculation is implemented. If not, that is, response 5 ‘no’ the loop terminates. Thus, the user controls termination by inputting a value for response. Now although the above algorithm works for certain cases, it is not foolproof. Depending on the values of the coefficients, the algorithm might not work. Here is what can happen: • If a 5 0, an immediate problem arises because of division by zero. In fact, close inspection of Eq. (E2.1.1) indicates that two different cases can arise. That is, If b ﬁ 0, the equation reduces to a linear equation with one real root, 2cyb. If b 5 0, then no solution exists. That is, the problem is trivial. • If a ﬁ 0, two possible cases occur depending on the value of the discriminant, d 5 b2 2 4ac. That is, If d $ 0, two real roots occur. If d , 0, two complex roots occur. Notice how we have used indentation to highlight the decisional structure that underlies the mathematics. This structure then readily translates to a set of coupled IF/THEN/ELSE structures that can be inserted in place of the shaded statements in the previous code to give the final algorithm: DO INPUT a, b, c r1 5 0: r2 5 0: i1 5 0: i2 5 0 IF a 5 0 THEN IF b ﬁ 0 THEN r1 5 2cyb ELSE DISPLAY "Trivial solution" ENDIF ELSE discr 5 b2 2 4 * a * c IF discr $ 0 THEN r1 5 (2b 1 Sqrt(discr))y(2 * a) r2 5 (2b 2 Sqrt(discr))y(2 * a) ELSE r1 5 2by(2 * a) r2 5 r1 i1 5 Sqrt(Abs(discr))y(2 * a) i2 5 2il ENDIF ENDIF DISPLAY r1, r2, i1, i2 DISPLAY 'Try again? Answer yes or no' INPUT response IF response 5 'no' EXIT ENDDO

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PROGRAMMING AND SOFTWARE

The approach in the foregoing example can be employed to develop an algorithm for the parachutist problem. Recall that, given an initial condition for time and velocity, the problem involved iteratively solving the formula yi11 5 yi 1

dyi ¢t dt

(2.1)

Now also remember that if we desired to attain good accuracy, we would need to employ small steps. Therefore, we would probably want to apply the formula repeatedly from the initial time to the final time. Consequently, an algorithm to solve the problem would be based on a loop. For example, suppose that we started the computation at t 5 0 and wanted to predict the velocity at t 5 4 s using a time step of Dt 5 0.5 s. We would, therefore, need to apply Eq. (2.1) eight times, that is, n5

4 58 0.5

where n 5 the number of iterations of the loop. Because this result is exact, that is, the ratio is an integer, we can use a count-controlled loop as the basis for the algorithm. Here is an example of the pseudocode: g 5 9.81 INPUT cd, m INPUT ti, vi, tf, dt t 5 ti v 5 vi n 5 (tf 2 ti) y dt DOFOR i 5 1 TO n dvdt 5 g 2 (cd y m) * v v 5 v 1 dvdt * dt t 5 t 1 dt ENDDO DISPLAY v

Although this scheme is simple to program, it is not foolproof. In particular, it will work only if the computation interval is evenly divisible by the time step.3 In order to cover such cases, a decision loop can be substituted in place of the shaded area in the previous pseudocode. The final result is g 5 9.81 INPUT cd, m INPUT ti, vi, tf, dt t 5 ti v 5 vi 3

This problem is compounded by the fact that computers use base-2 number representation for their internal math. Consequently, some apparently evenly divisible numbers do not yield integers when the division is implemented on a computer. We will cover this in Chap. 3.

2.3

MODULAR PROGRAMMING

37

h 5 dt DO IF t 1 dt . tf THEN h 5 tf 2 t ENDIF dvdt 5 g 2 (cd y m) * v v 5 v 1 dvdt * h t 5 t 1 h IF t $ tf EXIT ENDDO DISPLAY v

As soon as we enter the loop, we use an IF/THEN structure to test whether adding t 1 dt will take us beyond the end of the interval. If it does not, which would usually be the case at first, we do nothing. If it does, we would need to shorten the interval by setting the variable step h to t f 2 t. By doing this, we guarantee that the next step falls exactly on t f. After we implement this final step, the loop will terminate because the condition t $ t f will test true. Notice that before entering the loop, we assign the value of the time step, dt, to another variable, h. We create this dummy variable so that our routine does not change the given value of dt if and when we shorten the time step. We do this in anticipation that we might need to use the original value of dt somewhere else in the event that this code is integrated within a larger program. It should be noted that the algorithm is still not foolproof. For example, the user could have mistakenly entered a step size greater than the calculation interval, for example, t f 2 ti 5 5 and dt 5 20. Thus, you might want to include error traps in your code to catch such errors and to then allow the user to correct the mistake.

2.3

MODULAR PROGRAMMING Imagine how difficult it would be to study a textbook that had no chapters, sections, or paragraphs. Breaking complicated tasks or subjects into more manageable parts is one way to make them easier to handle. In the same spirit, computer programs can be divided into small subprograms, or modules, that can be developed and tested separately. This approach is called modular programming. The most important attribute of modules is that they be as independent and selfcontained as possible. In addition, they are typically designed to perform a specific, well-defined function and have one entry and one exit point. As such, they are usually short (generally 50 to 100 instructions in length) and highly focused. In standard high-level languages such as Fortran 90 or C, the primary programming element used to represent each module is the procedure. A procedure is a series of computer instructions that together perform a given task. Two types of procedures are commonly employed: functions and subroutines. The former usually returns a single result, whereas the latter returns several. In addition, it should be mentioned that much of the programming related to software packages like Excel and MATLAB involves the development of subprograms. Hence,

38

PROGRAMMING AND SOFTWARE

Excel macros and MATLAB functions are designed to receive some information, perform a calculation, and return results. Thus, modular thinking is also consistent with how programming is implemented in package environments. Modular programming has a number of advantages. The use of small, self-contained units makes the underlying logic easier to devise and to understand for both the developer and the user. Development is facilitated because each module can be perfected in isolation. In fact, for large projects, different programmers can work on individual parts. Modular design also increases the ease with which a program can be debugged and tested because errors can be more easily isolated. Finally, program maintenance and modification are facilitated. This is primarily due to the fact that new modules can be developed to perform additional tasks and then easily incorporated into the already coherent and organized scheme. While all these attributes are reason enough to use modules, the most important reason related to numerical engineering problem solving is that they allow you to maintain your own library of useful modules for later use in other programs. This will be the philosophy of this book: All the algorithms will be presented as modules. This approach is illustrated in Fig. 2.7, which shows a function developed to implement Euler’s method. Notice that this function application and the previous versions differ in how they handle input/output. In the former versions, input and output directly come from (via INPUT statements) and to (via DISPLAY statements) the user. In the function, the inputs are passed into the FUNCTION via its argument list Function Euler(dt, ti, tf, yi)

and the output is returned via the assignment statement y 5 Euler(dt, ti, tf, yi)

In addition, recognize how generic the routine has become. There are no references to the specifics of the parachutist problem. For example, rather than calling the dependent

FIGURE 2.7 Pseudocode for a function that solves a differential equation using Euler’s method.

FUNCTION Euler(dt, ti, tf, yi) t 5 ti y 5 yi h 5 dt DO IF t 1 dt . tf THEN h 5 tf 2 t ENDIF dydt 5 dy(t, y) y 5 y 1 dydt * h t 5 t 1 h IF t $ tf EXIT ENDDO Euler 5 y END Euler

2.4

EXCEL

39

variable y for velocity, the more generic label, y, is used within the function. Further, notice that the derivative is not computed within the function by an explicit equation. Rather, another function, dy, must be invoked to compute it. This acknowledges the fact that we might want to use this function for many different problems beyond solving for the parachutist’s velocity.

2.4

EXCEL Excel is the spreadsheet produced by Microsoft, Inc. Spreadsheets are a special type of mathematical software that allow the user to enter and perform calculations on rows and columns of data. As such, they are a computerized version of a large accounting worksheet on which large interconnected calculations can be implemented and displayed. Because the entire calculation is updated when any value on the sheet is changed, spreadsheets are ideal for “what if?” sorts of analysis. Excel has some built-in numerical capabilities including equation solving, curve fitting, and optimization. It also includes VBA as a macro language that can be used to implement numerical calculations. Finally, it has several visualization tools, such as graphs and three-dimensional surface plots, that serve as valuable adjuncts for numerical analysis. In the present section, we will show how these capabilities can be used to solve the parachutist problem. To do this, let us first set up a simple spreadsheet. As shown below, the first step involves entering labels and numbers into the spreadsheet cells.

Before we write a macro program to calculate the numerical value, we can make our subsequent work easier by attaching names to the parameter values. To do this, select cells A3:B5 (the easiest way to do this is by moving the mouse to A3, holding down the left mouse button and dragging down to B5). Next, go to the Formulas tab and in the Defined Names group, click Create from Selection. This will open the Create Names from Selection dialog box, where the Left column box should be automatically selected. Then click OK to create the names. To verify that this has worked properly, select cell B3 and check that the label “m” appears in the name box (located on the left side of the sheet just below the menu bars).

40

PROGRAMMING AND SOFTWARE

Move to cell C8 and enter the analytical solution (Eq. 1.9), =9.81*m/cd*(1−exp(−cd/m*A8))

When this formula is entered, the value 0 should appear in cell C8. Then copy the formula down to cell C9 to give a value of 16.405 m/s. All the above is typical of the standard use of Excel. For example, at this point you could change parameter values and see how the analytical solution changes. Now, we will illustrate how VBA macros can be used to extend the standard capabilities. Figure 2.8 lists pseudocode alongside Excel VBA code for all the control structures described in Sec. 2.2 (Figs. 2.3 through 2.6). Notice how, although the details differ, the structure of the pseudocode and the VBA code are identical. We can now use some of the constructs from Fig. 2.8 to write a macro function to numerically compute velocity. Open VBA by selecting4 Tools Macro Visual Basic Editor

Once inside the Visual Basic Editor (VBE), select Insert Module

and a new code window will open up. The following VBA function can be developed directly from the pseudocode in Fig. 2.7. Type it into the code window. Option Explicit Function Euler(dt, ti, tf, yi, m, cd) Dim h As Double, t As Double, y As Double, dydt As Double t = ti y = yi h = dt Do If t + dt > tf Then h = tf − t End If dydt = dy(t, y, m, cd) y = y + dydt * h t = t + h If t >= tf Then Exit Do Loop Euler = y End Function

Compare this macro with the pseudocode from Fig. 2.7 and recognize how similar they are. Also, see how we have expanded the function’s argument list to include the necessary parameters for the parachutist velocity model. The resulting velocity, y, is then passed back to the spreadsheet via the function name. 4

The hot key combination Alt-F11 is even quicker!

(a) Pseudocode

(b) Excel VBA

IF/THEN: IF condition THEN True block ENDIF

If b 0 Then r1 = −c / b End If

IF/THEN/ELSE: IF condition THEN True block ELSE False block ENDIF

If a < 0 Then b = Sqr(Abs(a)) Else b = Sqr(a) End If

IF/THEN/ELSEIF: IF condition1 THEN Block1 ELSEIF condition2 Block2 ELSEIF condition3 Block3 ELSE Block4 ENDIF

If class = 1 Then x = x + 8 ElseIf class < 1 Then x = x − 8 ElseIf class < 10 Then x = x − 32 Else x = x − 64 End If

CASE: SELECT CASE Test Expression CASE Value1 Block1 CASE Value2 Block2 CASE Value3 Block3 CASE ELSE Block4 END SELECT

Select Case a + b Case Is < −50 x = −5 Case Is < 0 x = −5 − (a + b) / 10 Case Is < 50 x = (a + b) / 10 Case Else x = 5 End Select

DOEXIT: DO Block1 IF condition EXIT Block2 ENDDO

FIGURE 2.8 The fundamental control structures in (a) pseudocode and (b) Excel VBA.

COUNT-CONTROLLED LOOP: DOFOR i = start, finish, step Block ENDDO

Do i = i + 1 If i >= 10 Then Exit Do j = i*x Loop For i = 1 To 10 Step 2 x = x + i Next i 41

42

PROGRAMMING AND SOFTWARE

Also notice how we have included another function to compute the derivative. This can be entered in the same module by typing it directly below the Euler function, Function dy(t, v, m, cd) Const g As Double = 9.81 dy = g − (cd / m) * v End Function

The final step is to return to the spreadsheet and invoke the function by entering the following formula in cell B9 =Euler(dt,A8,A9,B8,m,cd)

The result of the numerical integration, 16.531, will appear in cell B9. You should appreciate what has happened here. When you enter the function into the spreadsheet cell, the parameters are passed into the VBA program where the calculation is performed and the result is then passed back and displayed in the cell. In effect, the VBA macro language allows you to use Excel as your input/output mechanism. All sorts of benefits arise from this fact. For example, now that you have set up the calculation, you can play with it. Suppose that the jumper was much heavier, say, m 5 100 kg (about 220 lb). Enter 100 into cell B3 and the spreadsheet will update immediately to show a value of 17.438 in cell B9. Change the mass back to 68.1 kg and the previous result, 16.531, automatically reappears in cell B9. Now let us take the process one step further by filling in some additional numbers for the time. Enter the numbers 4, 6, . . . 16 in cells A10 through A16. Then copy the formulas from cells B9:C9 down to rows 10 through 16. Notice how the VBA program calculates the numerical result correctly for each new row. (To verify this, change dt to 2 and compare with the results previously computed by hand in Example 1.2.) An additional embellishment would be to develop an x-y plot of the results using the Excel Chart Wizard. The final spreadsheet is shown below. We now have created a pretty nice problemsolving tool. You can perform sensitivity analyses by changing the values for each of

2.5

MATLAB

43

the parameters. As each new value is entered, the computation and the graph would be automatically updated. It is this interactive nature that makes Excel so powerful. However, recognize that the ability to solve this problem hinges on being able to write the macro with VBA. It is the combination of the Excel environment with the VBA programming language that truly opens up a world of possibilities for engineering problem solving. In the coming chapters, we will illustrate how this is accomplished.

2.5

MATLAB MATLAB is the flagship software product of The MathWorks, Inc., which was cofounded by the numerical analysts Cleve Moler and John N. Little. As the name implies, MATLAB was originally developed as a matrix laboratory. To this day, the major element of MATLAB is still the matrix. Mathematical manipulations of matrices are very conveniently implemented in an easy-to-use, interactive environment. To these matrix manipulations, MATLAB has added a variety of numerical functions, symbolic computations, and visualization tools. As a consequence, the present version represents a fairly comprehensive technical computing environment. MATLAB has a variety of functions and operators that allow convenient implementation of many of the numerical methods developed in this book. These will be described in detail in the individual chapters that follow. In addition, programs can be written as so-called M-files that can be used to implement numerical calculations. Let us explore how this is done. First, you should recognize that normal MATLAB use is closely related to programming. For example, suppose that we wanted to determine the analytical solution to the parachutist problem. This could be done with the following series of MATLAB commands >> >> >> >> >>

g=9.81; m=68.1; cd=12.5; tf=2; v=g*m/cd*(1−exp(−cd/m*tf))

with the result being displayed as v = 16.4217

Thus, the sequence of commands is just like the sequence of instructions in a typical programming language. Now what if you want to deviate from the sequential structure. Although there are some neat ways to inject some nonsequential capabilities in the standard command mode, the inclusion of decisions and loops is best done by creating a MATLAB document called an M-file. To do this, make the menu selection File New Script

44

PROGRAMMING AND SOFTWARE

and a new window will open with a heading “MATLAB Editor/Debugger.” In this window, you can type and edit MATLAB programs. Type the following code there: g=9.81; m=68.1; cd=12.5; tf=2; v=g*m/cd*(1−exp(−cd/m*tf))

Notice how the commands are written in exactly the way as they would be written in the front end of MATLAB. Save the program with the name: analpara. MATLAB will automatically attach the extension .m to denote it as an M-file: analpara.m. To run the program, you must go back to the command mode. The most direct way to do this is to click on the “MATLAB Command Window” button on the task bar (which is usually at the bottom of the screen). The program can now be run by typing the name of the M-file, analpara, which should look like >> analpara

If you have done everything correctly, MATLAB should respond with the correct answer: v =

16.4217

Now one problem with the foregoing is that it is set up to compute one case only. You can make it more flexible by having the user input some of the variables. For example, suppose that you wanted to assess the impact of mass on the velocity at 2 s. The M-file could be rewritten as the following to accomplish this g=9.81; m=input('mass (kg): '); cd=12.5; tf=2; v=g*m/cd*(1−exp(−cd/m*tf))

Save this as analpara2.m. If you typed analpara2 while being in command mode, the prompt would show mass (kg):

The user could then enter a value like 100, and the result will be displayed as v =

17.3597

Now it should be pretty clear how we can program a numerical solution with an M-file. In order to do this, we must first understand how MATLAB handles logical and looping structures. Figure 2.9 lists pseudocode alongside MATLAB code for all the

2.5

FIGURE 2.9 The fundamental control structures in (a) pseudocode and (b) the MATLAB programming language.

MATLAB

45

(a) Pseudocode

(b) MATLAB

IF/THEN: IF condition THEN True block ENDIF

if b ~= 0 r1 = −c / b; end

IF/THEN/ELSE: IF condition THEN True block ELSE False block ENDIF

if a < 0 b = sqrt(abs(a)); else b 5 sqrt(a); end

IF/THEN/ELSEIF: IF condition1 THEN Block1 ELSEIF condition2 Block2 ELSEIF condition3 Block3 ELSE Block4 ENDIF

if class == 1 x = x + 8; elseif class < 1 x = x − 8; elseif class < 10 x = x − 32; else x = x − 64; end

CASE: SELECT CASE Test Expression CASE Value1 Block1 CASE Value2 Block2 CASE Value3 Block3 CASE ELSE Block4 END SELECT

switch a + b case 1 x = −25; case 2 x = −5 − (a + b) / 10; case 3 x = (a + b) / 10; otherwise x = 5; end

DOEXIT: DO Block1 IF condition EXIT Block2 ENDDO

while (1) i = i + 1; if i >= 10, break, end j = i*x; end

COUNT-CONTROLLED LOOP: DOFOR i = start, finish, step Block ENDDO

for i = 1:2:10 x = x + i; end

46

PROGRAMMING AND SOFTWARE

control structures from Sec. 2.2. Although the structures of the pseudocode and the MATLAB code are very similar, there are some slight differences that should be noted. In particular, look at how we have represented the DOEXIT structure. In place of the DO, we use the statement WHILE(1). Because MATLAB interprets the number 1 as corresponding to “true,” this statement will repeat infinitely in the same manner as the DO statement. The loop is terminated with a break command. This command transfers control to the statement following the end statement that terminates the loop. Also notice that the parameters of the count-controlled loop are ordered differently. For the pseudocode, the loop parameters are specified as start, finish, step. For MATLAB, the parameters are ordered as start:step:finish. The following MATLAB M-file can now be developed directly from the pseudocode in Fig. 2.7. Type it into the MATLAB Editor/Debugger: g=9.81; m=input('mass (kg): '); cd=12.5; ti=0; tf=2; vi=0; dt=0.1; t = ti; v = vi; h = dt; while (1) if t + dt > tf h = tf − t; end dvdt = g − (cd / m) * v; v = v + dvdt * h; t = t + h; if t >= tf, break, end end disp('velocity (m/s):') disp(v)

Save this file as numpara.m and return to the command mode and run it by entering: numpara. The following output should result: mass (kg): 100 velocity (m/s): 17.4559

As a final step in this development, let us take the above M-file and convert it into a proper function. This can be done in the following M-file based on the pseudocode from Fig. 2.7 function yy 5 euler(dt,ti,tf,yi,m,cd) t = ti; y = yi; h = dt;

2.6

MATHCAD

47

while (1) if t + dt > tf h = tf − t; end dydt = dy(t, y, m, cd); y = y + dydt * h; t = t + h; if t >= tf, break, end end yy = y;

Save this file as euler.m and then create another M-file to compute the derivative, function dydt = dy(t, v, m, cd) g = 9.81; dydt = g − (cd / m) * v;

Save this file as dy.m and return to the command mode. In order to invoke the function and see the result, you can type in the following commands >> >> >> >> >> >> >>

m=68.1; cd=12.5; ti=0; tf=2.; vi=0; dt=0.1; euler(dt,ti,tf,vi,m,cd)

When the last command is entered, the answer will be displayed as ans = 16.5478

It is the combination of the MATLAB environment with the M-file programming language that truly opens up a world of possibilities for engineering problem solving. In the coming chapters we will illustrate how this is accomplished.

2.6

MATHCAD Mathcad attempts to bridge the gap between spreadsheets like Excel and notepads. It was originally developed by Allen Razdow of MIT who cofounded Mathsoft, Inc., which published the first commercial version in 1986. Today, Mathsoft is part of Parametric Technology Corporation (PTC) and Mathcad is in version 15. Mathcad is essentially an interactive notepad that allows engineers and scientists to perform a number of common mathematical, data-handling, and graphical tasks. Information and equations are input to a “whiteboard” design environment that is similar in spirit to a page of paper. Unlike a programming tool or spreadsheet, Mathcad’s interface accepts and displays natural mathematical notation using keystrokes or menu palette clicks—with no programming required. Because the worksheets contain live calculations, a single keystroke that changes an input or equation instantly returns an updated result.

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PROGRAMMING AND SOFTWARE

Mathcad can perform tasks in either numeric or symbolic mode. In numeric mode, Mathcad functions and operators give numerical responses, whereas in symbolic mode results are given as general expressions or equations. Maple V, a comprehensive symbolic math package, is the basis of the symbolic mode and was incorporated into Mathcad in 1993. Mathcad has a variety of functions and operators that allow convenient implementation of many of the numerical methods developed in this book. These will be described in detail in succeeding chapters. In the event that you are unfamiliar with Mathcad, Appendix C also provides a primer on using this powerful software.

2.7

OTHER LANGUAGES AND LIBRARIES In Secs. 2.4 and 2.5, we showed how Excel and MATLAB function procedures for Euler’s method could be developed from an algorithm expressed as pseudocode. You should recognize that similar functions can be written in high-level languages like Fortran 90 and C++. For example, a Fortran 90 function for Euler’s method is Function Euler(dt, ti, tf, yi, m, cd) REAL dt, ti, tf, yi, m, cd Real h, t, y, dydt t = ti y = yi h = dt Do If (t + dt > h = tf − t End If dydt = dy(t, y = y + dydt t = t + h If (t >= tf) End Do Euler = y End Function

tf) Then y, m, cd) * h Exit

For C, the result would look quite similar to the MATLAB function. The point is that once a well-structured algorithm is developed in pseudocode form, it can be readily implemented in a variety of programming environments. In this book, our approach will be to provide you with well-structured procedures written as pseudocode. This collection of algorithms then constitutes a numerical library that can be accessed to perform specific numerical tasks in a range of software tools and programming languages. Beyond your own programs, you should be aware that commercial programming libraries contain many useful numerical procedures. For example, the Numerical Recipe library includes a large range of algorithms written in Fortran and C.5 These procedures are described in both book (for example, Press et al. 2007) and electronic form. 5

Numerical Recipe procedures are also available in book and electronic format for Pascal, MS BASIC, and MATLAB. Information on all the Numerical Recipe products can be found at http://www.nr.com/.

PROBLEMS

49

PROBLEMS 2.1 Write pseudocode to implement the flowchart depicted in Fig. P2.1. Make sure that proper indentation is included to make the structure clear.

2.4 The sine function can be evaluated by the following infinite series: sin x 5 x 2

x5 x7 x3 1 2 1p 3! 5! 7!

Write an algorithm to implement this formula so that it computes and prints out the values of sin x as each term in the series is added. In other words, compute and print in sequence the values for F

x < 100

F T

sin x 5 x

T

x < 50

T

x ≤ 500 F

x = 75

x=0

sin x 5 x 2

x3 3!

sin x 5 x 2

x5 x3 1 3! 5!

up to the order term n of your choosing. For each of the preceding, compute and display the percent relative error as % error 5

x = x – 50

FIGURE P2.1

2.2 Rewrite the following pseudocode using proper indentation DO j 5 j 1 1 x 5 x 1 5 IF x . 5 THEN y 5x ELSE y 50 ENDIF z 5x 1 y IF z . 50 EXIT ENDDO

2.3 Develop, debug, and document a program to determine the roots of a quadratic equation, ax2 1 bx 1 c, in either a high-level language or a macro language of your choice. Use a subroutine procedure to compute the roots (either real or complex). Perform test runs for the cases (a) a 5 1, b 5 6, c 5 2; (b) a 5 0, b 5 24, c 5 1.6; (c) a 5 3, b 5 2.5, c 5 7.

true 2 series approximation 3 100% true

Write the algorithm as (a) a structured flowchart and (b) pseudocode. 2.5 Develop, debug, and document a program for Prob. 2.4 in either a high-level language or a macro language of your choice. Employ the library function for the sine in your computer to determine the true value. Have the program print out the series approximation and the error at each step. As a test case, employ the program to compute sin(1.5) for up to and including the term x15/15!. Interpret your results. 2.6 The following algorithm is designed to determine a grade for a course that consists of quizzes, homework, and a final exam: Step 1: Input course number and name. Step 2: Input weighting factors for quizzes (WQ), homework (WH), and the final exam (WF). Step 3: Input quiz grades and determine an average quiz grade (AQ). Step 4: Input homework grades and determine an average homework grade (AH). Step 5: If this course has a final grade, continue to step 6. If not, go to step 9. Step 6: Input final exam grade (FE). Step 7: Determine average grade AG according to AG 5

WQ 3 AQ 1 WH 3 AH 1 WF 3 FE 3 100% WQ 1 WH 1 WF

Step 8: Go to step 10. Step 9: Determine average grade AG according to AG 5

WQ 3 AQ 1 WH 3 AH 3 100% WQ 1 WH

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PROGRAMMING AND SOFTWARE

Step 10: Print out course number, name, and average grade. Step 11: Terminate computation. (a) Write well-structured pseudocode to implement this algorithm. (b) Write, debug, and document a structured computer program based on this algorithm. Test it using the following data to calculate a grade without the final exam and a grade with the final exam: WQ 5 30; WH 5 40; WF 5 30; quizzes 5 98, 95, 90, 60, 99; homework 5 98, 95, 86, 100, 100, 77; and final exam 5 91. 2.7 The “divide and average” method, an old-time method for approximating the square root of any positive number a can be formulated as x5

x 1 ayx 2

(a) Write well-structured pseudocode to implement this algorithm as depicted in Fig. P2.7. Use proper indentation so that the structure is clear. (b) Develop, debug, and document a program to implement this equation in either a high-level language or a macro language of your choice. Structure your code according to Fig. P2.7.

F

T

a>0

tol = 106 x = a/2

SquareRoot = 0

y = (x + a/x)/2 e = |(y – x)/y| x=y

F

e < tol T SquareRoot = x

FIGURE P2.7

2.8 An amount of money P is invested in an account where interest is compounded at the end of the period. The future worth F yielded at an interest rate i after n periods may be determined from the following formula: F 5 P(1 1 i) n Write a program that will calculate the future worth of an investment for each year from 1 through n. The input to the function should include the initial investment P, the interest rate i (as a decimal), and the number of years n for which the future worth is to be calculated. The output should consist of a table with headings and columns for n and F. Run the program for P 5 $100,000, i 5 0.04, and n 5 11 years. 2.9 Economic formulas are available to compute annual payments for loans. Suppose that you borrow an amount of money P and agree to repay it in n annual payments at an interest rate of i. The formula to compute the annual payment A is A5P

i(1 1 i) n (1 1 i) n 2 1

Write a program to compute A. Test it with P 5 $55,000 and an interest rate of 6.6% (i 5 0.066). Compute results for n 5 1, 2, 3, 4, and 5 and display the results as a table with headings and columns for n and A. 2.10 The average daily temperature for an area can be approximated by the following function, T 5 Tmean 1 (Tpeak 2 Tmean ) cos (v(t 2 tpeak ) ) where Tmean 5 the average annual temperature, Tpeak 5 the peak temperature, v 5 the frequency of the annual variation (5 2p/365), and tpeak 5 day of the peak temperature (˘ 205 d). Develop a program that computes the average temperature between two days of the year for a particular city. Test it for (a) January–February (t 5 0 to 59) in Miami, Florida (Tmean 5 22.18C; Tpeak 5 28.38C), and (b) July–August (t 5 180 to 242) in Boston, Massachusetts (Tmean 5 10.78C; Tpeak 5 22.98C). 2.11 Develop, debug, and test a program in either a high-level language or a macro language of your choice to compute the velocity of the falling parachutist as outlined in Example 1.2. Design the program so that it allows the user to input values for the drag coefficient and mass. Test the program by duplicating the results from Example 1.2. Repeat the computation but employ step sizes of 1 and 0.5 s. Compare your results with the analytical solution obtained previously in Example 1.1. Does a smaller step size make the results better or worse? Explain your results. 2.12 The bubble sort is an inefficient, but easy-to-program, sorting technique. The idea behind the sort is to move down through an array comparing adjacent pairs and swapping the

PROBLEMS

51

values if they are out of order. For this method to sort the array completely, it may need to pass through it many times. As the passes proceed for an ascending-order sort, the smaller elements in the array appear to rise toward the top like bubbles. Eventually, there will be a pass through the array where no swaps are required. Then, the array is sorted. After the first pass, the largest value in the array drops directly to the bottom. Consequently, the second pass only has to proceed to the second-to-last value, and so on. Develop a program to set up an array of 20 random numbers and sort them in ascending order with the bubble sort (Fig. P2.12).

decisional control structures (like If/Then, ElseIf, Else, End If). Design the function so that it returns the volume for all cases where the depth is less than 3R. Return an error message (“Overtop”) if you overtop the tank, that is, d . 3R. Test it with the following data:

R

1

1

1

1

d

0.5

1.2

3.0

3.1

start

2R

m=n–1

d R

FIGURE P2.13

switch = false

T

i>m

i=1 i=i+1

F T

Not switch

F

ai > ai+1

II

T

I y

F ai

r

swap ai+1

x

m=m–1

switch = true

end

III

IV

FIGURE P2.12 FIGURE P2.14 2.13 Figure P2.13 shows a cylindrical tank with a conical base. If the liquid level is quite low in the conical part, the volume is simply the conical volume of liquid. If the liquid level is midrange in the cylindrical part, the total volume of liquid includes the filled conical part and the partially filled cylindrical part. Write a well-structured function procedure to compute the tank’s volume as a function of given values of R and d. Use

2.14 Two distances are required to specify the location of a point relative to an origin in two-dimensional space (Fig. P2.14): • The horizontal and vertical distances (x, y) in Cartesian coordinates • The radius and angle (r, u) in radial coordinates.

52

PROGRAMMING AND SOFTWARE

It is relatively straightforward to compute Cartesian coordinates (x, y) on the basis of polar coordinates (r, u). The reverse process is not so simple. The radius can be computed by the following formula: r 5 2x2 1 y2 If the coordinates lie within the first and fourth coordinates (i.e., x . 0), then a simple formula can be used to compute u y u 5 tan 21 a b x The difficulty arises for the other cases. The following table summarizes the possibilities:

x

y

,0 ,0 ,0 50 50 50

U 21

tan (y/x) 1 p tan21(y/x) 2 p p p/2 2p/2 0

.0 ,0 50 .0 ,0 50

(a) Write a well-structured flowchart for a subroutine procedure to calculate r and u as a function of x and y. Express the final results for u in degrees. (b) Write a well-structured function procedure based on your flowchart. Test your program by using it to fill out the following table:

x

y

1 1 0 21 21 21 0 1 0

0 1 1 1 0 21 21 21 0

r

U

2.15 Develop a well-structured function procedure that is passed a numeric grade from 0 to 100 and returns a letter grade according to the scheme:

Letter

Criteria

A B C D F

90 # numeric grade # 100 80 # numeric grade , 90 70 # numeric grade , 80 60 # numeric grade , 70 numeric grade , 60

2.16 Develop well-structured function procedures to determine (a) the factorial; (b) the minimum value in a vector; and (c) the average of the values in a vector. 2.17 Develop well-structured programs to (a) determine the square root of the sum of the squares of the elements of a two-dimensional array (i.e., a matrix) and (b) normalize a matrix by dividing each row by the maximum absolute value in the row so that the maximum element in each row is 1. 2.18 Piecewise functions are sometimes useful when the relationship between a dependent and an independent variable cannot be adequately represented by a single equation. For example, the velocity of a rocket might be described by 11t2 2 5t 1100 2 5t y(t) 5 e50t 1 2(t 2 20) 2 1520e20.2(t230) 0

0 # t # 10 10 # t # 20 20 # t # 30 t . 30 otherwise

Develop a well-structured function to compute v as a function of t. Then use this function to generate a table of v versus t for t 5 25 to 50 at increments of 0.5. 2.19 Develop a well-structured function to determine the elapsed days in a year. The function should be passed three values: mo 5 the month (1–12), da 5 the day (1–31) and leap 5 (0 for non–leap year and 1 for leap year). Test it for January 1, 1999; February 29, 2000; March 1, 2001; June 21, 2002; and December 31, 2004. Hint: a nice way to do this combines the for and the switch structures. 2.20 Develop a well-structured function to determine the elapsed days in a year. The first line of the function should be set up as function nd = days(mo, da, year)

where mo 5 the month (1–12), da 5 the day (1–31) and year 5 the year. Test it for January 1, 1999; February 29, 2000; March 1, 2001; June 21, 2002; and December 31, 2004. 2.21 Manning’s equation can be used to compute the velocity of water in a rectangular open channel, U5

2y3 2S BH b a n B 1 2H

PROBLEMS

53

where U 5 velocity (m/s), S 5 channel slope, n 5 roughness coefficient, B 5 width (m), and H 5 depth (m). The following data are available for five channels:

2.23 The volume V of liquid in a hollow horizontal cylinder of radius r and length L is related to the depth of the liquid h by V 5 c r 2 cos 21a

n

S

B

H

0.035 0.020 0.015 0.030 0.022

0.0001 0.0002 0.0010 0.0007 0.0003

10 8 20 24 15

2 1 1.5 3 2.5

Write a well-structured program that computes the velocity for each of these channels. Have the program display the input data along with the computed velocity in tabular form where velocity is the fifth column. Include headings on the table to label the columns. 2.22 A simply supported beam is loaded as shown in Fig. P2.22. Using singularity functions, the displacement along the beam can be expressed by the equation: uy (x) 5

25 15 [kx 2 0l4 2 kx 2 5l4 ] 1 kx 2 8l3 6 6 1 75 kx 2 7l2 1

57 3 x 2 238.25x 6

By definition, the singularity function can be expressed as follows: kx 2 aln 5 e

(x 2 a) n 0

when x . a f when x # a

Develop a program that creates a plot of displacement versus distance along the beam x. Note that x 5 0 at the left end of the beam.

r2h b 2 (r 2 h) 22rh 2 h2 d L r

Develop a well-structured function to create a plot of volume versus depth. Test the program for r 5 2 m and L 5 5 m. 2.24 Develop a well-structured program to compute the velocity of a parachutist as a function of time using Euler’s method. Test your program for the case where m 5 80 kg and c 5 10 kg/s. Perform the calculation from t 5 0 to 20 s with a step size of 2 s. Use an initial condition that the parachutist has an upward velocity of 20 m/s at t 5 0. At t 5 10 s, assume that the parachute is instantaneously deployed so that the drag coefficient jumps to 50 kg/s. 2.25 The pseudocode in Fig. P2.25 computes the factorial. Express this algorithm as a well-structured function in the language of your choice. Test it by computing 0! and 5!. In addition, test the error trap by trying to evaluate 22!. FUNCTION fac(n) IF n $ 0 THEN x5 1 DOFOR i 5 1, n x5 x? i END DO fac 5 x ELSE display error message terminate ENDIF END fac

FIGURE P2.25 20 kips/ft

150 kip-ft

15 kips

20.26 The height of a small rocket y can be calculated as a function of time after blastoff with the following piecewise function: y 5 38.1454t 1 0.13743t3

5’

FIGURE P2.22

2’

1’

2’

y 5 1036 1 130.909(t 2 15) 1 6.18425(t 2 15) 2 2 0.428(t 2 15) 3 y 5 2900 2 62.468(t 233) 216.9274(t 233) 2 1 0.41796(t 233) 3

0 # t , 15 15 # t , 33 t . 33

54

PROGRAMMING AND SOFTWARE

Develop a well-structured pseudocode function to compute y as a function of t. Note that if the user enters a negative value of t or if the rocket has hit the ground (y # 0) then return a value of zero for y. Also, the function should be invoked in the calling program as height(t). Write the algorithm as (a) pseudocode, or (b) in the high-level language of your choice. 20.27 As depicted in Fig. P2.27, a water tank consists of a cylinder topped by the frustum of a cone. Develop a wellstructured function in the high-level language or macro language of your choice to compute the volume given the water level h (m) above the tank’s bottom. Design the function so that it returns a value of zero for negative h’s and the value of the maximum filled volume for h’s greater than the tank’s maximum depth. Given the following parameters, H1 5 10 m, r1 5 4 m, H2 5 5 m, and r 2 5 6.5 m, test your function by using it to compute the volumes and generate a graph of the volume as a function of level from h 5 21 to 16 m.

r2

H2

r1

h H1

FIGURE P2.27

CHAPTER

3 Approximations and Round-Off Errors Because so many of the methods in this book are straightforward in description and application, it would be very tempting at this point for us to proceed directly to the main body of the text and teach you how to use these techniques. However, understanding the concept of error is so important to the effective use of numerical methods that we have chosen to devote the next two chapters to this topic. The importance of error was introduced in our discussion of the falling parachutist in Chap. 1. Recall that we determined the velocity of a falling parachutist by both analytical and numerical methods. Although the numerical technique yielded estimates that were close to the exact analytical solution, there was a discrepancy, or error, because the numerical method involved an approximation. Actually, we were fortunate in that case because the availability of an analytical solution allowed us to compute the error exactly. For many applied engineering problems, we cannot obtain analytical solutions. Therefore, we cannot compute exactly the errors associated with our numerical methods. In these cases, we must settle for approximations or estimates of the errors. Such errors are characteristic of most of the techniques described in this book. This statement might at first seem contrary to what one normally conceives of as sound engineering. Students and practicing engineers constantly strive to limit errors in their work. When taking examinations or doing homework problems, you are penalized, not rewarded, for your errors. In professional practice, errors can be costly and sometimes catastrophic. If a structure or device fails, lives can be lost. Although perfection is a laudable goal, it is rarely, if ever, attained. For example, despite the fact that the model developed from Newton’s second law is an excellent approximation, it would never in practice exactly predict the parachutist’s fall. A variety of factors such as winds and slight variations in air resistance would result in deviations from the prediction. If these deviations are systematically high or low, then we might need to develop a new model. However, if they are randomly distributed and tightly grouped around the prediction, then the deviations might be considered negligible and the model deemed adequate. Numerical approximations also introduce similar discrepancies into the analysis. Again, the question is: How much the next error is present in our calculations and is it tolerable? This chapter and Chap. 4 cover basic topics related to the identification, quantification, and minimization of these errors. In this chapter, general information concerned with the quantification of error is reviewed in the first sections. This is 55

56

APPROXIMATIONS AND ROUND-OFF ERRORS

followed by a section on one of the two major forms of numerical error: round-off error. Round-off error is due to the fact that computers can represent only quantities with a finite number of digits. Then Chap. 4 deals with the other major form: truncation error. Truncation error is the discrepancy introduced by the fact that numerical methods may employ approximations to represent exact mathematical operations and quantities. Finally, we briefly discuss errors not directly connected with the numerical methods themselves. These include blunders, formulation or model errors, and data uncertainty.

3.1

SIGNIFICANT FIGURES This book deals extensively with approximations connected with the manipulation of numbers. Consequently, before discussing the errors associated with numerical methods, it is useful to review basic concepts related to approximate representation of the numbers themselves. Whenever we employ a number in a computation, we must have assurance that it can be used with confidence. For example, Fig. 3.1 depicts a speedometer and odometer from an automobile. Visual inspection of the speedometer indicates that the car is traveling between 48 and 49 km/h. Because the indicator is higher than the midpoint between the markers on the gauge, we can say with assurance that the car is traveling at approximately 49 km/h. We have confidence in this result because two or more reasonable individuals reading this gauge would arrive at the same conclusion. However, let us say that we insist that the speed be estimated to one decimal place. For this case,

FIGURE 3.1 An automobile speedometer and odometer illustrating the concept of a signiﬁcant ﬁgure.

40

60 40

80

100

20

0

120 4 8 7 3 2 4 5

3.1

SIGNIFICANT FIGURES

57

one person might say 48.8, whereas another might say 48.9 km/h. Therefore, because of the limits of this instrument, only the first two digits can be used with confidence. Estimates of the third digit (or higher) must be viewed as approximations. It would be ludicrous to claim, on the basis of this speedometer, that the automobile is traveling at 48.8642138 km/h. In contrast, the odometer provides up to six certain digits. From Fig. 3.1, we can conclude that the car has traveled slightly less than 87,324.5 km during its lifetime. In this case, the seventh digit (and higher) is uncertain. The concept of a significant figure, or digit, has been developed to formally designate the reliability of a numerical value. The significant digits of a number are those that can be used with confidence. They correspond to the number of certain digits plus one estimated digit. For example, the speedometer and the odometer in Fig. 3.1 yield readings of three and seven significant figures, respectively. For the speedometer, the two certain digits are 48. It is conventional to set the estimated digit at one-half of the smallest scale division on the measurement device. Thus the speedometer reading would consist of the three significant figures: 48.5. In a similar fashion, the odometer would yield a sevensignificant-figure reading of 87,324.45. Although it is usually a straightforward procedure to ascertain the significant figures of a number, some cases can lead to confusion. For example, zeros are not always significant figures because they may be necessary just to locate a decimal point. The numbers 0.00001845, 0.0001845, and 0.001845 all have four significant figures. Similarly, when trailing zeros are used in large numbers, it is not clear how many, if any, of the zeros are significant. For example, at face value the number 45,300 may have three, four, or five significant digits, depending on whether the zeros are known with confidence. Such uncertainty can be resolved by using scientific notation, where 4.53 3 104, 4.530 3 104, 4.5300 3 104 designate that the number is known to three, four, and five significant figures, respectively. The concept of significant figures has two important implications for our study of numerical methods: 1. As introduced in the falling parachutist problem, numerical methods yield approximate results. We must, therefore, develop criteria to specify how confident we are in our approximate result. One way to do this is in terms of significant figures. For example, we might decide that our approximation is acceptable if it is correct to four significant figures. 2. Although quantities such as p, e, or 17 represent specific quantities, they cannot be expressed exactly by a limited number of digits. For example, p 5 3.141592653589793238462643 p ad infinitum. Because computers retain only a finite number of significant figures, such numbers can never be represented exactly. The omission of the remaining significant figures is called round-off error. Both round-off error and the use of significant figures to express our confidence in a numerical result will be explored in detail in subsequent sections. In addition, the concept of significant figures will have relevance to our definition of accuracy and precision in the next section.

APPROXIMATIONS AND ROUND-OFF ERRORS

3.2

ACCURACY AND PRECISION The errors associated with both calculations and measurements can be characterized with regard to their accuracy and precision. Accuracy refers to how closely a computed or measured value agrees with the true value. Precision refers to how closely individual computed or measured values agree with each other. These concepts can be illustrated graphically using an analogy from target practice. The bullet holes on each target in Fig. 3.2 can be thought of as the predictions of a numerical technique, whereas the bull’s-eye represents the truth. Inaccuracy (also called bias) is defined as systematic deviation from the truth. Thus, although the shots in Fig. 3.2c are more tightly grouped than those in Fig. 3.2a, the two cases are equally biased because they are both centered on the upper left quadrant of the target. Imprecision (also called uncertainty), on the other hand, refers to the magnitude of the scatter. Therefore, although Fig. 3.2b and d are equally accurate (that is, centered on the bull’s-eye), the latter is more precise because the shots are tightly grouped. Numerical methods should be sufficiently accurate or unbiased to meet the requirements of a particular engineering problem. They also should be precise enough for adequate

FIGURE 3.2 An example from marksmanship illustrating the concepts of accuracy and precision. (a) Inaccurate and imprecise; (b) accurate and imprecise; (c) inaccurate and precise; (d) accurate and precise. Increasing accuracy

Increasing precision

58

(a)

(b)

(c)

(d)

3.3

ERROR DEFINITIONS

59

engineering design. In this book, we will use the collective term error to represent both the inaccuracy and the imprecision of our predictions. With these concepts as background, we can now discuss the factors that contribute to the error of numerical computations.

3.3

ERROR DEFINITIONS Numerical errors arise from the use of approximations to represent exact mathematical operations and quantities. These include truncation errors, which result when approximations are used to represent exact mathematical procedures, and round-off errors, which result when numbers having limited significant figures are used to represent exact numbers. For both types, the relationship between the exact, or true, result and the approximation can be formulated as True value 5 approximation 1 error

(3.1)

By rearranging Eq. (3.1), we find that the numerical error is equal to the discrepancy between the truth and the approximation, as in Et 5 true value 2 approximation

(3.2)

where Et is used to designate the exact value of the error. The subscript t is included to designate that this is the “true” error. This is in contrast to other cases, as described shortly, where an “approximate” estimate of the error must be employed. A shortcoming of this definition is that it takes no account of the order of magnitude of the value under examination. For example, an error of a centimeter is much more significant if we are measuring a rivet rather than a bridge. One way to account for the magnitudes of the quantities being evaluated is to normalize the error to the true value, as in True fractional relative error 5

true error true value

where, as specified by Eq. (3.2), error 5 true value 2 approximation. The relative error can also be multiplied by 100 percent to express it as et 5

true error 100% true value

(3.3)

where et designates the true percent relative error. EXAMPLE 3.1

Calculation of Errors Problem Statement. Suppose that you have the task of measuring the lengths of a bridge and a rivet and come up with 9999 and 9 cm, respectively. If the true values are 10,000 and 10 cm, respectively, compute (a) the true error and (b) the true percent relative error for each case. Solution. (a) The error for measuring the bridge is [Eq. (3.2)] Et 5 10,000 2 9999 5 1 cm

60

APPROXIMATIONS AND ROUND-OFF ERRORS

and for the rivet it is Et 5 10 2 9 5 1 cm (b) The percent relative error for the bridge is [Eq. (3.3)] et 5

1 100% 5 0.01% 10,000

and for the rivet it is et 5

1 100% 5 10% 10

Thus, although both measurements have an error of 1 cm, the relative error for the rivet is much greater. We would conclude that we have done an adequate job of measuring the bridge, whereas our estimate for the rivet leaves something to be desired.

Notice that for Eqs. (3.2) and (3.3), E and e are subscripted with a t to signify that the error is normalized to the true value. In Example 3.1, we were provided with this value. However, in actual situations such information is rarely available. For numerical methods, the true value will be known only when we deal with functions that can be solved analytically. Such will typically be the case when we investigate the theoretical behavior of a particular technique for simple systems. However, in real-world applications, we will obviously not know the true answer a priori. For these situations, an alternative is to normalize the error using the best available estimate of the true value, that is, to the approximation itself, as in ea 5

approximate error 100% approximation

(3.4)

where the subscript a signifies that the error is normalized to an approximate value. Note also that for real-world applications, Eq. (3.2) cannot be used to calculate the error term for Eq. (3.4). One of the challenges of numerical methods is to determine error estimates in the absence of knowledge regarding the true value. For example, certain numerical methods use an iterative approach to compute answers. In such an approach, a present approximation is made on the basis of a previous approximation. This process is performed repeatedly, or iteratively, to successively compute (we hope) better and better approximations. For such cases, the error is often estimated as the difference between previous and current approximations. Thus, percent relative error is determined according to ea 5

current approximation 2 previous approximation 100% current approximation

(3.5)

This and other approaches for expressing errors will be elaborated on in subsequent chapters. The signs of Eqs. (3.2) through (3.5) may be either positive or negative. If the approximation is greater than the true value (or the previous approximation is greater than the current approximation), the error is negative; if the approximation is less than the true value, the error is positive. Also, for Eqs. (3.3) to (3.5), the denominator may

3.3

ERROR DEFINITIONS

61

be less than zero, which can also lead to a negative error. Often, when performing computations, we may not be concerned with the sign of the error, but we are interested in whether the percent absolute value is lower than a prespecified percent tolerance es. Therefore, it is often useful to employ the absolute value of Eqs. (3.2) through (3.5). For such cases, the computation is repeated until ZeaZ , es

(3.6)

If this relationship holds, our result is assumed to be within the prespecified acceptable level es. Note that for the remainder of this text, we will almost exclusively employ absolute values when we use relative errors. It is also convenient to relate these errors to the number of significant figures in the approximation. It can be shown (Scarborough, 1966) that if the following criterion is met, we can be assured that the result is correct to at least n significant figures. es 5 ( 0.5 3 1022n ) % EXAMPLE 3.2

(3.7)

Error Estimates for Iterative Methods Problem Statement. In mathematics, functions can often be represented by infinite series. For example, the exponential function can be computed using ex 5 1 1 x 1

x2 x3 xn 1 1p1 2 3! n!

(E3.2.1)

Thus, as more terms are added in sequence, the approximation becomes a better and better estimate of the true value of ex. Equation (E3.2.1) is called a Maclaurin series expansion. Starting with the simplest version, ex 5 1, add terms one at a time to estimate e0.5. After each new term is added, compute the true and approximate percent relative errors with Eqs. (3.3) and (3.5), respectively. Note that the true value is e0.5 5 1.648721 . . . . Add terms until the absolute value of the approximate error estimate ea falls below a prespecified error criterion es conforming to three significant figures. Solution. First, Eq. (3.7) can be employed to determine the error criterion that ensures a result is correct to at least three significant figures: es 5 ( 0.5 3 10223 ) % 5 0.05% Thus, we will add terms to the series until ea falls below this level. The first estimate is simply equal to Eq. (E3.2.1) with a single term. Thus, the first estimate is equal to 1. The second estimate is then generated by adding the second term, as in ex 5 1 1 x or for x 5 0.5, e0.5 5 1 1 0.5 5 1.5 This represents a true percent relative error of [Eq. (3.3)] et 5

1.648721 2 1.5 100% 5 9.02% 1.648721

62

APPROXIMATIONS AND ROUND-OFF ERRORS

Equation (3.5) can be used to determine an approximate estimate of the error, as in ea 5

1.5 2 1 100% 5 33.3% 1.5

Because ea is not less than the required value of es, we would continue the computation by adding another term, x2y2!, and repeating the error calculations. The process is continued until ea , es. The entire computation can be summarized as Terms

Result

Et (%)

Ea (%)

1 2 3 4 5 6

1 1.5 1.625 1.645833333 1.648437500 1.648697917

39.3 9.02 1.44 0.175 0.0172 0.00142

33.3 7.69 1.27 0.158 0.0158

Thus, after six terms are included, the approximate error falls below es 5 0.05% and the computation is terminated. However, notice that, rather than three significant figures, the result is accurate to five! This is because, for this case, both Eqs. (3.5) and (3.7) are conservative. That is, they ensure that the result is at least as good as they specify. Although, as discussed in Chap. 6, this is not always the case for Eq. (3.5), it is true most of the time.

3.3.1 Computer Algorithm for Iterative Calculations Many of the numerical methods described in the remainder of this text involve iterative calculations of the sort illustrated in Example 3.2. These all entail solving a mathematical problem by computing successive approximations to the solution starting from an initial guess. The computer implementation of such iterative solutions involves loops. As we saw in Sec. 2.1.1, these come in two basic flavors: count-controlled and decision loops. Most iterative solutions use decision loops. Thus, rather than employing a prespecified number of iterations, the process typically is repeated until an approximate error estimate falls below a stopping criterion, as in Example 3.2. A pseudocode for a generic iterative calculation is presented in Fig. 3.3. The function is passed a value (val) along with a stopping error criterion (es) and a maximum allowable number of iterations (maxit). The value is typically either (1) an initial value or (2) the value for which the iterative calculation is to be made. The function first initializes three variables. These include (1) a variable iter that keeps track of the number of iterations, (2) a variable sol that holds the current estimate of the solution, and (3) a variable ea that holds the approximate percent relative error. Note that ea is initially set to a value of 100 to ensure that the loop executes at least once. These initializations are followed by the decision loop that actually implements the iterative calculation. Prior to generating a new solution, sol is first assigned to solold. Then a new value of sol is computed and the iteration counter is incremented. If the new value of sol is nonzero, the percent relative error ea is determined. The stopping

3.3

ERROR DEFINITIONS

63

FUNCTION IterMeth(val, es, maxit) iter 5 1 sol 5 val ea 5 100 DO solold 5 sol sol 5 ... iter 5 iter 1 1 IF sol ﬁ 0 ea5abs((sol 2 solold)/sol)*100 IF ea # es OR iter $ maxit EXIT END DO IterMeth 5 sol END IterMeth

FIGURE 3.3 Pseudocode for a generic iterative calculation.

criteria are then tested. If both are false, the loop repeats. If either are true, the loop terminates and the final solution is sent back to the function call. The following example illustrates how the generic algorithm can be applied to a specific iterative calculation. EXAMPLE 3.3

Computer Implementation of an Iterative Calculation Problem Statement. Develop a computer program based on the pseudocode from Fig. 3.3 to implement the calculation from Example 3.2. Solution. A function to implement the Maclaurin series expansion for ex can be based on the general scheme in Fig. 3.3. To do this, we first formulate the series expansion as a formula: n

xn i50 n!

ex > a

Figure 3.4 shows functions to implement this series written in VBA and MATLAB software. Similar codes could be developed in other languages such a C11 or Fortran 95. Notice that whereas MATLAB has a built-in factorial function, it is necessary to compute the factorial as part of the VBA implementation with a simple product accumulator fac. When the programs are run, they generate an estimate for the exponential function. For the MATLAB version, the answer is returned along with the approximate error and the number of iterations. For example, e1 can be evaluated as >> format long >> [val, ea, iter] = IterMeth(1,1e−6,100) val = 2.718281826198493 ea = 9.216155641522974e−007 iter = 12

64

APPROXIMATIONS AND ROUND-OFF ERRORS

We can see that after 12 iterations, we obtain a result of 2.7182818 with an approximate error estimate of 5 9.2162 3 1027%. The result can be verified by using the built-in exp function to directly calculate the exact value and the true percent relative error, >> trueval=exp(1) trueval = 2.718281828459046 >> et=abs((trueval−val)/trueval)*100 et = 8.316108397236229e−008

As was the case with Example 3.2, we obtain the desirable outcome that the true error is less than the approximate error.

(a) VBA/Excel Function IterMeth(x, es, maxit) ’ initialization iter = 1 sol = 1 ea = 100 fac = 1 ’ iterative calculation Do solold = sol fac = fac * iter sol = sol + x ^ iter / fac iter = iter + 1 If sol 0 Then ea = Abs((sol − solold) / sol) * 100 End If If ea = maxit Then Exit Do Loop IterMeth = sol End Function

(b) MATLAB function [v,ea,iter] = IterMeth(x,es,maxit) % initialization iter = 1; sol = 1; ea = 100; % iterative calculation while (1) solold = sol; sol = sol + x ^ iter / factorial(iter); iter = iter + 1; if sol~=0 ea=abs((sol − solold)/sol)*100; end if ea=maxit,break,end end v = sol; end

FIGURE 3.4 (a) VBA/Excel and (b) MATLAB functions based on the pseudocode from Fig. 3.3.

With the preceding definitions as background, we can now proceed to the two types of error connected directly with numerical methods: round-off errors and truncation errors.

3.4

3.4

ROUND-OFF ERRORS

65

ROUND-OFF ERRORS As mentioned previously, round-off errors originate from the fact that computers retain only a fixed number of significant figures during a calculation. Numbers such as p, e, or 27 cannot be expressed by a fixed number of significant figures. Therefore, they cannot be represented exactly by the computer. In addition, because computers use a base-2 representation, they cannot precisely represent certain exact base-10 numbers. The discrepancy introduced by this omission of significant figures is called round-off error. 3.4.1 Computer Representation of Numbers Numerical round-off errors are directly related to the manner in which numbers are stored in a computer. The fundamental unit whereby information is represented is called a word. This is an entity that consists of a string of binary digits, or bits. Numbers are typically stored in one or more words. To understand how this is accomplished, we must first review some material related to number systems. Number Systems. A number system is merely a convention for representing quantities. Because we have 10 fingers and 10 toes, the number system that we are most familiar with is the decimal, or base-10, number system. A base is the number used as the reference for constructing the system. The base-10 system uses the 10 digits—0, 1, 2, 3, 4, 5, 6, 7, 8, 9—to represent numbers. By themselves, these digits are satisfactory for counting from 0 to 9. For larger quantities, combinations of these basic digits are used, with the position or place value specifying the magnitude. The right-most digit in a whole number represents a number from 0 to 9. The second digit from the right represents a multiple of 10. The third digit from the right represents a multiple of 100 and so on. For example, if we have the number 86,409 then we have eight groups of 10,000, six groups of 1000, four groups of 100, zero groups of 10, and nine more units, or

( 8 3 104 ) 1 ( 6 3 103 ) 1 ( 4 3 102 ) 1 ( 0 3 101 ) 1 ( 9 3 100 ) 5 86,409 Figure 3.5a provides a visual representation of how a number is formulated in the base-10 system. This type of representation is called positional notation. Because the decimal system is so familiar, it is not commonly realized that there are alternatives. For example, if human beings happened to have had eight fingers and eight toes, we would undoubtedly have developed an octal, or base-8, representation. In the same sense, our friend the computer is like a two-fingered animal who is limited to two states—either 0 or 1. This relates to the fact that the primary logic units of digital computers are on/off electronic components. Hence, numbers on the computer are represented with a binary, or base-2, system. Just as with the decimal system, quantities can be represented using positional notation. For example, the binary number 11 is equivalent to (1 3 21) 1 (1 3 20) 5 2 1 1 5 3 in the decimal system. Figure 3.5b illustrates a more complicated example. Integer Representation. Now that we have reviewed how base-10 numbers can be represented in binary form, it is simple to conceive of how integers are represented on a computer. The most straightforward approach, called the signed magnitude method, employs the first bit of a word to indicate the sign, with a 0 for positive and a 1 for

66

APPROXIMATIONS AND ROUND-OFF ERRORS

104 103 102 101 100 8

6

4

0

9 9 0 4 6 8

(a)

27

26

25

24

23

22

21

20

1

0

1

0

1

1

0

1

⫻ 1 ⫻ 10 ⫻ 100 ⫻ 1,000 ⫻ 10,000

1 0 1 1 0 1 0 1

(b)

⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻

1 2 4 8 16 32 64 128

= 9 = 0 = 400 = 6,000 = 80,000 86,409

= 1 = 0 = 4 = 8 = 0 = 32 = 0 = 128 173

FIGURE 3.5 How the (a) decimal (base-10) and the (b) binary (base-2) systems work. In (b), the binary number 10101101 is equivalent to the decimal number 173.

1

0

0

0

0

0

0

0

1

0

1

0

1

1

0

1

Number Sign

FIGURE 3.6 The representation of the decimal integer 2173 on a 16-bit computer using the signed magnitude method.

negative. The remaining bits are used to store the number. For example, the integer value of 2173 would be stored on a 16-bit computer, as in Fig. 3.6. EXAMPLE 3.4

Range of Integers Problem Statement. Determine the range of integers in base-10 that can be represented on a 16-bit computer.

3.4

ROUND-OFF ERRORS

67

Solution. Of the 16 bits, the first bit holds the sign. The remaining 15 bits can hold binary numbers from 0 to 111111111111111. The upper limit can be converted to a decimal integer, as in

( 1 3 214 ) 1 ( 1 3 213 ) 1 p 1 ( 1 3 21 ) 1 ( 1 3 20 ) which equals 32,767 (note that this expression can be simply evaluated as 215 2 1). Thus, a 16-bit computer word can store decimal integers ranging from 232,767 to 32,767. In addition, because zero is already defined as 0000000000000000, it is redundant to use the number 1000000000000000 to define a “minus zero.” Therefore, it is usually employed to represent an additional negative number: 232,768, and the range is from 232,768 to 32,767.

Note that the signed magnitude method described above is not used to represent integers on conventional computers. A preferred approach called the 2’s complement technique directly incorporates the sign into the number’s magnitude rather than providing a separate bit to represent plus or minus (see Chapra and Canale 1994). However, Example 3.4 still serves to illustrate how all digital computers are limited in their capability to represent integers. That is, numbers above or below the range cannot be represented. A more serious limitation is encountered in the storage and manipulation of fractional quantities as described next. Floating-Point Representation. Fractional quantities are typically represented in computers using floating-point form. In this approach, the number is expressed as a fractional part, called a mantissa or significand, and an integer part, called an exponent or characteristic, as in m # be where m 5 the mantissa, b 5 the base of the number system being used, and e 5 the exponent. For instance, the number 156.78 could be represented as 0.15678 3 103 in a floating-point base-10 system. Figure 3.7 shows one way that a floating-point number could be stored in a word. The first bit is reserved for the sign, the next series of bits for the signed exponent, and the last bits for the mantissa.

FIGURE 3.7 The manner in which a ﬂoating-point number is stored in a word. Signed exponent Mantissa Sign

68

APPROXIMATIONS AND ROUND-OFF ERRORS

Note that the mantissa is usually normalized if it has leading zero digits. For example, suppose the quantity 1y34 5 0.029411765 . . . was stored in a floating-point base10 system that allowed only four decimal places to be stored. Thus, 1y34 would be stored as 0.0294 3 100 However, in the process of doing this, the inclusion of the useless zero to the right of the decimal forces us to drop the digit 1 in the fifth decimal place. The number can be normalized to remove the leading zero by multiplying the mantissa by 10 and lowering the exponent by 1 to give 0.2941 3 1021 Thus, we retain an additional significant figure when the number is stored. The consequence of normalization is that the absolute value of m is limited. That is, 1 #m , 1 b

(3.8)

where b 5 the base. For example, for a base-10 system, m would range between 0.1 and 1, and for a base-2 system, between 0.5 and 1. Floating-point representation allows both fractions and very large numbers to be expressed on the computer. However, it has some disadvantages. For example, floating-point numbers take up more room and take longer to process than integer numbers. More significantly, however, their use introduces a source of error because the mantissa holds only a finite number of significant figures. Thus, a round-off error is introduced. EXAMPLE 3.5

Hypothetical Set of Floating-Point Numbers Problem Statement. Create a hypothetical floating-point number set for a machine that stores information using 7-bit words. Employ the first bit for the sign of the number, the next three for the sign and the magnitude of the exponent, and the last three for the magnitude of the mantissa (Fig. 3.8).

FIGURE 3.8 The smallest possible positive ﬂoating-point number from Example 3.5. 21 20 2–1 2–2 2–3 0

1

1

Sign of Sign of number exponent

1

1

0

0

Magnitude of mantissa

Magnitude of exponent

3.4

ROUND-OFF ERRORS

69

Solution. The smallest possible positive number is depicted in Fig. 3.8. The initial 0 indicates that the quantity is positive. The 1 in the second place designates that the exponent has a negative sign. The 1’s in the third and fourth places give a maximum value to the exponent of 1 3 21 1 1 3 20 5 3 Therefore, the exponent will be 23. Finally, the mantissa is specified by the 100 in the last three places, which conforms to 1 3 221 1 0 3 222 1 0 3 223 5 0.5 Although a smaller mantissa is possible (e.g., 000, 001, 010, 011), the value of 100 is used because of the limit imposed by normalization [Eq. (3.8)]. Thus, the smallest possible positive number for this system is 10.5 3 223, which is equal to 0.0625 in the base-10 system. The next highest numbers are developed by increasing the mantissa, as in 0111101 5 ( 1 3 221 1 0 3 222 1 1 3 223 ) 3 223 5 (0.078125) 10 0111110 5 ( 1 3 221 1 1 3 222 1 0 3 223 ) 3 223 5 (0.093750) 10 0111111 5 ( 1 3 221 1 1 3 222 1 1 3 223 ) 3 223 5 (0.109375) 10 Notice that the base-10 equivalents are spaced evenly with an interval of 0.015625. At this point, to continue increasing, we must decrease the exponent to 10, which gives a value of 1 3 21 1 0 3 20 5 2 The mantissa is decreased back to its smallest value of 100. Therefore, the next number is 0110100 5 ( 1 3 221 1 0 3 222 1 0 3 223 ) 3 222 5 (0.125000) 10 This still represents a gap of 0.125000 2 0.109375 5 0.015625. However, now when higher numbers are generated by increasing the mantissa, the gap is lengthened to 0.03125, 0110101 5 ( 1 3 221 1 0 3 222 1 1 3 223 ) 3 222 5 (0.156250) 10 0110110 5 ( 1 3 221 1 1 3 222 1 0 3 223 ) 3 222 5 (0.187500) 10 0110111 5 ( 1 3 221 1 1 3 222 1 1 3 223 ) 3 222 5 (0.218750) 10 This pattern is repeated as each larger quantity is formulated until a maximum number is reached, 0011111 5 ( 1 3 221 1 1 3 222 1 1 3 223 ) 3 23 5 (7) 10 The final number set is depicted graphically in Fig. 3.9. Figure 3.9 manifests several aspects of floating-point representation that have significance regarding computer round-off errors: 1. There Is a Limited Range of Quantities That May Be Represented. Just as for the integer case, there are large positive and negative numbers that cannot be represented. Attempts to employ numbers outside the acceptable range will result in what is called

70

APPROXIMATIONS AND ROUND-OFF ERRORS Chopping

Rounding

x – ⌬x

⌬x

x – ⌬x

⌬x/2 ⌬x/2

x + ⌬x

7 0

Overflow

0

Underflow “hole” at zero

FIGURE 3.9 The hypothetical number system developed in Example 3.5. Each value is indicated by a tick mark. Only the positive numbers are shown. An identical set would also extend in the negative direction.

an overflow error. However, in addition to large quantities, the floating-point representation has the added limitation that very small numbers cannot be represented. This is illustrated by the underflow “hole” between zero and the first positive number in Fig. 3.9. It should be noted that this hole is enlarged because of the normalization constraint of Eq. (3.8). 2. There Are Only a Finite Number of Quantities That Can Be Represented within the Range. Thus, the degree of precision is limited. Obviously, irrational numbers cannot be represented exactly. Furthermore, rational numbers that do not exactly match one of the values in the set also cannot be represented precisely. The errors introduced by approximating both these cases are referred to as quantizing errors. The actual approximation is accomplished in either of two ways: chopping or rounding. For example, suppose that the value of p 5 3.14159265358 . . . is to be stored on a base10 number system carrying seven significant figures. One method of approximation would be to merely omit, or “chop off,” the eighth and higher terms, as in p 5 3.141592, with the introduction of an associated error of [Eq. (3.2)] Et 5 0.00000065 p This technique of retaining only the significant terms was originally dubbed “truncation” in computer jargon. We prefer to call it chopping to distinguish it from the truncation errors discussed in Chap. 4. Note that for the base-2 number system

3.4

ROUND-OFF ERRORS

71

in Fig. 3.9, chopping means that any quantity falling within an interval of length Dx will be stored as the quantity at the lower end of the interval. Thus, the upper error bound for chopping is Dx. Additionally, a bias is introduced because all errors are positive. The shortcomings of chopping are attributable to the fact that the higher terms in the complete decimal representation have no impact on the shortened version. For instance, in our example of p, the first discarded digit is 6. Thus, the last retained digit should be rounded up to yield 3.141593. Such rounding reduces the error to Et 5 20.00000035 p Consequently, rounding yields a lower absolute error than chopping. Note that for the base-2 number system in Fig. 3.9, rounding means that any quantity falling within an interval of length Dx will be represented as the nearest allowable number. Thus, the upper error bound for rounding is Dxy2. Additionally, no bias is introduced because some errors are positive and some are negative. Some computers employ rounding. However, this adds to the computational overhead, and, consequently, many machines use simple chopping. This approach is justified under the supposition that the number of significant figures is large enough that resulting round-off error is usually negligible. 3. The Interval between Numbers, Dx, Increases as the Numbers Grow in Magnitude. It is this characteristic, of course, that allows floating-point representation to preserve significant digits. However, it also means that quantizing errors will be proportional to the magnitude of the number being represented. For normalized floating-point numbers, this proportionality can be expressed, for cases where chopping is employed, as Z¢xZ

ZxZ

#e

(3.9)

and, for cases where rounding is employed, as Z¢xZ

ZxZ

#

e 2

(3.10)

where % is referred to as the machine epsilon, which can be computed as e 5 b12t

(3.11)

where b is the number base and t is the number of significant digits in the mantissa. Notice that the inequalities in Eqs. (3.9) and (3.10) signify that these are error bounds. That is, they specify the worst cases. EXAMPLE 3.6

Machine Epsilon Problem Statement. Determine the machine epsilon and verify its effectiveness in characterizing the errors of the number system from Example 3.5. Assume that chopping is used. Solution. The hypothetical floating-point system from Example 3.5 employed values of the base b 5 2, and the number of mantissa bits t 5 3. Therefore, the machine epsilon would be [Eq. (3.11)] e 5 2123 5 0.25

72

APPROXIMATIONS AND ROUND-OFF ERRORS

Largest relative error

FIGURE 3.10 The largest quantizing error will occur for those values falling just below the upper bound of the ﬁrst of a series of equispaced intervals.

Consequently, the relative quantizing error should be bounded by 0.25 for chopping. The largest relative errors should occur for those quantities that fall just below the upper bound of the first interval between successive equispaced numbers (Fig. 3.10). Those numbers falling in the succeeding higher intervals would have the same value of Dx but a greater value of x and, hence, would have a lower relative error. An example of a maximum error would be a value falling just below the upper bound of the interval between (0.125000)10 and (0.156250)10. For this case, the error would be less than 0.03125 5 0.25 0.125000 Thus, the error is as predicted by Eq. (3.9).

The magnitude dependence of quantizing errors has a number of practical applications in numerical methods. Most of these relate to the commonly employed operation of testing whether two numbers are equal. This occurs when testing convergence of quantities as well as in the stopping mechanism for iterative processes (recall Example 3.2). For these cases, it should be clear that, rather than test whether the two quantities are equal, it is advisable to test whether their difference is less than an acceptably small tolerance. Further, it should also be evident that normalized rather than absolute difference should be compared, particularly when dealing with numbers of large magnitude. In addition, the machine epsilon can be employed in formulating stopping or convergence criteria. This ensures that programs are portable—that is, they are not dependent on the computer on which they are implemented. Figure 3.11 lists pseudocode to automatically determine the machine epsilon of a binary computer. epsilon 5 1 DO IF (epsilon11 # 1)EXIT epsilon 5 epsilon/2 END DO epsilon 5 2 3 epsilon

FIGURE 3.11 Pseudocode to determine machine epsilon for a binary computer.

Extended Precision. It should be noted at this point that, although round-off errors can be important in contexts such as testing convergence, the number of significant digits carried on most computers allows most engineering computations to be performed with more than acceptable precision. For example, the hypothetical number system in Fig. 3.9 is a gross exaggeration that was employed for illustrative purposes. Commercial computers use much larger words and, consequently, allow numbers to be expressed with more than adequate precision. For example, computers that use IEEE format allow 24 bits to be used for the mantissa, which translates into about seven significant base-10 digits of precision1 with a range of about 10238 to 1039. 1

Note that only 23 bits are actually used to store the mantissa. However, because of normalization, the first bit of the mantissa is always 1 and is, therefore, not stored. Thus, this first bit together with the 23 stored bits gives the 24 total bits of precision for the mantissa.

3.4

ROUND-OFF ERRORS

73

With this acknowledged, there are still cases where round-off error becomes critical. For this reason most computers allow the specification of extended precision. The most common of these is double precision, in which the number of words used to store floating-point numbers is doubled. It provides about 15 to 16 decimal digits of precision and a range of approximately 102308 to 10308. In many cases, the use of double-precision quantities can greatly mitigate the effect of round-off errors. However, a price is paid for such remedies in that they also require more memory and execution time. The difference in execution time for a small calculation might seem insignificant. However, as your programs become larger and more complicated, the added execution time could become considerable and have a negative impact on your effectiveness as a problem solver. Therefore, extended precision should not be used frivolously. Rather, it should be selectively employed where it will yield the maximum benefit at the least cost in terms of execution time. In the following sections, we will look closer at how round-off errors affect computations, and in so doing provide a foundation of understanding to guide your use of the double-precision capability. Before proceeding, it should be noted that some of the commonly used software packages (for example, Excel, Mathcad) routinely use double precision to represent numerical quantities. Thus, the developers of these packages decided that mitigating round-off errors would take precedence over any loss of speed incurred by using extended precision. Others, like MATLAB software, allow you to use extended precision, if you desire. 3.4.2 Arithmetic Manipulations of Computer Numbers Aside from the limitations of a computer’s number system, the actual arithmetic manipulations involving these numbers can also result in round-off error. In the following section, we will first illustrate how common arithmetic operations affect round-off errors. Then we will investigate a number of particular manipulations that are especially prone to round-off errors. Common Arithmetic Operations. Because of their familiarity, normalized base-10 numbers will be employed to illustrate the effect of round-off errors on simple addition, subtraction, multiplication, and division. Other number bases would behave in a similar fashion. To simplify the discussion, we will employ a hypothetical decimal computer with a 4-digit mantissa and a 1-digit exponent. In addition, chopping is used. Rounding would lead to similar though less dramatic errors. When two floating-point numbers are added, the mantissa of the number with the smaller exponent is modified so that the exponents are the same. This has the effect of aligning the decimal points. For example, suppose we want to add 0.1557 ? 101 1 0.4381 ? 1021. The decimal of the mantissa of the second number is shifted to the left a number of places equal to the difference of the exponents [1 2 (21) 5 2], as in 0.4381 # 1021 S 0.004381 # 101 Now the numbers can be added, 0.1557 # 101 0.004381 # 101 0.160081 # 101 and the result chopped to 0.1600 ? 101. Notice how the last two digits of the second number that were shifted to the right have essentially been lost from the computation.

74

APPROXIMATIONS AND ROUND-OFF ERRORS

Subtraction is performed identically to addition except that the sign of the subtrahend is reversed. For example, suppose that we are subtracting 26.86 from 36.41. That is, 0.3641 # 102 20.2686 # 102 0.0955 # 102 For this case the result is not normalized, and so we must shift the decimal one place to the right to give 0.9550 ? 101 5 9.550. Notice that the zero added to the end of the mantissa is not significant but is merely appended to fill the empty space created by the shift. Even more dramatic results would be obtained when the numbers are very close, as in 0.7642 # 103 20.7641 # 103 0.0001 # 103 which would be converted to 0.1000 ? 100 5 0.1000. Thus, for this case, three nonsignificant zeros are appended. This introduces a substantial computational error because subsequent manipulations would act as if these zeros were significant. As we will see in a later section, the loss of significance during the subtraction of nearly equal numbers is among the greatest source of round-off error in numerical methods. Multiplication and division are somewhat more straightforward than addition or subtraction. The exponents are added and the mantissas multiplied. Because multiplication of two n-digit mantissas will yield a 2n-digit result, most computers hold intermediate results in a double-length register. For example, 0.1363 # 103 3 0.6423 # 1021 5 0.08754549 # 102 If, as in this case, a leading zero is introduced, the result is normalized, 0.08754549 # 102 S 0.8754549 # 101 and chopped to give 0.8754 # 101 Division is performed in a similar manner, but the mantissas are divided and the exponents are subtracted. Then the results are normalized and chopped. Large Computations. Certain methods require extremely large numbers of arithmetic manipulations to arrive at their final results. In addition, these computations are often interdependent. That is, the later calculations are dependent on the results of earlier ones. Consequently, even though an individual round-off error could be small, the cumulative effect over the course of a large computation can be significant. EXAMPLE 3.7

Large Numbers of Interdependent Computations Problem Statement. Investigate the effect of round-off error on large numbers of interdependent computations. Develop a program to sum a number 100,000 times. Sum the number 1 in single precision, and 0.00001 in single and double precision. Solution. Figure 3.12 shows a Fortran 90 program that performs the summation. Whereas the single-precision summation of 1 yields the expected result, the single-precision

3.4 FIGURE 3.12 Fortran 90 program to sum a number 105 times. The case sums the number 1 in single precision and the number 1025 in single and double precision.

ROUND-OFF ERRORS

75

PROGRAM fig0312 IMPLICIT none INTEGER::i REAL::sum1, sum2, x1, x2 DOUBLE PRECISION::sum3, x3 sum1=0. sum2=0. sum3=0. x1=1. x2=1.e−5 x3=1.d−5 DO i=1,100000 sum1=sum1+x1 sum2=sum2+x2 sum3=sum3+x3 END DO PRINT *, sum1 PRINT *, sum2 PRINT *, sum3 END output: 100000.000000 1.000990 9.999999999980838E-001

summation of 0.00001 yields a large discrepancy. This error is reduced significantly when 0.00001 is summed in double precision. Quantizing errors are the source of the discrepancies. Because the integer 1 can be represented exactly within the computer, it can be summed exactly. In contrast, 0.00001 cannot be represented exactly and is quantized by a value that is slightly different from its true value. Whereas this very slight discrepancy would be negligible for a small computation, it accumulates after repeated summations. The problem still occurs in double precision but is greatly mitigated because the quantizing error is much smaller.

Note that the type of error illustrated by the previous example is somewhat atypical in that all the errors in the repeated operation are of the same sign. In most cases the errors of a long computation alternate sign in a random fashion and, thus, often cancel out. However, there are also instances where such errors do not cancel but, in fact, lead to a spurious final result. The following sections are intended to provide insight into ways in which this may occur. Adding a Large and a Small Number. Suppose we add a small number, 0.0010, to a large number, 4000, using a hypothetical computer with the 4-digit mantissa and the 1-digit exponent. We modify the smaller number so that its exponent matches the larger, # 104 0.4000 0.0000001 # 104 0.4000001 # 104

76

APPROXIMATIONS AND ROUND-OFF ERRORS

which is chopped to 0.4000 ? 104. Thus, we might as well have not performed the addition! This type of error can occur in the computation of an infinite series. The initial terms in such series are often relatively large in comparison with the later terms. Thus, after a few terms have been added, we are in the situation of adding a small quantity to a large quantity. One way to mitigate this type of error is to sum the series in reverse order—that is, in ascending rather than descending order. In this way, each new term will be of comparable magnitude to the accumulated sum (see Prob. 3.5). Subtractive Cancellation. This term refers to the round-off induced when subtracting two nearly equal floating-point numbers. One common instance where this can occur involves finding the roots of a quadratic equation or parabola with the quadratic formula, x1 2b6 2b2 24ac 5 x2 2a

(3.12)

For cases where b2 W 4ac, the difference in the numerator can be very small. In such cases, double precision can mitigate the problem. In addition, an alternative formulation can be used to minimize subtractive cancellation, x1 22c 5 x2 b 6 2b2 2 4ac

(3.13)

An illustration of the problem and the use of this alternative formula are provided in the following example. EXAMPLE 3.8

Subtractive Cancellation Problem Statement. Compute the values of the roots of a quadratic equation with a 5 1, b 5 3000.001, and c 5 3. Check the computed values versus the true roots of x1 5 20.001 and x2 5 23000. Solution. Figure 3.13 shows an Excel/VBA program that computes the roots x1 and x2 on the basis of the quadratic formula [(Eq. (3.12)]. Note that both single- and double-precision versions are given. Whereas the results for x2 are adequate, the percent relative errors for x1 are poor for the single-precision version, et 5 2.4%. This level could be inadequate for many applied engineering problems. This result is particularly surprising because we are employing an analytical formula to obtain our solution! The loss of significance occurs in the line of both programs where two relatively large numbers are subtracted. Similar problems do not occur when the same numbers are added. On the basis of the above, we can draw the general conclusion that the quadratic formula will be susceptible to subtractive cancellation whenever b2 W 4ac. One way to circumvent this problem is to use double precision. Another is to recast the quadratic formula in the format of Eq. (3.13). As in the program output, both options give a much smaller error because the subtractive cancellation is minimized or avoided.

3.4

ROUND-OFF ERRORS

Option Explicit Sub Dim Dim Dim Dim Dim Dim Dim

77

'Display results Sheets("sheet1").Select Range("b2").Select ActiveCell.Value = x1 ActiveCell.Offset(1, 0).Select ActiveCell.Value = x2 ActiveCell.Offset(2, 0).Select ActiveCell.Value = x11 ActiveCell.Offset(1, 0).Select ActiveCell.Value = x22 ActiveCell.Offset(2, 0).Select ActiveCell.Value = x1r End Sub

fig0313() a As Single, b As Single c As Single, d As Single x1 As Single, x2 As Single x1r As Single aa As Double, bb As Double cc As Double, dd As Double x11 As Double, x22 As Double

'Single precision: a = 1: b = 3000.001: c = 3 d = Sqr(b * b − 4 * a * c) x1 = (−b + d) / (2 * a) x2 = (−b − d) / (2 * a)

OUTPUT:

'Double precision: aa = 1: bb = 3000.001: cc = 3 dd = Sqr(bb * bb − 4 * aa * cc) x11 = (−bb + dd) / (2 * aa) x22 = (−bb − dd) / (2 * aa) 'Modified formula for first root 'single precision: x1r = −2 * c / (b + d) FIGURE 3.13 Excel/VBA program to determine the roots of a quadratic.

Note that, as in the foregoing example, there are times when subtractive cancellation can be circumvented by using a transformation. However, the only general remedy is to employ extended precision. Smearing. Smearing occurs whenever the individual terms in a summation are larger than the summation itself. As in the following example, one case where this occurs is in series of mixed signs. EXAMPLE 3.9

Evaluation of ex using Inﬁnite Series Problem Statement. The exponential function y 5 ex is given by the infinite series y511x1

x3 x2 1 1p 2 3!

Evaluate this function for x 5 10 and x 5 210, and be attentive to the problems of round-off error. Solution. Figure 3.14a gives an Excel/VBA program that uses the infinite series to evaluate ex. The variable i is the number of terms in the series, term is the value of the

78

APPROXIMATIONS AND ROUND-OFF ERRORS

(b) Evaluation of e10

(a) Program Option Explicit Sub fig0314() Dim term As Single, test As Single Dim sum As Single, x As Single Dim i As Integer i = 0: term = 1#: sum = 1#: test = 0# Sheets("sheet1").Select Range("b1").Select x = ActiveCell.Value Range("a3:c1003").ClearContents Range("a3").Select Do If sum = test Then Exit Do ActiveCell.Value = i ActiveCell.Offset(0, 1).Select ActiveCell.Value = term ActiveCell.Offset(0, 1).Select ActiveCell.Value = sum ActiveCell.Offset(1, -2).Select i = i + 1 test = sum term = x ^ i / _ Application.WorksheetFunction.Fact(i) sum = sum + term Loop ActiveCell.Offset(0, 1).Select ActiveCell.Value = "Exact value = " ActiveCell.Offset(0, 1).Select ActiveCell.Value = Exp(x) End Sub

(c) Evaluation of e10

FIGURE 3.14 (a) An Excel/VBA program to evaluate ex using an inﬁnite series. (b) Evaluation of ex. (c) Evaluation of e2x.

current term added to the series, and sum is the accumulative value of the series. The variable test is the preceding accumulative value of the series prior to adding term. The series is terminated when the computer cannot detect the difference between test and sum. Figure 3.14b shows the results of running the program for x 5 10. Note that this case is completely satisfactory. The final result is achieved in 31 terms with the series identical to the library function value within seven significant figures. Figure 3.14c shows similar results for x 5 210. However, for this case, the results of the series calculation are not even the same sign as the true result. As a matter of fact, the negative results are open to serious question because ex can never be less than zero. The problem here is caused by round-off error. Note that many of the terms that make up the

PROBLEMS

79

sum are much larger than the final result of the sum. Furthermore, unlike the previous case, the individual terms vary in sign. Thus, in effect we are adding and subtracting large numbers (each with some small error) and placing great significance on the differences—that is, subtractive cancellation. Thus, we can see that the culprit behind this example of smearing is, in fact, subtractive cancellation. For such cases it is appropriate to seek some other computational strategy. For example, one might try to compute y 5 e10 as y 5 (e21)10. Other than such a reformulation, the only general recourse is extended precision.

Inner Products. As should be clear from the last sections, some infinite series are particularly prone to round-off error. Fortunately, the calculation of series is not one of the more common operations in numerical methods. A far more ubiquitous manipulation is the calculation of inner products, as in n

p 1 xn yn a xi yi 5 x1 y1 1 x2 y2 1

i51

This operation is very common, particularly in the solution of simultaneous linear algebraic equations. Such summations are prone to round-off error. Consequently, it is often desirable to compute such summations in extended precision. Although the foregoing sections should provide rules of thumb to mitigate round-off error, they do not provide a direct means beyond trial and error to actually determine the effect of such errors on a computation. In Chap. 4, we will introduce the Taylor series, which will provide a mathematical approach for estimating these effects.

PROBLEMS 3.1 Convert the following base-2 numbers to base-10: (a) 101101, (b) 101.011, and (c) 0.01101. 3.2 Convert the following base-8 numbers to base-10: 71,263 and 3.147. 3.3 Compose your own program based on Fig. 3.11 and use it to determine your computer’s machine epsilon. 3.4 In a fashion similar to that in Fig. 3.11, write a short program to determine the smallest number, xmin, used on the computer you will be employing along with this book. Note that your computer will be unable to reliably distinguish between zero and a quantity that is smaller than this number. 3.5 The infinite series n 1 f (n) 5 a 4 i51 i 4

converges on a value of f (n) 5 p y90 as n approaches infinity. Write a program in single precision to calculate f (n) for n 5 10,000 by computing the sum from i 5 1 to 10,000. Then repeat the calculation but in reverse order—that is, from i 5 10,000 to 1 using increments of 21. In each case, compute the true percent relative error. Explain the results.

3.6 Evaluate e25 using two approaches e2x 5 1 2 x 1

x2 x3 2 1p 2 3!

and e2x 5

1 5 ex

1 x2 x3 11x1 1 1p 2 3!

and compare with the true value of 6.737947 3 1023. Use 20 terms to evaluate each series and compute true and approximate relative errors as terms are added. 3.7 The derivative of f (x) 5 1y(1 2 3x2) is given by 6x ( 1 2 3x2 ) 2 Do you expect to have difficulties evaluating this function at x 5 0.577? Try it using 3- and 4-digit arithmetic with chopping. 3.8 (a) Evaluate the polynomial y 5 x3 2 5x2 1 6x 1 0.55

80

APPROXIMATIONS AND ROUND-OFF ERRORS

at x 5 1.37. Use 3-digit arithmetic with chopping. Evaluate the percent relative error. (b) Repeat (a) but express y as y 5 ( (x 2 5)x 1 6)x 1 0.55 Evaluate the error and compare with part (a). 3.9 Calculate the random access memory (RAM) in megabytes necessary to store a multidimensional array that is 20 3 40 3 120. This array is double precision, and each value requires a 64-bit word. Recall that a 64-bit word 5 8 bytes and 1 kilobyte 5 210 bytes. Assume that the index starts at 1. 3.10 Determine the number of terms necessary to approximate cos x to 8 significant figures using the Maclaurin series approximation cos x 5 1 2

x2 x4 x6 x8 1 2 1 2p 2 4! 6! 8!

Calculate the approximation using a value of x 5 0.3p. Write a program to determine your result.

3.11 Use 5-digit arithmetic with chopping to determine the roots of the following equation with Eqs. (3.12) and (3.13) x2 2 5000.002x 1 10 Compute percent relative errors for your results. 3.12 How can the machine epsilon be employed to formulate a stopping criterion es for your programs? Provide an example. 3.13 The “divide and average” method, an old-time method for approximating the square root of any positive number a, can be formulated as x5

x 1 ayx 2

Write a well-structured function to implement this algorithm based on the algorithm outlined in Fig. 3.3.

4

CHAPTER

Truncation Errors and the Taylor Series Truncation errors are those that result from using an approximation in place of an exact mathematical procedure. For example, in Chap. 1 we approximated the derivative of velocity of a falling parachutist by a finite-divided-difference equation of the form [Eq. (1.11)] y(ti11 ) 2 y(ti ) dy ¢y > 5 dt ¢t ti11 2 ti

(4.1)

A truncation error was introduced into the numerical solution because the difference equation only approximates the true value of the derivative (recall Fig. 1.4). In order to gain insight into the properties of such errors, we now turn to a mathematical formulation that is used widely in numerical methods to express functions in an approximate fashion— the Taylor series.

4.1

THE TAYLOR SERIES Taylor’s theorem (Box 4.1) and its associated formula, the Taylor series, is of great value in the study of numerical methods. In essence, the Taylor series provides a means to predict a function value at one point in terms of the function value and its derivatives at another point. In particular, the theorem states that any smooth function can be approximated as a polynomial. A useful way to gain insight into the Taylor series is to build it term by term. For example, the first term in the series is f(xi11 ) > f(xi )

(4.2)

This relationship, called the zero-order approximation, indicates that the value of f at the new point is the same as its value at the old point. This result makes intuitive sense because if xi and xi+1 are close to each other, it is likely that the new value is probably similar to the old value. Equation (4.2) provides a perfect estimate if the function being approximated is, in fact, a constant. However, if the function changes at all over the interval, additional terms 81

82

TRUNCATION ERRORS AND THE TAYLOR SERIES

Box 4.1

Taylor’s Theorem

Taylor’s Theorem If the function f and its first n 1 1 derivatives are continuous on an interval containing a and x, then the value of the function at x is given by f(x) 5 f(a) 1 f ¿(a) (x 2 a) 1 1

f (3) (a) (x 2 a) 3 1 p 3!

1

f (n) (a) (x 2 a) n 1 Rn n!

f –(a) (x 2 a) 2 2!

Second Theorem of Mean for Integrals (B4.1.1)

where the remainder Rn is defined as Rn 5

#

a

x

If the functions g and h are continuous and integrable on an interval containing a and x, and h does not change sign in the interval, then there exists a point j between a and x such that x

(x 2 t) n (n11) f (t)dt n!

(B4.1.2)

where t 5 a dummy variable. Equation (B4.1.1) is called the Taylor series or Taylor’s formula. If the remainder is omitted, the right side of Eq. (B4.1.1) is the Taylor polynomial approximation to f(x). In essence, the theorem states that any smooth function can be approximated as a polynomial. Equation (B4.1.2) is but one way, called the integral form, by which the remainder can be expressed. An alternative formulation can be derived on the basis of the integral mean-value theorem.

First Theorem of Mean for Integrals

x

x

# g(t)h(t)dt 5 g(j) # h(t) dt a

(B4.1.4)

a

Thus, Eq. (B4.1.3) is equivalent to Eq. (B4.1.4) with h(t) 5 1. The second theorem can be applied to Eq. (B4.1.2) with g(t) 5 f (n11) (t) h(t) 5

(x 2 t) n n!

As t varies from a to x, h(t) is continuous and does not change sign. Therefore, if f (n11)(t) is continuous, then the integral mean-value theorem holds and Rn 5

If the function g is continuous and integrable on an interval containing a and x, then there exists a point j between a and x such that

# g(t) dt 5 g(j) (x 2 a)

In other words, this theorem states that the integral can be represented by an average value for the function g(j) times the interval length x 2 a. Because the average must occur between the minimum and maximum values for the interval, there is a point x 5 j at which the function takes on the average value. The first theorem is in fact a special case of a second meanvalue theorem for integrals.

f (n11) (j) (x 2 a) n11 (n 1 1)!

This equation is referred to as the derivative or Lagrange form of the remainder.

(B4.1.3)

a

of the Taylor series are required to provide a better estimate. For example, the first-order approximation is developed by adding another term to yield f(xi11 ) > f(xi ) 1 f ¿(xi )(xi11 2 xi )

(4.3)

The additional first-order term consists of a slope f 9(xi) multiplied by the distance between xi and xi+1. Thus, the expression is now in the form of a straight line and is capable of predicting an increase or decrease of the function between xi and xi+1. Although Eq. (4.3) can predict a change, it is exact only for a straight-line, or linear, trend. Therefore, a second-order term is added to the series to capture some of the curvature that the function might exhibit: f(xi11 ) > f(xi ) 1 f ¿(xi )(xi11 2 xi ) 1

f –(xi ) (xi11 2 xi ) 2 2!

(4.4)

4.1

THE TAYLOR SERIES

83

In a similar manner, additional terms can be included to develop the complete Taylor series expansion: f –(xi ) (xi11 2 xi ) 2 2! f (n) (xi ) 2 xi ) 3 1 p 1 (xi11 2 xi ) n 1 Rn n!

f(xi11 ) 5 f(xi ) 1 f ¿(xi )(xi11 2 xi ) 1 1

f (3) (xi ) (xi11 3!

(4.5)

Note that because Eq. (4.5) is an infinite series, an equal sign replaces the approximate sign that was used in Eqs. (4.2) through (4.4). A remainder term is included to account for all terms from n 1 1 to infinity: Rn 5

f (n11) (j) (xi11 2 xi ) n11 (n 1 1)!

(4.6)

where the subscript n connotes that this is the remainder for the nth-order approximation and j is a value of x that lies somewhere between xi and xi+1. The introduction of the j is so important that we will devote an entire section (Sec. 4.1.1) to its derivation. For the time being, it is sufficient to recognize that there is such a value that provides an exact determination of the error. It is often convenient to simplify the Taylor series by defining a step size h 5 xi+1 2 xi and expressing Eq. (4.5) as f(xi11 ) 5 f(xi ) 1 f ¿(xi )h 1

f –(xi ) 2 f (3) (xi ) 3 p f n (xi ) n h 1 h 1 h 1 Rn 1 2! 3! n!

(4.7)

where the remainder term is now Rn 5

EXAMPLE 4.1

f (n11) (j) n11 h (n 1 1)!

(4.8)

Taylor Series Approximation of a Polynomial Problem Statement. Use zero- through fourth-order Taylor series expansions to approximate the function f (x) 5 20.1x4 2 0.15x3 2 0.5x2 2 0.25x 1 1.2 from xi 5 0 with h 5 1. That is, predict the function’s value at xi+1 5 1. Solution. Because we are dealing with a known function, we can compute values for f(x) between 0 and 1. The results (Fig. 4.1) indicate that the function starts at f (0) 5 1.2 and then curves downward to f (1) 5 0.2. Thus, the true value that we are trying to predict is 0.2. The Taylor series approximation with n 5 0 is [Eq. (4.2)] f(xi11 ) . 1.2

84

TRUNCATION ERRORS AND THE TAYLOR SERIES

f (x)

f (xi)

Zero order First o rder

1.0

f (xi + 1) ⯝ f (xi)

Se co nd

or de r

f (xi + 1) ⯝ f (xi) + f ⬘(xi)h +

ue Tr

0.5

f (xi + 1) ⯝ f (xi) + f ⬘(xi)h

f ⬙(xi) 2 h 2!

f (xi + 1) 0

xi = 0

xi + 1 = 1

x

h

FIGURE 4.1 The approximation of f (x) 5 20.1x4 2 0.15x3 2 0.5x2 2 0.25x 1 1.2 at x 5 1 by zero-order, ﬁrst-order, and second-order Taylor series expansions.

Thus, as in Fig. 4.1, the zero-order approximation is a constant. Using this formulation results in a truncation error [recall Eq. (3.2)] of Et 5 0.2 2 1.2 5 21.0 at x 5 1. For n 5 1, the first derivative must be determined and evaluated at x 5 0: f ¿(0) 5 20.4(0.0) 3 2 0.45(0.0) 2 2 1.0(0.0) 2 0.25 5 20.25 Therefore, the first-order approximation is [Eq. (4.3)] f(xi11 ) . 1.2 2 0.25h which can be used to compute f(1) 5 0.95. Consequently, the approximation begins to capture the downward trajectory of the function in the form of a sloping straight line (Fig. 4.1). This results in a reduction of the truncation error to Et 5 0.2 2 0.95 5 20.75 For n 5 2, the second derivative is evaluated at x 5 0: f –(0) 5 21.2(0.0) 2 2 0.9(0.0) 2 1.0 5 21.0 Therefore, according to Eq. (4.4), f(xi11 ) . 1.2 2 0.25h 2 0.5h2 and substituting h 5 1, f (1) 5 0.45. The inclusion of the second derivative now adds some downward curvature resulting in an improved estimate, as seen in Fig. 4.1. The truncation error is reduced further to 0.2 2 0.45 5 20.25.

4.1

THE TAYLOR SERIES

85

Additional terms would improve the approximation even more. In fact, the inclusion of the third and the fourth derivatives results in exactly the same equation we started with: f(x) 5 1.2 2 0.25h 2 0.5h2 2 0.15h3 2 0.1h4 where the remainder term is R4 5

f (5) (j) 5 h 50 5!

because the fifth derivative of a fourth-order polynomial is zero. Consequently, the Taylor series expansion to the fourth derivative yields an exact estimate at xi+1 5 1: f(1) 5 1.2 2 0.25(1) 2 0.5(1) 2 2 0.15(1) 3 2 0.1(1) 4 5 0.2

In general, the nth-order Taylor series expansion will be exact for an nth-order polynomial. For other differentiable and continuous functions, such as exponentials and sinusoids, a finite number of terms will not yield an exact estimate. Each additional term will contribute some improvement, however slight, to the approximation. This behavior will be demonstrated in Example 4.2. Only if an infinite number of terms are added will the series yield an exact result. Although the above is true, the practical value of Taylor series expansions is that, in most cases, the inclusion of only a few terms will result in an approximation that is close enough to the true value for practical purposes. The assessment of how many terms are required to get “close enough” is based on the remainder term of the expansion. Recall that the remainder term is of the general form of Eq. (4.8). This relationship has two major drawbacks. First, j is not known exactly but merely lies somewhere between xi and xi+1. Second, to evaluate Eq. (4.8), we need to determine the (n 1 1)th derivative of f(x). To do this, we need to know f(x). However, if we knew f(x), there would be no need to perform the Taylor series expansion in the present context! Despite this dilemma, Eq. (4.8) is still useful for gaining insight into truncation errors. This is because we do have control over the term h in the equation. In other words, we can choose how far away from x we want to evaluate f(x), and we can control the number of terms we include in the expansion. Consequently, Eq. (4.8) is usually expressed as Rn 5 O ( hn11 ) where the nomenclature O(hn11) means that the truncation error is of the order of hn11. That is, the error is proportional to the step size h raised to the (n 1 l)th power. Although this approximation implies nothing regarding the magnitude of the derivatives that multiply hn11, it is extremely useful in judging the comparative error of numerical methods based on Taylor series expansions. For example, if the error is O(h), halving the step size will halve the error. On the other hand, if the error is O(h2), halving the step size will quarter the error. In general, we can usually assume that the truncation error is decreased by the addition of terms to the Taylor series. In many cases, if h is sufficiently small, the first- and other lower-order terms usually account for a disproportionately high percent of the error. Thus, only a few terms are required to obtain an adequate estimate. This property is illustrated by the following example.

86

TRUNCATION ERRORS AND THE TAYLOR SERIES

EXAMPLE 4.2

Use of Taylor Series Expansion to Approximate a Function with an Inﬁnite Number of Derivatives Problem Statement. Use Taylor series expansions with n 5 0 to 6 to approximate f(x) 5 cos x at xi+1 5 py3 on the basis of the value of f(x) and its derivatives at xi 5 py4. Note that this means that h 5 py3 2 py4 5 py12. Solution. As with Example 4.1, our knowledge of the true function means that we can determine the correct value f(py3) 5 0.5. The zero-order approximation is [Eq. (4.3)] p p f a b > cos a b 5 0.707106781 3 4 which represents a percent relative error of et 5

0.5 2 0.707106781 100% 5 241.4% 0.5

For the first-order approximation, we add the first derivative term where f 9(x) 5 2sin x: p p p p f a b > cos a b 2 sin a b a b 5 0.521986659 3 4 4 12 which has et 5 24.40 percent. For the second-order approximation, we add the second derivative term where f 0(x) 5 2cos x: cos(py4) p 2 p p p p a b 5 0.497754491 f a b > cos a b 2 sin a b a b 2 3 4 4 12 2 12 with et 5 0.449 percent. Thus, the inclusion of additional terms results in an improved estimate. The process can be continued and the results listed, as in Table 4.1. Notice that the derivatives never go to zero, as was the case with the polynomial in Example 4.1. Therefore, each additional term results in some improvement in the estimate. However, also notice how most of the improvement comes with the initial terms. For this case, by the time we have added the third-order term, the error is reduced to 2.62 3 1022 percent, TABLE 4.1 Taylor series approximation of f(x) 5 cos x at xi11 5 p/3 using a base point of p/4. Values are shown for various orders (n) of approximation. Order n 0 1 2 3 4 5 6

f (n)(x) cos 2sin 2cos sin cos 2sin 2cos

x x x x x x x

f (P/3) 0.707106781 0.521986659 0.497754491 0.499869147 0.500007551 0.500000304 0.499999988

Et 241.4 24.4 0.449 2.62 3 21.51 3 26.08 3 2.44 3

1022 1023 1025 1026

4.1

THE TAYLOR SERIES

87

which means that we have attained 99.9738 percent of the true value. Consequently, although the addition of more terms will reduce the error further, the improvement becomes negligible.

4.1.1 The Remainder for the Taylor Series Expansion Before demonstrating how the Taylor series is actually used to estimate numerical errors, we must explain why we included the argument j in Eq. (4.8). A mathematical derivation is presented in Box 4.1. We will now develop an alternative exposition based on a somewhat more visual interpretation. Then we can extend this specific case to the more general formulation. Suppose that we truncated the Taylor series expansion [Eq. (4.7)] after the zeroorder term to yield f(xi11 ) > f(xi ) A visual depiction of this zero-order prediction is shown in Fig. 4.2. The remainder, or error, of this prediction, which is also shown in the illustration, consists of the infinite series of terms that were truncated: R0 5 f ¿(xi )h 1

f –(xi ) 2 f (3) (xi ) 3 p h 1 h 1 2! 3!

It is obviously inconvenient to deal with the remainder in this infinite series format. One simplification might be to truncate the remainder itself, as in R0 > f ¿(xi )h

(4.9)

FIGURE 4.2 Graphical depiction of a zero-order Taylor series prediction and remainder.

f (x)

ic red tp c a Ex

n tio R0

Zero-order prediction f (xi) xi

xi + 1 h

x

88

TRUNCATION ERRORS AND THE TAYLOR SERIES

Although, as stated in the previous section, lower-order derivatives usually account for a greater share of the remainder than the higher-order terms, this result is still inexact because of the neglected second- and higher-order terms. This “inexactness” is implied by the approximate equality symbol (>) employed in Eq. (4.9). An alternative simplification that transforms the approximation into an equivalence is based on a graphical insight. As in Fig. 4.3, the derivative mean-value theorem states that if a function f(x) and its first derivative are continuous over an interval from xi to xi+1, then there exists at least one point on the function that has a slope, designated by f9(j), that is parallel to the line joining f(xi) and f(xi+1). The parameter j marks the x value where this slope occurs (Fig. 4.3). A physical illustration of this theorem is that, if you travel between two points with an average velocity, there will be at least one moment during the course of the trip when you will be moving at that average velocity. By invoking this theorem it is simple to realize that, as illustrated in Fig. 4.3, the slope f9(j) is equal to the rise R0 divided by the run h, or f ¿(j) 5

R0 h

which can be rearranged to give R0 5 f ¿(j)h

(4.10)

Thus, we have derived the zero-order version of Eq. (4.8). The higher-order versions are merely a logical extension of the reasoning used to derive Eq. (4.10). The first-order version is R1 5

f –(j) 2 h 2!

(4.11)

FIGURE 4.3 Graphical depiction of the derivative mean-value theorem.

f (x) Slope = f ⬘( )

Slope =

xi

R0

R0 h

xi + 1 h

x

4.1

THE TAYLOR SERIES

89

For this case, the value of j conforms to the x value corresponding to the second derivative that makes Eq. (4.11) exact. Similar higher-order versions can be developed from Eq. (4.8). 4.1.2 Using the Taylor Series to Estimate Truncation Errors Although the Taylor series will be extremely useful in estimating truncation errors throughout this book, it may not be clear to you how the expansion can actually be applied to numerical methods. In fact, we have already done so in our example of the falling parachutist. Recall that the objective of both Examples 1.1 and 1.2 was to predict velocity as a function of time. That is, we were interested in determining y(t). As specified by Eq. (4.5), y(t) can be expanded in a Taylor series: y(ti11 ) 5 y(ti ) 1 y¿(ti )(ti11 2 ti ) 1

y–(ti ) (ti11 2 ti ) 2 1 p 1 Rn 2!

(4.12)

Now let us truncate the series after the first derivative term: y(ti11 ) 5 y(ti ) 1 y¿(ti )(ti11 2 ti ) 1 R1

(4.13)

Equation (4.13) can be solved for y¿(ti ) 5

y(ti11 ) 2 y(ti ) R1 2 ti11 2 ti ti11 2 ti First-order approximation

(4.14)

Truncation error

The first part of Eq. (4.14) is exactly the same relationship that was used to approximate the derivative in Example 1.2 [Eq. (1.11)]. However, because of the Taylor series approach, we have now obtained an estimate of the truncation error associated with this approximation of the derivative. Using Eqs. (4.6) and (4.14) yields y–(j) R1 5 (ti11 2 ti ) ti11 2 ti 2!

(4.15)

R1 5 O(ti11 2 ti ) ti11 2 ti

(4.16)

or

Thus, the estimate of the derivative [Eq. (1.11) or the first part of Eq. (4.14)] has a truncation error of order ti11 2 ti. In other words, the error of our derivative approximation should be proportional to the step size. Consequently, if we halve the step size, we would expect to halve the error of the derivative. EXAMPLE 4.3

The Effect of Nonlinearity and Step Size on the Taylor Series Approximation Problem Statement. Figure 4.4 is a plot of the function f(x) 5 xm

(E4.3.1)

for m 5 1, 2, 3, and 4 over the range from x 5 1 to 2. Notice that for m 5 1 the function is linear, and as m increases, more curvature or nonlinearity is introduced into the function.

TRUNCATION ERRORS AND THE TAYLOR SERIES

f (x)

15

3

m=

4

10

=

5

m

90

m=

2

m=1

0

2

1

x

FIGURE 4.4 Plot of the function f (x) 5 xm for m 5 1, 2, 3, and 4. Notice that the function becomes more nonlinear as m increases.

Employ the first-order Taylor series to approximate this function for various values of the exponent m and the step size h. Solution. Equation (E4.3.1) can be approximated by a first-order Taylor series expansion, as in f(xi11 ) 5 f(xi ) 1 mxm21 h i

(E4.3.2)

which has a remainder R1 5

f –(xi ) 2 f (3) (xi ) 3 f (4) (xi ) 4 p h 1 h 1 h 1 2! 3! 4!

First, we can examine how the approximation performs as m increases—that is, as the function becomes more nonlinear. For m 5 1, the actual value of the function at x 5 2 is 2.

4.1

THE TAYLOR SERIES

91

The Taylor series yields f(2) 5 1 1 1(1) 5 2 and R1 5 0 The remainder is zero because the second and higher derivatives of a linear function are zero. Thus, as expected, the first-order Taylor series expansion is perfect when the underlying function is linear. For m 5 2, the actual value is f (2) 5 22 5 4. The first-order Taylor series approximation is f(2) 5 1 1 2(1) 5 3 and R1 5 22 (1) 2 1 0 1 0 1 p 5 1 Thus, because the function is a parabola, the straight-line approximation results in a discrepancy. Note that the remainder is determined exactly. For m 5 3, the actual value is f(2) 5 23 5 8. The Taylor series approximation is f(2) 5 1 1 3(1) 2 (1) 5 4 and R1 5 62 (1) 2 1 66 (1) 3 1 0 1 0 1 p 5 4 Again, there is a discrepancy that can be determined exactly from the Taylor series. For m 5 4, the actual value is f(2) 5 24 5 16. The Taylor series approximation is f(2) 5 1 1 4(1) 3 (1) 5 5 and 4 p 5 11 R1 5 122 (1) 2 1 246 (1) 3 1 24 24 (1) 1 0 1 0 1

On the basis of these four cases, we observe that R1 increases as the function becomes more nonlinear. Furthermore, R1 accounts exactly for the discrepancy. This is because Eq. (E4.3.1) is a simple monomial with a finite number of derivatives. This permits a complete determination of the Taylor series remainder. Next, we will examine Eq. (E4.3.2) for the case m 5 4 and observe how R1 changes as the step size h is varied. For m 5 4, Eq. (E4.3.2) is f(x 1 h) 5 f(x) 1 4x3i h

If x 5 1, f(1) 5 1 and this equation can be expressed as f(1 1 h) 5 1 1 4h with a remainder of R1 5 6h2 1 4h3 1 h4

92

TRUNCATION ERRORS AND THE TAYLOR SERIES

R1 10

1

0.1 兩Slope兩 = 2

0.01

0.001

1

0.1

0.01 h

FIGURE 4.5 Log-log plot of the remainder R1 of the ﬁrst-order Taylor series approximation of the function f (x) 5 x4 versus step size h. A line with a slope of 2 is also shown to indicate that as h decreases, the error becomes proportional to h2. TABLE 4.2 Comparison of the exact value of the function f(x) 5 x4 with the ﬁrst-order Taylor series approximation. Both the function and the approximation are evaluated at x 1 h, where x 5 1. h

True

First-Order Approximation

R1

1 0.5 0.25 0.125 0.0625 0.03125 0.015625

16 5.0625 2.441406 1.601807 1.274429 1.130982 1.063980

5 3 2 1.5 1.25 1.125 1.0625

11 2.0625 0.441406 0.101807 0.024429 0.005982 0.001480

This leads to the conclusion that the discrepancy will decrease as h is reduced. Also, at sufficiently small values of h, the error should become proportional to h2. That is, as h is halved, the error will be quartered. This behavior is confirmed by Table 4.2 and Fig. 4.5. Thus, we conclude that the error of the first-order Taylor series approximation decreases as m approaches 1 and as h decreases. Intuitively, this means that the Taylor

4.1

THE TAYLOR SERIES

93

series becomes more accurate when the function we are approximating becomes more like a straight line over the interval of interest. This can be accomplished either by reducing the size of the interval or by “straightening” the function by reducing m. Obviously, the latter option is usually not available in the real world because the functions we analyze are typically dictated by the physical problem context. Consequently, we do not have control of their lack of linearity, and our only recourse is reducing the step size or including additional terms in the Taylor series expansion.

4.1.3 Numerical Differentiation Equation (4.14) is given a formal label in numerical methods—it is called a finite divided difference. It can be represented generally as f ¿(xi ) 5

f(xi11 ) 2 f(xi ) 1 O(xi11 2 xi ) xi11 2 xi

(4.17)

f ¿(xi ) 5

¢fi 1 O(h) h

(4.18)

or

where D fi is referred to as the first forward difference and h is called the step size, that is, the length of the interval over which the approximation is made. It is termed a “forward” difference because it utilizes data at i and i 1 1 to estimate the derivative (Fig. 4.6a). The entire term D fyh is referred to as a first finite divided difference. This forward divided difference is but one of many that can be developed from the Taylor series to approximate derivatives numerically. For example, backward and centered difference approximations of the first derivative can be developed in a fashion similar to the derivation of Eq. (4.14). The former utilizes values at xi21 and xi (Fig. 4.6b), whereas the latter uses values that are equally spaced around the point at which the derivative is estimated (Fig. 4.6c). More accurate approximations of the first derivative can be developed by including higher-order terms of the Taylor series. Finally, all the above versions can also be developed for second, third, and higher derivatives. The following sections provide brief summaries illustrating how some of these cases are derived. Backward Difference Approximation of the First Derivative. The Taylor series can be expanded backward to calculate a previous value on the basis of a present value, as in f(xi21 ) 5 f(xi ) 2 f ¿(xi )h 1

f –(xi ) 2 p h 2 2!

(4.19)

Truncating this equation after the first derivative and rearranging yields f ¿(xi ) >

f(xi ) 2 f(xi21 ) §fi 5 h h

(4.20)

where the error is O(h), and = fi is referred to as the first backward difference. See Fig. 4.6b for a graphical representation.

TRUNCATION ERRORS AND THE TAYLOR SERIES

f (x) e Tru

d

va eri

e

tiv

xim Appro

ation

h

xi+1

xi

(a) f (x)

d

e tiv

pp r

ox im

at

io n

e Tru

va eri

x

A

94

h

xi

xi –1

x

(b)

f (x)

e Tru

Ap

d

va eri

o pr

tiv

xim

e

ati

on

2h

xi –1

(c)

xi+1

x

FIGURE 4.6 Graphical depiction of (a) forward, (b) backward, and (c) centered ﬁnite-divided-difference approximations of the ﬁrst derivative.

4.1

THE TAYLOR SERIES

95

Centered Difference Approximation of the First Derivative. A third way to approximate the first derivative is to subtract Eq. (4.19) from the forward Taylor series expansion: f (xi11 ) 5 f (xi ) 1 f ¿(xi )h 1

f –(xi ) 2 p h 1 2!

(4.21)

to yield f (xi11 ) 5 f (xi21 ) 1 2f ¿(xi )h 1

2f (3) (xi ) 3 p h 1 3!

which can be solved for f ¿(xi ) 5

f (3) (xi ) 2 p f (xi11 ) 2 f (xi21 ) 2 h 2 2h 6

f ¿(xi ) 5

f (xi11 ) 2 f (xi21 ) 2 O(h2 ) 2h

or

(4.22)

Equation (4.22) is a centered difference representation of the first derivative. Notice that the truncation error is of the order of h2 in contrast to the forward and backward approximations that were of the order of h. Consequently, the Taylor series analysis yields the practical information that the centered difference is a more accurate representation of the derivative (Fig. 4.6c). For example, if we halve the step size using a forward or backward difference, we would approximately halve the truncation error, whereas for the central difference, the error would be quartered. EXAMPLE 4.4

Finite-Divided-Difference Approximations of Derivatives Problem Statement. Use forward and backward difference approximations of O(h) and a centered difference approximation of O(h2) to estimate the first derivative of f (x) 5 20.1x4 2 0.15x3 2 0.5x2 2 0.25x 1 1.25

at x 5 0.5 using a step size h 5 0.5. Repeat the computation using h 5 0.25. Note that the derivative can be calculated directly as f ¿(x) 5 20.4x3 2 0.45x2 2 1.0x 2 0.25 and can be used to compute the true value as f9(0.5) 5 20.9125. Solution. For h 5 0.5, the function can be employed to determine xi21 5 0 xi 5 0.5 xi11 5 1.0

f (xi21 ) 5 1.2 f (xi ) 5 0.925 f (xi11 ) 5 0.2

These values can be used to compute the forward divided difference [Eq. (4.17)], f ¿(0.5) >

0.2 2 0.925 5 21.45 Zet Z 5 58.9% 0.5

96

TRUNCATION ERRORS AND THE TAYLOR SERIES

the backward divided difference [Eq. (4.20)], 0.925 2 1.2 5 20.55 Zet Z 5 39.7% 0.5

f ¿(0.5) >

and the centered divided difference [Eq. (4.22)], 0.2 2 1.2 5 21.0 Zet Z 5 9.6% 1.0

f ¿(0.5) > For h 5 0.25,

xi21 5 0.25 xi 5 0.5 xi11 5 0.75

f (xi21 ) 5 1.10351563 f (xi ) 5 0.925 f (xi11 ) 5 0.63632813

which can be used to compute the forward divided difference, 0.63632813 2 0.925 5 21.155 Zet Z 5 26.5% 0.25

f ¿(0.5) >

the backward divided difference, 0.925 2 1.10351563 5 20.714 Zet Z 5 21.7% 0.25

f ¿(0.5) >

and the centered divided difference, 0.63632813 2 1.10351563 5 20.934 Zet Z 5 2.4% 0.5

f ¿(0.5) >

For both step sizes, the centered difference approximation is more accurate than forward or backward differences. Also, as predicted by the Taylor series analysis, halving the step size approximately halves the error of the backward and forward differences and quarters the error of the centered difference.

Finite Difference Approximations of Higher Derivatives. Besides first derivatives, the Taylor series expansion can be used to derive numerical estimates of higher derivatives. To do this, we write a forward Taylor series expansion for f(xi12) in terms of f(xi): f (xi12 ) 5 f (xi ) 1 f ¿(xi )(2h) 1

f –(xi ) (2h) 2 1 p 2!

(4.23)

Equation (4.21) can be multiplied by 2 and subtracted from Eq. (4.23) to give f (xi12 ) 2 2f (xi11 ) 5 2f (xi ) 1 f –(xi )h2 1 p

which can be solved for f –(xi ) 5

f (xi12 ) 2 2f (xi11 ) 1 f (xi )

2

h

1 O(h)

(4.24)

4.2

ERROR PROPAGATION

97

This relationship is called the second forward finite divided difference. Similar manipulations can be employed to derive a backward version f (xi ) 2 2f (xi21 ) 1 f (xi22 )

f –(xi ) 5

1 O(h)

2

h

and a centered version f (xi11 ) 2 2f (xi ) 1 f (xi21 )

f –(xi ) 5

1 O(h2 )

2

h

As was the case with the first-derivative approximations, the centered case is more accurate. Notice also that the centered version can be alternatively expressed as f (xi11 ) 2 f (xi ) f (xi ) 2 f (xi21 ) 2 h h f –(xi ) > h

Thus, just as the second derivative is a derivative of a derivative, the second divided difference approximation is a difference of two first divided differences. We will return to the topic of numerical differentiation in Chap. 23. We have introduced you to the topic at this point because it is a very good example of why the Taylor series is important in numerical methods. In addition, several of the formulas introduced in this section will be employed prior to Chap. 23.

4.2

ERROR PROPAGATION The purpose of this section is to study how errors in numbers can propagate through mathematical functions. For example, if we multiply two numbers that have errors, we would like to estimate the error in the product. 4.2.1 Functions of a Single Variable Suppose that we have a function f(x) that is dependent on a single independent variable x. Assume that x˜ is an approximation of x. We, therefore, would like to assess the effect of the discrepancy between x and x˜ on the value of the function. That is, we would like to estimate ¢ f (x˜ ) 5 Z f (x) 2 f (x˜)Z

The problem with evaluating ¢ f(x˜ ) is that f(x) is unknown because x is unknown. We can overcome this difficulty if x˜ is close to x and f(x˜) is continuous and differentiable. If these conditions hold, a Taylor series can be employed to compute f(x) near f(x˜), as in f(x) 5 f(x˜) 1 f ¿(x˜) (x 2 x˜) 1

f –(x˜ ) (x 2 x˜ ) 2 1 p 2

Dropping the second- and higher-order terms and rearranging yields f (x) 2 f (x˜) > f ¿(x˜) (x 2 x˜)

98

TRUNCATION ERRORS AND THE TAYLOR SERIES

f (x)

True error 兩 f ⬘(x)兩⌬x Estimated error

FIGURE 4.7 Graphical depiction of ﬁrstorder error propagation.

x

x

x

⌬x

or ¢ f (x˜ ) 5 Z f ¿(x˜ ) Z¢x˜

(4.25)

where ¢ f (x˜ ) 5 Z f (x) 2 f (x˜ ) Z represents an estimate of the error of the function and ¢x˜ 5 Zx 2 x˜ Z represents an estimate of the error of x. Equation (4.25) provides the capability to approximate the error in f(x) given the derivative of a function and an estimate of the error in the independent variable. Figure 4.7 is a graphical illustration of the operation.

EXAMPLE 4.5

Error Propagation in a Function of a Single Variable Problem Statement. Given a value of x˜ 5 2.5 with an error of ¢x˜ 5 0.01, estimate the resulting error in the function f(x) 5 x3. Solution. Using Eq. (4.25), ¢ f(x˜) > 3(2.5) 2 (0.01) 5 0.1875 Because f(2.5) 5 15.625, we predict that f (2.5) 5 15.625 6 0.1875

or that the true value lies between 15.4375 and 15.8125. In fact, if x were actually 2.49, the function could be evaluated as 15.4382, and if x were 2.51, it would be 15.8132. For this case, the first-order error analysis provides a fairly close estimate of the true error.

4.2

ERROR PROPAGATION

99

4.2.2 Functions of More than One Variable The foregoing approach can be generalized to functions that are dependent on more than one independent variable. This is accomplished with a multivariable version of the Taylor series. For example, if we have a function of two independent variables u and y, the Taylor series can be written as f(ui11, yi11 ) 5 f(ui, yi ) 1

0f 0f (ui11 2 ui ) 1 (yi11 2 yi ) 0u 0y

02 f 1 02 f c 2 (ui11 2 ui ) 2 1 2 (ui11 2 ui )(yi11 2 yi ) 2! 0u 0u0y 02 f 1 2 (yi11 2 yi ) 2 d 1 p 0y

1

(4.26)

where all partial derivatives are evaluated at the base point i. If all second-order and higher terms are dropped, Eq. (4.26) can be solved for ¢ f (u˜ , y˜ ) 5 `

0f 0f ` ¢u˜ 1 ` ` ¢y˜ 0u 0y

where ¢u˜ and ¢y˜ 5 estimates of the errors in u and y, respectively. For n independent variables x˜ 1, x˜ 2, p , x˜ n having errors ¢x˜ 1, ¢x˜ 2, p , ¢xn the following general relationship holds: ¢f (x˜1, x˜2, p , x˜n ) > `

EXAMPLE 4.6

0f 0f 0f ` ¢x˜ 1 1 ` ` ¢x˜ 2 1 p 1 ` ` ¢x˜ n 0x1 0x2 0xn

(4.27)

Error Propagation in a Multivariable Function Problem Statement. The deflection y of the top of a sailboat mast is y5

FL4 8EI

where F 5 a uniform side loading (N/m), L 5 height (m), E 5 the modulus of elasticity (N/m2), and I 5 the moment of inertia (m4). Estimate the error in y given the following data: ¢F˜ 5 30 N/m F˜ 5 750 N/m L˜ 5 9 m ¢L˜ 5 0.03 m 9 2 ˜ E 5 7.5 3 10 N/m ¢E˜ 5 5 3 107 N/m2 4 I˜ 5 0.0005 m ¢I˜ 5 0.000005 m4 Solution. Employing Eq. (4.27) gives ¢y(F˜, L˜ , E˜, I˜) 5 `

0y 0y 0y 0y ` ¢F˜ 1 ` ` ¢L˜ 1 ` ` ¢E˜ 1 ` ` ¢I˜ 0F 0L 0E 0I

or L˜ 4 F˜ L˜ 3 F˜L˜ 4 ˜ F˜ L˜ 4 ˜ ¢y(F˜ , L˜ , E˜ , I˜ 2 > ˜ ˜ ¢F˜ 1 ˜ ˜ ¢L˜ 1 ¢E 1 ¢I 8E I 2E I 8E˜ 2 I˜ 8E˜ I˜ 2

100

TRUNCATION ERRORS AND THE TAYLOR SERIES

Substituting the appropriate values gives ¢y 5 0.006561 1 0.002187 1 0.001094 1 0.00164 5 0.011482 Therefore, y 5 0.164025 6 0.011482. In other words, y is between 0.152543 and 0.175507 m. The validity of these estimates can be verified by substituting the extreme values for the variables into the equation to generate an exact minimum of ymin 5

720 ( 8.97 ) 4 8 ( 7.55 3 109 ) 0.000505

5 0.152818

and ymax 5

780 ( 9.03 ) 4 8 ( 7.45 3 109 ) 0.000495

5 0.175790

Thus, the first-order estimates are reasonably close to the exact values.

Equation (4.27) can be employed to define error propagation relationships for common mathematical operations. The results are summarized in Table 4.3. We will leave the derivation of these formulas as a homework exercise. 4.2.3 Stability and Condition The condition of a mathematical problem relates to its sensitivity to changes in its input values. We say that a computation is numerically unstable if the uncertainty of the input values is grossly magnified by the numerical method. These ideas can be studied using a first-order Taylor series f (x) 5 f (x˜) 1 f ¿(x˜) (x 2 x˜)

This relationship can be employed to estimate the relative error of f(x) as in f (x) 2 f (x˜) f ¿(x˜ ) (x 2 x˜) > f (x) f (x˜ )

The relative error of x is given by x 2 x˜ x˜ TABLE 4.3 Estimated error bounds associated with common mathematical operations using inexact numbers u˜ and v˜. Operation Addition Subtraction Multiplication Division

Estimated Error ¢(u˜ 1 v˜) ¢(u˜ 2 v˜) ¢(u˜ 3 v˜) u˜ ¢a b v˜

¢u˜ 1 ¢v˜ ¢u˜ 1 ¢v˜ Z u˜ Z¢v˜ 1 Z v˜ Z¢u˜ Z u˜ Z¢v˜ 1 Z v˜Z¢u˜ Zv˜ Z2

4.3

TOTAL NUMERICAL ERROR

101

A condition number can be defined as the ratio of these relative errors Condition number 5

x˜ f ¿(x˜) f (x˜)

(4.28)

The condition number provides a measure of the extent to which an uncertainty in x is magnified by f(x). A value of 1 tells us that the function’s relative error is identical to the relative error in x. A value greater than 1 tells us that the relative error is amplified, whereas a value less than 1 tells us that it is attenuated. Functions with very large values are said to be ill-conditioned. Any combination of factors in Eq. (4.28) that increases the numerical value of the condition number will tend to magnify uncertainties in the computation of f(x). EXAMPLE 4.7

Condition Number Problem Statement. Compute and interpret the condition number for f(x) 5 tan x for x˜ 5

p p 1 0.1 a b 2 2

f(x) 5 tan x for x˜ 5

p p 1 0.01 a b 2 2

Solution. The condition number is computed as Condition number 5

x˜(1ycos2 x) tan x˜

For x˜ 5 py2 1 0.1(py2), Condition number 5

1.7279(40.86) 5 211.2 26.314

Thus, the function is ill-conditioned. For x˜ 5 py2 1 0.01(py2), the situation is even worse: Condition number 5

1.5865(4053) 5 2101 263.66

For this case, the major cause of ill conditioning appears to be the derivative. This makes sense because in the vicinity of py2, the tangent approaches both positive and negative infinity.

4.3

TOTAL NUMERICAL ERROR The total numerical error is the summation of the truncation and round-off errors. In general, the only way to minimize round-off errors is to increase the number of significant figures of the computer. Further, we have noted that round-off error will increase due to subtractive cancellation or due to an increase in the number of computations in an analysis. In contrast, Example 4.4 demonstrated that the truncation error can be reduced by decreasing the step size. Because a decrease in step size can lead to subtractive cancellation or to an increase in computations, the truncation errors are decreased as the round-off

TRUNCATION ERRORS AND THE TAYLOR SERIES

Point of diminishing returns log error

102

Tota l er ror

n Tru

ca

n tio

err

or

Rou ndo

ff e rror

log step size

FIGURE 4.8 A graphical depiction of the trade-off between round-off and truncation error that sometimes comes into play in the course of a numerical method. The point of diminishing returns is shown, where round-off error begins to negate the beneﬁts of step-size reduction.

errors are increased. Therefore, we are faced by the following dilemma: The strategy for decreasing one component of the total error leads to an increase of the other component. In a computation, we could conceivably decrease the step size to minimize truncation errors only to discover that in doing so, the round-off error begins to dominate the solution and the total error grows! Thus, our remedy becomes our problem (Fig. 4.8). One challenge that we face is to determine an appropriate step size for a particular computation. We would like to choose a large step size in order to decrease the amount of calculations and round-off errors without incurring the penalty of a large truncation error. If the total error is as shown in Fig. 4.8, the challenge is to identify the point of diminishing returns where round-off error begins to negate the benefits of step-size reduction. In actual cases, however, such situations are relatively uncommon because most computers carry enough significant figures that round-off errors do not predominate. Nevertheless, they sometimes do occur and suggest a sort of “numerical uncertainty principle” that places an absolute limit on the accuracy that may be obtained using certain computerized numerical methods. We explore such a case in the following section. 4.3.1 Error Analysis of Numerical Differentiation As described in the Sec. 4.1.3, a centered difference approximation of the first derivative can be written as (Eq. 4.22): f ¿(xi ) 5 True value

f(xi11 ) 2 f(xi21 ) f (3) (j) 2 h 2 2h 6 Finite-difference Truncation approximation error

(4.29)

4.3

TOTAL NUMERICAL ERROR

103

Thus, if the two function values in the numerator of the finite-difference approximation have no round-off error, the only error is due to truncation. However, because we are using digital computers, the function values do include round-off error as in f(xi21 ) 5 f˜(xi21 ) 1 ei21 f(xi11 ) 5 f˜(xi11 ) 1 ei11 where the f˜ ’s are the rounded function values and the e’s are the associated round-off errors. Substituting these values into Eq. (4.29) gives f˜(xi11 ) 2 f˜(xi21 ) f (3) (j) 2 ei11 2 ei21 f ¿(xi ) 5 1 2 h 2h 2h 6 True value

Finite-difference approximation

Round-off error

Truncation error

We can see that the total error of the finite-difference approximation consists of a roundoff error which increases with step size and a truncation error that decreases with step size. Assuming that the absolute value of each component of the round-off error has an upper bound of e, the maximum possible value of the difference ei+1 2 ei will be 2e. Further, assume that the third derivative has a maximum absolute value of M. An upper bound on the absolute value of the total error can therefore be represented as f˜(xi11 ) 2 f˜(xi21 ) e h2M Total error 5 ` f ¿(xi ) 2 ` # 1 (4.30) 2h h 6

An optimal step size can be determined by differentiating Eq. (4.30), setting the result equal to zero and solving for hopt 5

EXAMPLE 4.8

3e BM 3

(4.31)

Round-off and Truncation Errors in Numerical Differentiation Problem Statement. In Example 4.4, we used a centered difference approximation of O(h2) to estimate the first derivative of the following function at x 5 0.5, f(x) 5 20.1x4 2 0.15x3 2 0.5x2 2 0.25x 1 1.2 Perform the same computation starting with h 5 1. Then progressively divide the step size by a factor of 10 to demonstrate how round-off becomes dominant as the step size is reduced. Relate your results to Eq. (4.31). Recall that the true value of the derivative is 20.9125. Solution. We can develop a program to perform the computations and plot the results. For the present example, we have done this with a MATLAB software M-file. Notice that we pass both the function and its analytical derivative as arguments. In addition, the function generates a plot of the results.

104

TRUNCATION ERRORS AND THE TAYLOR SERIES

function diffex(func,dfunc,x,n) format long dftrue=dfunc(x); h=1; H(1)=h; D(1)=(func(x+h)−func(x−h))/(2*h); E(1)=abs(dftrue−D(1)); for i=2:n h=h/10; H(i)=h; D(i)=(func(x+h)−func(x−h))/(2*h); E(i)=abs(dftrue−D(i)); end L=[H' D' E']'; fprintf(' step size finite difference true error\n'); fprintf('%14.10f %16.14f %16.13f\n',L); loglog(H,E),xlabel('Step Size'),ylabel('Error') title('Plot of Error Versus Step Size') format short

The M-file can then be run using the following commands: >> [email protected](x) −0.1*x^4−0.15*x^3−0.5*x^2−0.25*x+1.2; >> [email protected](x) −0.4*x^3−0.45*x^2−x−0.25; >> diffex(ff,df,0.5,11)

When the function is run, the following numeric output is generated along with the plot (Fig. 4.9): step size 1.0000000000 0.1000000000 0.0100000000 0.0010000000 0.0001000000 0.0000100000 0.0000010000 0.0000001000 0.0000000100 0.0000000010 0.0000000001

finite difference −1.26250000000000 −0.91600000000000 −0.91253500000000 −0.91250035000001 −0.91250000349985 −0.91250000003318 −0.91250000000542 −0.91249999945031 −0.91250000333609 −0.91250001998944 −0.91250007550059

true error 0.3500000000000 0.0035000000000 0.0000350000000 0.0000003500000 0.0000000034998 0.0000000000332 0.0000000000054 0.0000000005497 0.0000000033361 0.0000000199894 0.0000000755006

The results are as expected. At first, round-off is minimal and the estimate is dominated by truncation error. Hence, as in Eq. (4.30), the total error drops by a factor of 100 each time we divide the step by 10. However, starting at h 5 0.0001, we see round-off error begin to creep in and erode the rate at which the error diminishes. A minimum error is reached at h 5 1026. Beyond this point, the error increases as round-off dominates. Because we are dealing with an easily differentiable function, we can also investigate whether these results are consistent with Eq. (4.31). First, we can estimate M by evaluating the function’s third derivative as M 5 Z f 3 (0.5) Z 5 Z 22.4(0.5) 2 0.9Z 5 2.1

4.3

TOTAL NUMERICAL ERROR

105

Plot of error versus step size 100 10–2

Error

10–4 10–6 10–8 10–10 10–12 10–10

10–8

10–6

10–4

10–2

10–0

Step size

FIGURE 4.9 Plot of error versus step size.

Because MATLAB has a precision of about 15 to 16 base-10 digits, a rough estimate of the upper bound on round-off would be about e 5 0.5 3 10216. Substituting these values into Eq. (4.31) gives hopt 5

3 ( 0.5 3 10 216 ) 5 4.3 3 10 26 B 2.1 3

which is on the same order as the result of 1 3 1026 obtained with our computer program.

4.3.2 Control of Numerical Errors For most practical cases, we do not know the exact error associated with numerical methods. The exception, of course, is when we have obtained the exact solution that makes our numerical approximations unnecessary. Therefore, for most engineering applications we must settle for some estimate of the error in our calculations. There are no systematic and general approaches to evaluating numerical errors for all problems. In many cases, error estimates are based on the experience and judgment of the engineer. Although error analysis is to a certain extent an art, there are several practical programming guidelines we can suggest. First and foremost, avoid subtracting two nearly equal numbers. Loss of significance almost always occurs when this is done. Sometimes you can rearrange or reformulate the problem to avoid subtractive cancellation. If this is not possible, you may want to use extended-precision arithmetic. Furthermore, when adding and

106

TRUNCATION ERRORS AND THE TAYLOR SERIES

subtracting numbers, it is best to sort the numbers and work with the smallest numbers first. This avoids loss of significance. Beyond these computational hints, one can attempt to predict total numerical errors using theoretical formulations. The Taylor series is our primary tool for analysis of both truncation and round-off errors. Several examples have been presented in this chapter. Prediction of total numerical error is very complicated for even moderately sized problems and tends to be pessimistic. Therefore, it is usually attempted for only small-scale tasks. The tendency is to push forward with the numerical computations and try to estimate the accuracy of your results. This can sometimes be done by seeing if the results satisfy some condition or equation as a check. Or it may be possible to substitute the results back into the original equation to check that it is actually satisfied. Finally you should be prepared to perform numerical experiments to increase your awareness of computational errors and possible ill-conditioned problems. Such experiments may involve repeating the computations with a different step size or method and comparing the results. We may employ sensitivity analysis to see how our solution changes when we change model parameters or input values. We may want to try different numerical algorithms that have different theoretical foundations, are based on different computational strategies, or have different convergence properties and stability characteristics. When the results of numerical computations are extremely critical and may involve loss of human life or have severe economic ramifications, it is appropriate to take special precautions. This may involve the use of two or more independent groups to solve the same problem so that their results can be compared. The roles of errors will be a topic of concern and analysis in all sections of this book. We will leave these investigations to specific sections.

4.4

BLUNDERS, FORMULATION ERRORS, AND DATA UNCERTAINTY Although the following sources of error are not directly connected with most of the numerical methods in this book, they can sometimes have great impact on the success of a modeling effort. Thus, they must always be kept in mind when applying numerical techniques in the context of real-world problems. 4.4.1 Blunders We are all familiar with gross errors, or blunders. In the early years of computers, erroneous numerical results could sometimes be attributed to malfunctions of the computer itself. Today, this source of error is highly unlikely, and most blunders must be attributed to human imperfection. Blunders can occur at any stage of the mathematical modeling process and can contribute to all the other components of error. They can be avoided only by sound knowledge of fundamental principles and by the care with which you approach and design your solution to a problem. Blunders are usually disregarded in discussions of numerical methods. This is no doubt due to the fact that, try as we may, mistakes are to a certain extent unavoidable. However, we believe that there are a number of ways in which their occurrence can be

4.4

BLUNDERS, FORMULATION ERRORS, AND DATA UNCERTAINTY

107

minimized. In particular, the good programming habits that were outlined in Chap. 2 are extremely useful for mitigating programming blunders. In addition, there are usually simple ways to check whether a particular numerical method is working properly. Throughout this book, we discuss ways to check the results of numerical calculations. 4.4.2 Formulation Errors Formulation, or model, errors relate to bias that can be ascribed to incomplete mathematical models. An example of a negligible formulation error is the fact that Newton’s second law does not account for relativistic effects. This does not detract from the adequacy of the solution in Example 1.1 because these errors are minimal on the time and space scales associated with the falling parachutist problem. However, suppose that air resistance is not linearly proportional to fall velocity, as in Eq. (1.7), but is a function of the square of velocity. If this were the case, both the analytical and numerical solutions obtained in the Chap. 1 would be erroneous because of formulation error. Further consideration of formulation error is included in some of the engineering applications in the remainder of the book. You should be cognizant of these problems and realize that, if you are working with a poorly conceived model, no numerical method will provide adequate results. 4.4.3 Data Uncertainty Errors sometimes enter into an analysis because of uncertainty in the physical data upon which a model is based. For instance, suppose we wanted to test the falling parachutist model by having an individual make repeated jumps and then measuring his or her velocity after a specified time interval. Uncertainty would undoubtedly be associated with these measurements, since the parachutist would fall faster during some jumps than during others. These errors can exhibit both inaccuracy and imprecision. If our instruments consistently underestimate or overestimate the velocity, we are dealing with an inaccurate, or biased, device. On the other hand, if the measurements are randomly high and low, we are dealing with a question of precision. Measurement errors can be quantified by summarizing the data with one or more well-chosen statistics that convey as much information as possible regarding specific characteristics of the data. These descriptive statistics are most often selected to represent (1) the location of the center of the distribution of the data and (2) the degree of spread of the data. As such, they provide a measure of the bias and imprecision, respectively. We will return to the topic of characterizing data uncertainty in Part Five. Although you must be cognizant of blunders, formulation errors, and uncertain data, the numerical methods used for building models can be studied, for the most part, independently of these errors. Therefore, for most of this book, we will assume that we have not made gross errors, we have a sound model, and we are dealing with error-free measurements. Under these conditions, we can study numerical errors without complicating factors.

108

TRUNCATION ERRORS AND THE TAYLOR SERIES

PROBLEMS 4.1 The following infinite series can be used to approximate ex: 2

n

3

x x x 1 1p1 2 3! n!

ex 5 1 1 x 1

(a) Prove that this Maclaurin series expansion is a special case of the Taylor series expansion [(Eq. (4.7)] with xi 5 0 and h 5 x. (b) Use the Taylor series to estimate f(x) 5 e2x at xi11 5 1 for xi 5 0.2. Employ the zero-, first-, second-, and third-order versions and compute the ZetZ for each case. 4.2 The Maclaurin series expansion for cos x is cos x 5 1 2

x2 x4 x6 x8 1 2 1 2p 2 4! 6! 8!

Starting with the simplest version, cos x 5 1, add terms one at a time to estimate cos(py3). After each new term is added, compute the true and approximate percent relative errors. Use your pocket calculator to determine the true value. Add terms until the absolute value of the approximate error estimate falls below an error criterion conforming to two significant figures. 4.3 Perform the same computation as in Prob. 4.2, but use the Maclaurin series expansion for the sin x to estimate sin(py3). sin x 5 x 2

x3 x5 x7 1 2 1p 3! 5! 7!

4.4 The Maclaurin series expansion for the arctangent of x is defined for Z x Z # 1 as q (21) n 2n11 arctan x 5 a x n50 2n 1 1

(a) Write out the first four terms (n 5 0, . . . , 3). (b) Starting with the simplest version, arctan x 5 x, add terms one at a time to estimate arctan(py6). After each new term is added, compute the true and approximate percent relative errors. Use your calculator to determine the true value. Add terms until the absolute value of the approximate error estimate falls below an error criterion conforming to two significant figures. 4.5 Use zero- through third-order Taylor series expansions to predict f (3) for f (x) 5 25x3 2 6x2 1 7x 2 88 using a base point at x 5 1. Compute the true percent relative error et for each approximation. 4.6 Use zero- through fourth-order Taylor series expansions to predict f(2.5) for f(x) 5 ln x using a base point at x 5 1. Compute the true percent relative error et for each approximation. Discuss the meaning of the results.

4.7 Use forward and backward difference approximations of O(h) and a centered difference approximation of O(h2) to estimate the first derivative of the function examined in Prob. 4.5. Evaluate the derivative at x 5 2 using a step size of h 5 0.2. Compare your results with the true value of the derivative. Interpret your results on the basis of the remainder term of the Taylor series expansion. 4.8 Use a centered difference approximation of O(h2) to estimate the second derivative of the function examined in Prob. 4.5. Perform the evaluation at x 5 2 using step sizes of h 5 0.25 and 0.125. Compare your estimates with the true value of the second derivative. Interpret your results on the basis of the remainder term of the Taylor series expansion. 4.9 The Stefan-Boltzmann law can be employed to estimate the rate of radiation of energy H from a surface, as in H 5 AesT 4

where H is in watts, A 5 the surface area (m2), e 5 the emissivity that characterizes the emitting properties of the surface (dimensionless), s 5 a universal constant called the Stefan-Boltzmann constant (5 5.67 3 1028 W m22 K24), and T 5 absolute temperature (K). Determine the error of H for a steel plate with A 5 0.15 m2, e 5 0.90, and T 5 650 6 20. Compare your results with the exact error. Repeat the computation but with T 5 650 6 40. Interpret your results. 4.10 Repeat Prob. 4.9 but for a copper sphere with radius 5 0.15 6 0.01 m, e 5 0.90 6 0.05, and T 5 550 6 20. 4.11 Recall that the velocity of the falling parachutist can be computed by [Eq. (1.10)], y(t) 5

gm ( 1 2 e 2 (cym)t ) c

Use a first-order error analysis to estimate the error of v at t 5 6, if g 5 9.81 and m 5 50 but c 5 12.5 6 1.5. 4.12 Repeat Prob. 4.11 with g 5 9.81, t 5 6, c 5 12.5 6 1.5, and m 5 50 6 2. 4.13 Evaluate and interpret the condition numbers for (a) f (x) 5 1Zx 2 1Z 1 1 for x 5 1.00001 (b) f(x) 5 e 2x for x 5 10 (c) f (x) 5 2x2 1 1 2 x for x 5 300 e 2x 2 1 (d) f (x) 5 for x 5 0.001 x sin x for x 5 1.0001p (e) f (x) 5 1 1 cos x 4.14 Employing ideas from Sec. 4.2, derive the relationships from Table 4.3. 4.15 Prove that Eq. (4.4) is exact for all values of x if f(x) 5 ax2 1 bx 1 c.

PROBLEMS 4.16 Manning’s formula for a rectangular channel can be written as Q5

1 (BH) 5y3 1S n (B 1 2H) 2y3

mum desired altitude is aR where R is the radius of the earth. The laws of mechanics can be used to show that sin f0 5 (1 1 a)

where Q 5 flow (m3/s), n 5 a roughness coefficient, B 5 width (m), H 5 depth (m), and S 5 slope. You are applying this formula to a stream where you know that the width 5 20 m and the depth 5 0.3 m. Unfortunately, you know the roughness and the slope to only a 6 10% precision. That is, you know that the roughness is about 0.03 with a range from 0.027 to 0.033 and the slope is 0.0003 with a range from 0.00027 to 0.00033. Use a first-order error analysis to determine the sensitivity of the flow prediction to each of these two factors. Which one should you attempt to measure with more precision? 4.17 If Z x Z , 1, it is known that 1 5 1 1 x 1 x2 1 x3 1 p 12x Repeat Prob. 4.1 for this series for x 5 0.1. 4.18 A missile leaves the ground with an initial velocity y0 forming an angle f0 with the vertical as shown in Fig. P4.18. The maxi-

0 v0

R

FIGURE P4.18

109

B

12

ye 2 a a b 1 1 a y0

where ye 5 the escape velocity of the missile. It is desired to fire the missile and reach the design maximum altitude within an accuracy of 62%. Determine the range of values for f0 if ye yy0 5 2 and a 5 0.25. 4.19 To calculate a planet’s space coordinates, we have to solve the function f (x) 5 x 2 1 2 0.5 sin x Let the base point be a 5 xi 5 py2 on the interval [0, p]. Determine the highest-order Taylor series expansion resulting in a maximum error of 0.015 on the specified interval. The error is equal to the absolute value of the difference between the given function and the specific Taylor series expansion. (Hint: Solve graphically.) 4.20 Consider the function f(x) 5 x3 2 2x 1 4 on the interval [22, 2] with h 5 0.25. Use the forward, backward, and centered finite difference approximations for the first and second derivatives so as to graphically illustrate which approximation is most accurate. Graph all three first derivative finite difference approximations along with the theoretical, and do the same for the second derivative as well. 4.21 Derive Eq. (4.31). 4.22 Repeat Example 4.8, but for f(x) 5 cos(x) at x 5 py6. 4.23 Repeat Example 4.8, but for the forward divided difference (Eq. 4.17). 4.24 Develop a well-structured program to compute the Maclaurin series expansion for the cosine function as described in Prob. 4.2. The function should have the following features: • Iterate until the relative error falls below a stopping criterion (es) or exceeds a maximum number of iterations (maxit). Allow the user to specify values for these parameters. • Include default values of es (5 0.000001) and maxit (5 100) in the event that they are not specified by the user. • Return the estimate of cos(x), the approximate relative error, the number of iterations, and the true relative error (that you can calculate based on the built-in cosine function).

110

EPILOGUE: PART ONE

EPILOGUE: PART ONE PT1.4

TRADE-OFFS Numerical methods are scientific in the sense that they represent systematic techniques for solving mathematical problems. However, there is a certain degree of art, subjective judgment, and compromise associated with their effective use in engineering practice. For each problem, you may be confronted with several alternative numerical methods and many different types of computers. Thus, the elegance and efficiency of different approaches to problems is highly individualistic and correlated with your ability to choose wisely among options. Unfortunately, as with any intuitive process, the factors influencing this choice are difficult to communicate. Only by experience can these skills be fully comprehended and honed. However, because these skills play such a prominent role in the effective implementation of the methods, we have included this section as an introduction to some of the trade-offs that you must consider when selecting a numerical method and the tools for implementing the method. It is hoped that the discussion that follows will influence your orientation when approaching subsequent material. Also, it is hoped that you will refer back to this material when you are confronted with choices and trade-offs in the remainder of the book. 1. Type of Mathematical Problem. As delineated previously in Fig. PT1.2, several types of mathematical problems are discussed in this book: (a) Roots of equations. (b) Systems of simultaneous linear algebraic equations. (c) Optimization. (d) Curve fitting. (e) Numerical integration. (f) Ordinary differential equations. (g) Partial differential equations. You will probably be introduced to the applied aspects of numerical methods by confronting a problem in one of the above areas. Numerical methods will be required because the problem cannot be solved efficiently using analytical techniques. You should be cognizant of the fact that your professional activities will eventually involve problems in all the above areas. Thus, the study of numerical methods and the selection of automatic computation equipment should, at the minimum, consider these basic types of problems. More advanced problems may require capabilities of handling areas such as functional approximation, integral equations, etc. These areas typically demand greater computation power or advanced methods not covered in this text. Other references such as Carnahan, Luther, and Wilkes (1969); Hamming (1973); Ralston and Rabinowitz (1978); Burden and Faires (2005); and Moler (2004) should be consulted for problems beyond the scope of this book. In addition, at the end of each part of this text, we include a brief summary

110

PT1.4

TRADE-OFFS

111

and references for advanced methods to provide you with avenues for pursuing further studies of numerical methods. 2. Type, Availability, Precision, Cost, and Speed of Computer. You may have the option of working with a variety of computation tools. These range from pocket calculators to large mainframe computers. Of course, any of the tools can be used to implement any numerical method (including simple paper and pencil). It is usually not a question of ultimate capability but rather of cost, convenience, speed, dependability, repeatability, and precision. Although each of the tools will continue to have utility, the recent rapid advances in the performance of personal computers have already had a major impact on the engineering profession. We expect this revolution will spread as technological improvements continue because personal computers offer an excellent compromise in convenience, cost, precision, speed, and storage capacity. Furthermore, they can be readily applied to most practical engineering problems. 3. Program Development Cost versus Software Cost versus Run-Time Cost. Once the types of mathematical problems to be solved have been identified and the computer system has been selected, it is appropriate to consider software and run-time costs. Software development may represent a substantial effort in many engineering projects and may therefore be a significant cost. In this regard, it is particularly important that you be very well acquainted with the theoretical and practical aspects of the relevant numerical methods. In addition, you should be familiar with professionally developed software. Low-cost software is widely available to implement numerical methods that may be readily adapted to a broad variety of problems. 4. Characteristics of the Numerical Method. When computer hardware and software costs are high, or if computer availability is limited (for example, on some timeshare systems), it pays to choose carefully the numerical method to suit the situation. On the other hand, if the problem is still at the exploratory stage and computer access and cost are not problems, it may be appropriate for you to select a numerical method that always works but may not be the most computationally efficient. The numerical methods available to solve any particular type of problem involve the types of tradeoffs just discussed and others: (a) Number of Initial Guesses or Starting Points. Some of the numerical methods for finding roots of equations or solving differential equations require the user to specify initial guesses or starting points. Simple methods usually require one value, whereas complicated methods may require more than one value. The advantages of complicated methods that are computationally efficient may be offset by the requirement for multiple starting points. You must use your experience and judgment to assess the trade-offs for each particular problem. (b) Rate of Convergence. Certain numerical methods converge more rapidly than others. However, this rapid convergence may require more refined initial guesses and more complex programming than a method with slower convergence. Again, you must use your judgment in selecting a method. Faster is not always better. (c) Stability. Some numerical methods for finding roots of equations or solutions for systems of linear equations may diverge rather than converge on the correct answer for certain problems. Why would you tolerate this possibility when confronted with design or planning problems? The answer is that these methods may be highly efficient when they work. Thus, trade-offs again emerge. You must decide

112

EPILOGUE: PART ONE

(d)

(e)

(f)

(g)

if your problem requirements justify the effort needed to apply a method that may not always converge. Accuracy and Precision. Some numerical methods are simply more accurate or precise than others. Good examples are the various equations available for numerical integration. Usually, the performance of low-accuracy methods can be improved by decreasing the step size or increasing the number of applications over a given interval. Is it better to use a low-accuracy method with small step sizes or a high-accuracy method with large step sizes? This question must be addressed on a case-by-case basis taking into consideration the additional factors such as cost and ease of programming. In addition, you must also be concerned with round-off errors when you are using multiple applications of low-accuracy methods and when the number of computations becomes large. Here the number of significant figures handled by the computer may be the deciding factor. Breadth of Application. Some numerical methods can be applied to only a limited class of problems or to problems that satisfy certain mathematical restrictions. Other methods are not affected by such limitations. You must evaluate whether it is worth your effort to develop programs that employ techniques that are appropriate for only a limited number of problems. The fact that such techniques may be widely used suggests that they have advantages that will often outweigh their disadvantages. Obviously, trade-offs are occurring. Special Requirements. Some numerical techniques attempt to increase accuracy and rate of convergence using additional or special information. An example would be to use estimated or theoretical values of errors to improve accuracy. However, these improvements are generally not achieved without some inconvenience in terms of added computing costs or increased program complexity. Programming Effort Required. Efforts to improve rates of convergence, stability, and accuracy can be creative and ingenious. When improvements can be made without increasing the programming complexity, they may be considered elegant and will probably find immediate use in the engineering profession. However, if they require more complicated programs, you are again faced with a trade-off situation that may or may not favor the new method.

It is clear that the above discussion concerning a choice of numerical methods reduces to one of cost and accuracy. The costs are those involved with computer time and program development. Appropriate accuracy is a question of professional judgment and ethics. 5. Mathematical Behavior of the Function, Equation, or Data. In selecting a particular numerical method, type of computer, and type of software, you must consider the complexity of your functions, equations, or data. Simple equations and smooth data may be appropriately handled by simple numerical algorithms and inexpensive computers. The opposite is true for complicated equations and data exhibiting discontinuities. 6. Ease of Application (User-Friendly?). Some numerical methods are easy to apply; others are difficult. This may be a consideration when choosing one method over

PT1.6

ADVANCED METHODS AND ADDITIONAL REFERENCES

113

another. This same idea applies to decisions regarding program development costs versus professionally developed software. It may take considerable effort to convert a difficult program to one that is user-friendly. Ways to do this were introduced in Chap. 2 and are elaborated throughout the book. 7. Maintenance. Programs for solving engineering problems require maintenance because during application, difficulties invariably occur. Maintenance may require changing the program code or expanding the documentation. Simple programs and numerical algorithms are simpler to maintain. The chapters that follow involve the development of various types of numerical methods for various types of mathematical problems. Several alternative methods will be given in each chapter. These various methods (rather than a single method chosen by the authors) are presented because there is no single “best” method. There is no best method because there are many trade-offs that must be considered when applying the methods to practical problems. A table that highlights the trade-offs involved in each method will be found at the end of each part of the book. This table should assist you in selecting the appropriate numerical procedure for your particular problem context.

PT1.5

IMPORTANT RELATIONSHIPS AND FORMULAS Table PT1.2 summarizes important information that was presented in Part One. The table can be consulted to quickly access important relationships and formulas. The epilogue of each part of the book will contain such a summary.

PT1.6

ADVANCED METHODS AND ADDITIONAL REFERENCES The epilogue of each part of the book will also include a section designed to facilitate and encourage further studies of numerical methods. This section will reference other books on the subject as well as material related to more advanced methods.1 To extend the background provided in Part One, numerous manuals on computer programming are available. It would be difficult to reference all the excellent books and manuals pertaining to specific languages and computers. In addition, you probably already have material from your previous exposure to programming. However, if this is your first experience with computers, your instructor and fellow students should also be able to advise you regarding good reference books for the machines and languages available at your school. As for error analysis, any good introductory calculus book will include supplementary material related to subjects such as the Taylor series expansion. Texts by Swokowski (1979), Thomas and Finney (1979), and Simmons (1985) provide very readable discussions of these subjects. In addition, Taylor (1982) presents a nice introduction to error analysis. Finally, although we hope that our book serves you well, it is always good to consult other sources when trying to master a new subject. Burden and Faires (2005); Ralston 1

Books are referenced only by author here; a complete bibliography will be found at the back of this text.

114

EPILOGUE: PART ONE TABLE PT1.2 Summary of important information presented in Part One. Error Deﬁnitions True error

Et 5 true value 2 approximation

True percent relative error

et 5

Approximate percent relative error

ea 5

Stopping criterion

Terminate computation when

true value 2 approximation true value

100%

present approximation 2 previous approximation present approximation

100%

ea , es where es is the desired percent relative error Taylor Series Taylor series expansion

f (xi11 ) 5 f (xi ) 1 f ' (xi )h 1 1

f '' (xi ) 2 h 2!

f ''' (xi ) 3 f (n) (xi ) n h 1p1 h 1 Rn 3! n!

where Remainder

Rn 5

f (n11) (j) (n 1 1)!

hn11

or Rn 5 O(hn11 ) Numerical Differentiation First forward ﬁnite divided difference

f ' (xi ) 5

f (xi11 ) 2 f (xi ) h

1 O(h)

(Other divided differences are summarized in Chaps. 4 and 23.) Error Propagation For n independent variables x1, x2, . . . , xn having errors ¢x˜1, ¢x˜2 , p ,¢x˜n, the error in the function f can be estimated via ¢f 5 `

0f 0f 0f ` ¢x˜1 1 ` ` ¢x˜2 1 p 1 ` ` ¢x˜n 0x1 0x2 0xn

and Rabinowitz (1978); Hoffman (1992); and Carnahan, Luther, and Wilkes (1969) provide comprehensive discussions of most numerical methods. Other enjoyable books on the subject are Gerald and Wheatley (2004), and Cheney and Kincaid (2008). In addition, Press et al. (2007) include algorithms to implement a variety of methods, and Moler (2004) and Chapra (2007) are devoted to numerical methods with MATLAB software.

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PART TWO

ROOTS OF EQUATIONS PT2.1

MOTIVATION Years ago, you learned to use the quadratic formula x5

2b 6 2b2 2 4ac 2a

(PT2.1)

to solve f (x) 5 ax2 1 bx 1 c 5 0

(PT2.2)

The values calculated with Eq. (PT2.1) are called the “roots” of Eq. (PT2.2). They represent the values of x that make Eq. (PT2.2) equal to zero. Thus, we can define the root of an equation as the value of x that makes f (x) 5 0. For this reason, roots are sometimes called the zeros of the equation. Although the quadratic formula is handy for solving Eq. (PT2.2), there are many other functions for which the root cannot be determined so easily. For these cases, the numerical methods described in Chaps. 5, 6, and 7 provide efficient means to obtain the answer. PT2.1.1 Noncomputer Methods for Determining Roots Before the advent of digital computers, there were several ways to solve for roots of algebraic and transcendental equations. For some cases, the roots could be obtained by direct methods, as was done with Eq. (PT2.1). Although there were equations like this that could be solved directly, there were many more that could not. For example, even an apparently simple function such as f (x) 5 e2x 2 x cannot be solved analytically. In such instances, the only alternative is an approximate solution technique. One method to obtain an approximate solution is to plot the function and determine where it crosses the x axis. This point, which represents the x value for which f (x) 5 0, is the root. Graphical techniques are discussed at the beginning of Chaps. 5 and 6. Although graphical methods are useful for obtaining rough estimates of roots, they are limited because of their lack of precision. An alternative approach is to use trial and error. This “technique” consists of guessing a value of x and evaluating whether f (x) is zero. If not (as is almost always the case), another guess is made, and f (x) is again evaluated to determine whether the new value provides a better estimate of the root. The process is repeated until a guess is obtained that results in an f (x) that is close to zero. Such haphazard methods are obviously inefficient and inadequate for the requirements of engineering practice. The techniques described in Part Two represent alternatives that are also approximate but employ systematic strategies to home in on the true root. As elaborated on in the following pages, the combination of these systematic methods and computers makes the solution of most applied roots-of-equations problems a simple and efficient task. 117

118

ROOTS OF EQUATIONS

PT2.1.2 Roots of Equations and Engineering Practice Although they arise in other problem contexts, roots of equations frequently occur in the area of engineering design. Table PT2.1 lists several fundamental principles that are routinely used in design work. As introduced in Chap. 1, mathematical equations or models derived from these principles are employed to predict dependent variables as a function of independent variables, forcing functions, and parameters. Note that in each case, the dependent variables reflect the state or performance of the system, whereas the parameters represent its properties or composition. An example of such a model is the equation, derived from Newton’s second law, used in Chap. 1 for the parachutist’s velocity: y5

gm ( 1 2 e 2 (cym)t ) c

(PT2.3)

where velocity y 5 the dependent variable, time t 5 the independent variable, the gravitational constant g 5 the forcing function, and the drag coefficient c and mass m 5 parameters. If the parameters are known, Eq. (PT2.3) can be used to predict the parachutist’s velocity as a function of time. Such computations can be performed directly because y is expressed explicitly as a function of time. That is, it is isolated on one side of the equal sign. TABLE PT2.1 Fundamental principles used in engineering design problems. Fundamental Principle

Dependent Variable

Independent Variable

Parameters

Heat balance

Temperature

Time and position

Mass balance

Concentration or quantity of mass

Time and position

Force balance

Magnitude and direction of forces

Time and position

Energy balance

Changes in the kineticand potential-energy states of the system Acceleration, velocity, or location

Time and position

Thermal properties of material and geometry of system Chemical behavior of material, mass transfer coefﬁcients, and geometry of system Strength of material, structural properties, and geometry of system Thermal properties, mass of material, and system geometry Mass of material, system geometry, and dissipative parameters such as friction or drag Electrical properties of systems such as resistance, capacitance, and inductance

Newton’s laws of motion

Kirchhoff’s laws

Currents and voltages in electric circuits

Time and position

Time

PT2.2

MATHEMATICAL BACKGROUND

119

However, suppose we had to determine the drag coefficient for a parachutist of a given mass to attain a prescribed velocity in a set time period. Although Eq. (PT2.3) provides a mathematical representation of the interrelationship among the model variables and parameters, it cannot be solved explicitly for the drag coefficient. Try it. There is no way to rearrange the equation so that c is isolated on one side of the equal sign. In such cases, c is said to be implicit. This represents a real dilemma, because many engineering design problems involve specifying the properties or composition of a system (as represented by its parameters) to ensure that it performs in a desired manner (as represented by its variables). Thus, these problems often require the determination of implicit parameters. The solution to the dilemma is provided by numerical methods for roots of equations. To solve the problem using numerical methods, it is conventional to reexpress Eq. (PT2.3). This is done by subtracting the dependent variable y from both sides of the equation to give f (c) 5

gm ( 1 2 e 2 (cym)t ) 2 y c

(PT2.4)

The value of c that makes f (c) 5 0 is, therefore, the root of the equation. This value also represents the drag coefficient that solves the design problem. Part Two of this book deals with a variety of numerical and graphical methods for determining roots of relationships such as Eq. (PT2.4). These techniques can be applied to engineering design problems that are based on the fundamental principles outlined in Table PT2.1 as well as to many other problems confronted routinely in engineering practice.

PT2.2

MATHEMATICAL BACKGROUND For most of the subject areas in this book, there is usually some prerequisite mathematical background needed to successfully master the topic. For example, the concepts of error estimation and the Taylor series expansion discussed in Chaps. 3 and 4 have direct relevance to our discussion of roots of equations. Additionally, prior to this point we have mentioned the terms “algebraic” and “transcendental” equations. It might be helpful to formally define these terms and discuss how they relate to the scope of this part of the book. By definition, a function given by y 5 f (x) is algebraic if it can be expressed in the form fn yn 1 fn21 yn21 1 p 1 f1 y 1 f0 5 0

(PT2.5)

where fi 5 an ith-order polynomial in x. Polynomials are a simple class of algebraic functions that are represented generally by fn (x) 5 a0 1 a1x 1 a2x2 1 p 1 anxn

(PT2.6)

where n 5 the order of the polynomial and the a’s 5 constants. Some specific examples are f2 (x) 5 1 2 2.37x 1 7.5x2

(PT2.7)

f6 (x) 5 5x2 2 x3 1 7x6

(PT2.8)

and

120

ROOTS OF EQUATIONS

A transcendental function is one that is nonalgebraic. These include trigonometric, exponential, logarithmic, and other, less familiar, functions. Examples are f (x) 5 ln x2 2 1

(PT2.9)

and f(x) 5 e 20.2x sin (3x 2 0.5)

(PT2.10)

Roots of equations may be either real or complex. Although there are cases where complex roots of nonpolynomials are of interest, such situations are less common than for polynomials. As a consequence, the standard methods for locating roots typically fall into two somewhat related but primarily distinct problem areas: 1. The determination of the real roots of algebraic and transcendental equations. These techniques are usually designed to determine the value of a single real root on the basis of foreknowledge of its approximate location. 2. The determination of all real and complex roots of polynomials. These methods are specifically designed for polynomials. They systematically determine all the roots of the polynomial rather than determining a single real root given an approximate location. In this book we discuss both. Chapters 5 and 6 are devoted to the first category. Chapter 7 deals with polynomials.

PT2.3

ORIENTATION Some orientation is helpful before proceeding to the numerical methods for determining roots of equations. The following is intended to give you an overview of the material in Part Two. In addition, some objectives have been included to help you focus your efforts when studying the material. PT2.3.1 Scope and Preview Figure PT2.1 is a schematic representation of the organization of Part Two. Examine this figure carefully, starting at the top and working clockwise. After the present introduction, Chap. 5 is devoted to bracketing methods for finding roots. These methods start with guesses that bracket, or contain, the root and then systematically reduce the width of the bracket. Two specific methods are covered: bisection and false position. Graphical methods are used to provide visual insight into the techniques. Error formulations are developed to help you determine how much computational effort is required to estimate the root to a prespecified level of precision. Chapter 6 covers open methods. These methods also involve systematic trial-anderror iterations but do not require that the initial guesses bracket the root. We will discover that these methods are usually more computationally efficient than bracketing methods but that they do not always work. One-point iteration, Newton-Raphson, and secant methods are described. Graphical methods are used to provide geometric insight into cases where the open methods do not work. Formulas are developed that provide an idea of how fast open methods home in on the root. An advanced approach, Brent’s method, that combines the reliability of bracketing with the speed of open methods is

PT2.3

ORIENTATION

PT 2.1 Motivation

PT 2.2 Mathematical background

121

PT 2.3 Orientation

5.1 Graphical methods

PART 2 Roots of Equations

PT 2.6 Advanced methods

5.2 Bisection

5.3 False position

PT 2.5 Important formulas

CHAPTER 5 Bracketing Methods

EPILOGUE PT 2.4 Trade-offs

5.4 Incremental searches 6.1 Fixed-point iteration

8.4 Mechanical engineering

8.3 Electrical engineering

6.2 NewtonRaphson

6.3 Secant

CHAPTER 6 Open Methods

CHAPTER 8 Engineering Case Studies

6.4 Brent’s method

8.2 Civil engineering

8.1 Chemical engineering

7.7 Software packages

6.5 Multiple roots

CHAPTER 7 Roots of Polynomials

7.1 Polynomials in engineering

7.6 Other methods 7.5 Bairstow's method

6.6 Nonlinear systems

7.2 Computing with polynomials

7.4 .. Muller's method

7.3 Conventional methods

FIGURE PT2.1 Schematic of the organization of the material in Part Two: Roots of Equations.

described. In addition, an approach to extend the Newton-Raphson method to systems of nonlinear equations is explained. Chapter 7 is devoted to finding the roots of polynomials. After background sections on polynomials, the use of conventional methods (in particular the open methods from Chap. 6) are discussed. Then two special methods for locating polynomial roots are

122

ROOTS OF EQUATIONS

described: Müller’s and Bairstow’s methods. The chapter ends with information related to finding roots with Excel, MATLAB software, and Mathcad. Chapter 8 extends the above concepts to actual engineering problems. Engineering case studies are used to illustrate the strengths and weaknesses of each method and to provide insight into the application of the techniques in professional practice. The applications also highlight the trade-offs (as discussed in Part One) associated with the various methods. An epilogue is included at the end of Part Two. It contains a detailed comparison of the methods discussed in Chaps. 5, 6, and 7. This comparison includes a description of trade-offs related to the proper use of each technique. This section also provides a summary of important formulas, along with references for some numerical methods that are beyond the scope of this text. PT2.3.2 Goals and Objectives Study Objectives. After completing Part Two, you should have sufficient information to successfully approach a wide variety of engineering problems dealing with roots of equations. In general, you should have mastered the techniques, have learned to assess their reliability, and be capable of choosing the best method (or methods) for any particular problem. In addition to these general goals, the specific concepts in Table PT2.2 should be assimilated for a comprehensive understanding of the material in Part Two. Computer Objectives. The book provides you with software and simple computer algorithms to implement the techniques discussed in Part Two. All have utility as learning tools. Pseudocodes for several methods are also supplied directly in the text. This information will allow you to expand your software library to include programs that are more efficient than the bisection method. For example, you may also want to have your own software for the false-position, Newton-Raphson, and secant techniques, which are often more efficient than the bisection method. Finally, packages such as Excel, MATLAB, and Mathcad have powerful capabilities for locating roots. You can use this part of the book to become familiar with these capabilities. TABLE PT2.2 Speciﬁc study objectives for Part Two. 1. Understand the graphical interpretation of a root 2. Know the graphical interpretation of the false-position method and why it is usually superior to the bisection method 3. Understand the difference between bracketing and open methods for root location 4. Understand the concepts of convergence and divergence; use the two-curve graphical method to provide a visual manifestation of the concepts 5. Know why bracketing methods always converge, whereas open methods may sometimes diverge 6. Realize that convergence of open methods is more likely if the initial guess is close to the true root 7. Understand the concepts of linear and quadratic convergence and their implications for the efﬁciencies of the ﬁxed-point-iteration and Newton-Raphson methods 8. Know the fundamental difference between the false-position and secant methods and how it relates to convergence 9. Understand how Brent’s method combines the reliability of bisection with the speed of open methods 10. Understand the problems posed by multiple roots and the modiﬁcations available to mitigate them 11. Know how to extend the single-equation Newton-Raphson approach to solve systems of nonlinear equations

5

CHAPTER

Bracketing Methods

This chapter on roots of equations deals with methods that exploit the fact that a function typically changes sign in the vicinity of a root. These techniques are called bracketing methods because two initial guesses for the root are required. As the name implies, these guesses must “bracket,” or be on either side of, the root. The particular methods described herein employ different strategies to systematically reduce the width of the bracket and, hence, home in on the correct answer. As a prelude to these techniques, we will briefly discuss graphical methods for depicting functions and their roots. Beyond their utility for providing rough guesses, graphical techniques are also useful for visualizing the properties of the functions and the behavior of the various numerical methods.

5.1

GRAPHICAL METHODS A simple method for obtaining an estimate of the root of the equation f(x) 5 0 is to make a plot of the function and observe where it crosses the x axis. This point, which represents the x value for which f(x) 5 0, provides a rough approximation of the root.

EXAMPLE 5.1

The Graphical Approach Problem Statement. Use the graphical approach to determine the drag coefficient c needed for a parachutist of mass m 5 68.1 kg to have a velocity of 40 m/s after freefalling for time t 5 10 s. Note: The acceleration due to gravity is 9.81 m/s2. Solution. This problem can be solved by determining the root of Eq. (PT2.4) using the parameters t 5 10, g 5 9.81, y 5 40, and m 5 68.1: f(c) 5

9.81(68.1) ( 1 2 e2(cy68.1)10 ) 2 40 c

f(c) 5

668.06 ( 1 2 e20.146843c ) 2 40 c

or (E5.1.1)

Various values of c can be substituted into the right-hand side of this equation to compute 123

124

BRACKETING METHODS

c

f (c)

4 8 12 16 20

34.190 17.712 6.114 22.230 28.368

These points are plotted in Fig. 5.1. The resulting curve crosses the c axis between 12 and 16. Visual inspection of the plot provides a rough estimate of the root of 14.75. The validity of the graphical estimate can be checked by substituting it into Eq. (E5.1.1) to yield f(14.75) 5

668.06 ( 1 2 e 20.146843(14.75) ) 2 40 5 0.100 14.75

which is close to zero. It can also be checked by substituting it into Eq. (PT2.3) along with the parameter values from this example to give y5

9.81(68.1) ( 1 2 e 2 (14.75y68.1)10 ) 5 40.100 14.75

which is very close to the desired fall velocity of 40 m/s. FIGURE 5.1 The graphical approach for determining the roots of an equation. f (c) 40

20

Root 0

–10

4

8

12

20 c

5.1

f (x)

x

(a) f (x)

x

(b) f (x)

x

(c) f (x)

x

xl

xu (d)

FIGURE 5.2 Illustration of a number of general ways that a root may occur in an interval prescribed by a lower bound xl and an upper bound xu. Parts (a) and (c) indicate that if both f(xl) and f(xu) have the same sign, either there will be no roots or there will be an even number of roots within the interval. Parts (b) and (d) indicate that if the function has different signs at the end points, there will be an odd number of roots in the interval.

GRAPHICAL METHODS

125

Graphical techniques are of limited practical value because they are not precise. However, graphical methods can be utilized to obtain rough estimates of roots. These estimates can be employed as starting guesses for numerical methods discussed in this and the next chapter. Aside from providing rough estimates of the root, graphical interpretations are important tools for understanding the properties of the functions and anticipating the pitfalls of the numerical methods. For example, Fig. 5.2 shows a number of ways in which roots can occur (or be absent) in an interval prescribed by a lower bound xl and an upper bound xu. Figure 5.2b depicts the case where a single root is bracketed by negative and positive values of f(x). However, Fig. 5.2d, where f(xl) and f(xu) are also on opposite sides of the x axis, shows three roots occurring within the interval. In general, if f(xl) and f(xu) have opposite signs, there are an odd number of roots in the interval. As indicated by Fig. 5.2a and c, if f(xl) and f(xu) have the same sign, there are either no roots or an even number of roots between the values. Although these generalizations are usually true, there are cases where they do not hold. For example, functions that are tangential to the x axis (Fig. 5.3a) and discontinuous functions (Fig. 5.3b) can violate these principles. An example of a function that is tangential to the axis is the cubic equation f(x) 5 (x 2 2)(x 2 2)(x 2 4). Notice that x 5 2 makes two terms in this polynomial equal to zero. Mathematically, x 5 2 is called a multiple root. At the end of Chap. 6, we will present techniques that are expressly designed to locate multiple roots. The existence of cases of the type depicted in Fig. 5.3 makes it difficult to develop general computer algorithms guaranteed to locate all the roots in an interval. However, when used in conjunction with graphical approaches, the methods described in the

FIGURE 5.3 Illustration of some exceptions to the general cases depicted in Fig. 5.2. (a) Multiple root that occurs when the function is tangential to the x axis. For this case, although the end points are of opposite signs, there are an even number of axis intersections for the interval. (b) Discontinuous function where end points of opposite sign bracket an even number of roots. Special strategies are required for determining the roots for these cases.

f (x)

x

(a) f (x)

x

xl

(b)

xu

126

BRACKETING METHODS

following sections are extremely useful for solving many roots of equations problems confronted routinely by engineers and applied mathematicians. EXAMPLE 5.2

Use of Computer Graphics to Locate Roots Problem Statement. Computer graphics can expedite and improve your efforts to locate roots of equations. The function f(x) 5 sin 10x 1 cos 3x has several roots over the range x 5 0 to x 5 5. Use computer graphics to gain insight into the behavior of this function. Solution. Packages such as Excel and MATLAB software can be used to generate plots. Figure 5.4a is a plot of f(x) from x 5 0 to x 5 5. This plot suggests the presence of several roots, including a possible double root at about x 5 4.2 where f(x) appears to be

FIGURE 5.4 The progressive enlargement of f(x) 5 sin 10x 1 cos 3x by the computer. Such interactive graphics permits the analyst to determine that two distinct roots exist between x 5 4.2 and x 5 4.3. 2

Y

2

Y

0

–2

0

–2

5

2.5 X

0

3

4 X

(a)

(b) .15

Y

0

– .15 4.2

4.25 X

(c)

4.3

5

5.2

THE BISECTION METHOD

127

tangent to the x axis. A more detailed picture of the behavior of f(x) is obtained by changing the plotting range from x 5 3 to x 5 5, as shown in Fig. 5.4b. Finally, in Fig. 5.4c, the vertical scale is narrowed further to f(x) 5 20.15 to f(x) 5 0.15 and the horizontal scale is narrowed to x 5 4.2 to x 5 4.3. This plot shows clearly that a double root does not exist in this region and that in fact there are two distinct roots at about x 5 4.23 and x 5 4.26. Computer graphics will have great utility in your studies of numerical methods. This capability will also find many other applications in your other classes and professional activities as well.

5.2

THE BISECTION METHOD When applying the graphical technique in Example 5.1, you have observed (Fig. 5.1) that f(x) changed sign on opposite sides of the root. In general, if f(x) is real and continuous in the interval from xl to xu and f(xl) and f(xu) have opposite signs, that is, f(xl ) f(xu ) , 0

(5.1)

then there is at least one real root between xl and xu. Incremental search methods capitalize on this observation by locating an interval where the function changes sign. Then the location of the sign change (and consequently, the root) is identified more precisely by dividing the interval into a number of subintervals. Each of these subintervals is searched to locate the sign change. The process is repeated and the root estimate refined by dividing the subintervals into finer increments. We will return to the general topic of incremental searches in Sec. 5.4. The bisection method, which is alternatively called binary chopping, interval halving, or Bolzano’s method, is one type of incremental search method in which the interval is always divided in half. If a function changes sign over an interval, the function value at the midpoint is evaluated. The location of the root is then determined as lying at the midpoint of the subinterval within which the sign change occurs. The process is repeated to obtain refined estimates. A simple algorithm for the bisection calculation is listed in Fig. 5.5, and a graphical depiction of the method is provided in Fig. 5.6. The following example goes through the actual computations involved in the method.

FIGURE 5.5 Step 1: Choose lower xl and upper xu guesses for the root such that the function changes sign over the interval. This can be checked by ensuring that f(xl)f(xu) , 0. Step 2: An estimate of the root xr is determined by xr 5

xl 1 xu 2

Step 3: Make the following evaluations to (a) If f(xl)f(xr) , 0, the root lies in to step 2. (b) If f(xl)f(xr) . 0, the root lies in to step 2. (c) If f(xl)f(xr) 5 0, the root equals

determine in which subinterval the root lies: the lower subinterval. Therefore, set xu 5 xr and return the upper subinterval. Therefore, set xl 5 xr and return xr; terminate the computation.

128

BRACKETING METHODS

12

16

14

16

15

FIGURE 5.6 A graphical depiction of the bisection method. This plot conforms to the ﬁrst three iterations from Example 5.3.

EXAMPLE 5.3

14

Bisection Problem Statement. in Example 5.1.

Use bisection to solve the same problem approached graphically

Solution. The first step in bisection is to guess two values of the unknown (in the present problem, c) that give values for f(c) with different signs. From Fig. 5.1, we can see that the function changes sign between values of 12 and 16. Therefore, the initial estimate of the root xr lies at the midpoint of the interval xr 5

12 1 16 5 14 2

This estimate represents a true percent relative error of et 5 5.3% (note that the true value of the root is 14.8011). Next we compute the product of the function value at the lower bound and at the midpoint: f(12) f(14) 5 6.114 (1.611) 5 9.850 which is greater than zero, and hence no sign change occurs between the lower bound and the midpoint. Consequently, the root must be located between 14 and 16. Therefore, we create a new interval by redefining the lower bound as 14 and determining a revised root estimate as xr 5

14 1 16 5 15 2

which represents a true percent error of et 5 1.3%. The process can be repeated to obtain refined estimates. For example, f(14) f(15) 5 1.611(20.384) 5 20.619

5.2

THE BISECTION METHOD

129

Therefore, the root is between 14 and 15. The upper bound is redefined as 15, and the root estimate for the third iteration is calculated as xr 5

14 1 15 5 14.5 2

which represents a percent relative error of et 5 2.0%. The method can be repeated until the result is accurate enough to satisfy your needs.

In the previous example, you may have noticed that the true error does not decrease with each iteration. However, the interval within which the root is located is halved with each step in the process. As discussed in the next section, the interval width provides an exact estimate of the upper bound of the error for the bisection method. 5.2.1 Termination Criteria and Error Estimates We ended Example 5.3 with the statement that the method could be continued to obtain a refined estimate of the root. We must now develop an objective criterion for deciding when to terminate the method. An initial suggestion might be to end the calculation when the true error falls below some prespecified level. For instance, in Example 5.3, the relative error dropped to 2.0 percent during the course of the computation. We might decide that we should terminate when the error drops below, say, 0.1 percent. This strategy is flawed because the error estimates in the example were based on knowledge of the true root of the function. This would not be the case in an actual situation because there would be no point in using the method if we already knew the root. Therefore, we require an error estimate that is not contingent on foreknowledge of the root. As developed previously in Sec. 3.3, an approximate percent relative error ea can be calculated, as in [recall Eq. (3.5)] ea 5 `

xrnew 2 xold r ` 100% xrnew

(5.2)

where xrnew is the root for the present iteration and xold r is the root from the previous iteration. The absolute value is used because we are usually concerned with the magnitude of ea rather than with its sign. When ea becomes less than a prespecified stopping criterion es, the computation is terminated. EXAMPLE 5.4

Error Estimates for Bisection Problem Statement. Continue Example 5.3 until the approximate error falls below a stopping criterion of es 5 0.5%. Use Eq. (5.2) to compute the errors. Solution. The results of the first two iterations for Example 5.3 were 14 and 15. Substituting these values into Eq. (5.2) yields ZeaZ 5 `

15 2 14 ` 100% 5 6.667% 15

130

BRACKETING METHODS

Recall that the true percent relative error for the root estimate of 15 was 1.3%. Therefore, ea is greater than et. This behavior is manifested for the other iterations: Iteration

xl

xu

xr

1 2 3 4 5 6

12 14 14 14.5 14.75 14.75

16 16 15 15 15 14.875

14 15 14.5 14.75 14.875 14.8125

Ea (%)

et (%)

6.667 3.448 1.695 0.840 0.422

5.413 1.344 2.035 0.345 0.499 0.077

Thus, after six iterations ea finally falls below es 5 0.5%, and the computation can be terminated. These results are summarized in Fig. 5.7. The “ragged” nature of the true error is due to the fact that, for bisection, the true root can lie anywhere within the bracketing interval. The true and approximate errors are far apart when the interval happens to be centered on the true root. They are close when the true root falls at either end of the interval.

Although the approximate error does not provide an exact estimate of the true error, Fig. 5.7 suggests that ea captures the general downward trend of et . In addition, the plot exhibits the extremely attractive characteristic that ea is always greater than et . Thus,

FIGURE 5.7 Errors for the bisection method. True and estimated errors are plotted versus the number of iterations.

10

Percent relative error

Approximate

1.0

True

0.1 0

2

4 Iterations

6

5.2

THE BISECTION METHOD

131

when ea falls below es, the computation could be terminated with confidence that the root is known to be at least as accurate as the prespecified acceptable level. Although it is always dangerous to draw general conclusions from a single example, it can be demonstrated that ea will always be greater than et for the bisection method. This is because each time an approximate root is located using bisection as xr 5 (xl 1 xu)y2, we know that the true root lies somewhere within an interval of (xu 2 xl)y2 5 Dxy2. Therefore, the root must lie within 6Dxy2 of our estimate (Fig. 5.8). For instance, when Example 5.3 was terminated, we could make the definitive statement that xr 5 14.5 6 0.5 Because ¢xy2 5 xrnew 2 xrold (Fig. 5.9), Eq. (5.2) provides an exact upper bound on the true error. For this bound to be exceeded, the true root would have to fall outside the bracketing interval, which, by definition, could never occur for the bisection method. As illustrated in a subsequent example (Example 5.6), other root-locating techniques do not always behave as nicely. Although bisection is generally slower than other methods,

FIGURE 5.8 Three ways in which the interval may bracket the root. In (a) the true value lies at the center of the interval, whereas in (b) and (c) the true value lies near the extreme. Notice that the discrepancy between the true value and the midpoint of the interval never exceeds half the interval length, or Dxy2.

(a)

(b)

xl

xl

xr

xr

xu

xu

xl

(c)

xr

⌬x /2

⌬x /2 True root

FIGURE 5.9 Graphical depiction of why the error estimate for bisection (Dxy2) is equivalent to the root estimate for the present iteration (xrnew) minus the root estimate for the previous iteration (xrold).

xnew – xold r r

Previous iteration xold r xnew r Present iteration

⌬x/2

xu

132

BRACKETING METHODS

the neatness of its error analysis is certainly a positive aspect that could make it attractive for certain engineering applications. Before proceeding to the computer program for bisection, we should note that the relationships (Fig. 5.9) xrnew 2 xold r 5

xu 2 xl 2

and xrnew 5

xl 1 xu 2

can be substituted into Eq. (5.2) to develop an alternative formulation for the approximate percent relative error ea 5 `

xu 2 xl ` 100% xu 1 xl

(5.3)

This equation yields identical results to Eq. (5.2) for bisection. In addition, it allows us to calculate an error estimate on the basis of our initial guesses—that is, on our first iteration. For instance, on the first iteration of Example 5.2, an approximate error can be computed as ea 5 `

16 2 12 ` 100% 5 14.29% 16 1 12

Another benefit of the bisection method is that the number of iterations required to attain an absolute error can be computed a priori—that is, before starting the iterations. This can be seen by recognizing that before starting the technique, the absolute error is Ea0 5 x0u 2 x0l 5 ¢x0 where the superscript designates the iteration. Hence, before starting the method, we are at the “zero iteration.” After the first iteration, the error becomes Ea1 5

¢x0 2

Because each succeeding iteration halves the error, a general formula relating the error and the number of iterations n is Ean 5

¢x0 2n

(5.4)

If Ea,d is the desired error, this equation can be solved for n5

log ( ¢x0 yEa,d ) log 2

5 log2 a

¢x0 b Ea,d

(5.5)

Let us test the formula. For Example 5.4, the initial interval was Dx0 5 16 2 12 5 4. After six iterations, the absolute error was Ea 5

Z14.875 2 14.75Z 2

5 0.0625

5.2

THE BISECTION METHOD

133

We can substitute these values into Eq. (5.5) to give n5

log(4y0.0625) 56 log 2

Thus, if we knew beforehand that an error of less than 0.0625 was acceptable, the formula tells us that six iterations would yield the desired result. Although we have emphasized the use of relative errors for obvious reasons, there will be cases where (usually through knowledge of the problem context) you will be able to specify an absolute error. For these cases, bisection along with Eq. (5.5) can provide a useful root-location algorithm. We will explore such applications in the end-of-chapter problems. 5.2.2 Bisection Algorithm The algorithm in Fig. 5.5 can now be expanded to include the error check (Fig. 5.10). The algorithm employs user-defined functions to make root location and function evaluation more efficient. In addition, an upper limit is placed on the number of iterations. Finally, an error check is included to avoid division by zero during the error evaluation. Such would be the case when the bracketing interval is centered on zero. For this situation, Eq. (5.2) becomes infinite. If this occurs, the program skips over the error evaluation for that iteration. The algorithm in Fig. 5.10 is not user-friendly; it is designed strictly to come up with the answer. In Prob. 5.14 at the end of this chapter, you will have the task of making it easier to use and understand.

FIGURE 5.10 Pseudocode for function to implement bisection.

FUNCTION Bisect(xl, xu, es, imax, xr, iter, ea) iter 5 0 DO xrold 5 xr xr 5 (xl 1 xu) / 2 iter 5 iter 1 1 IF xr ? 0 THEN ea 5 ABS((xr 2 xrold) / xr) * 100 END IF test 5 f(xl) * f(xr) IF test , 0 THEN xu 5 xr ELSE IF test . 0 THEN xl 5 xr ELSE ea 5 0 END IF IF ea , es OR iter $ imax EXIT END DO Bisect 5 xr END Bisect

134

BRACKETING METHODS

5.2.3 Minimizing Function Evaluations The bisection algorithm in Fig. 5.10 is just fine if you are performing a single root evaluation for a function that is easy to evaluate. However, there are many instances in engineering when this is not the case. For example, suppose that you develop a computer program that must locate a root numerous times. In such cases you could call the algorithm from Fig. 5.10 thousands and even millions of times in the course of a single run. Further, in its most general sense, a univariate function is merely an entity that returns a single value in return for a single value you send to it. Perceived in this sense, functions are not always simple formulas like the one-line equations solved in the preceding examples in this chapter. For example, a function might consist of many lines of code that could take a significant amount of execution time to evaluate. In some cases, the function might even represent an independent computer program. Because of both these factors, it is imperative that numerical algorithms minimize function evaluations. In this light, the algorithm from Fig. 5.10 is deficient. In particular, notice that in making two function evaluations per iteration, it recalculates one of the functions that was determined on the previous iteration. Figure 5.11 provides a modified algorithm that does not have this deficiency. We have highlighted the lines that differ from Fig. 5.10. In this case, only the new function value at

FIGURE 5.11 Pseudocode for bisection subprogram which minimizes function evaluations.

FUNCTION Bisect(xl, xu, es, imax, xr, iter, ea) iter 5 0 fl 5 f(xl) DO xrold 5 xr xr 5 (xl 1 xu) / 2 fr 5 f(xr) iter 5 iter 1 1 IF xr ? 0 THEN ea 5 ABS((xr 2 xrold) / xr) * 100 END IF test 5 fl * fr IF test , 0 THEN xu 5 xr ELSE IF test . 0 THEN xl 5 xr fl 5 fr ELSE ea 5 0 END IF IF ea , es OR iter $ imax EXIT END DO Bisect 5 xr END Bisect

5.3

THE FALSE-POSITION METHOD

135

the root estimate is calculated. Previously calculated values are saved and merely reassigned as the bracket shrinks. Thus, n 1 1 function evaluations are performed, rather than 2n.

5.3

THE FALSE-POSITION METHOD Although bisection is a perfectly valid technique for determining roots, its “brute-force” approach is relatively inefficient. False position is an alternative based on a graphical insight. A shortcoming of the bisection method is that, in dividing the interval from xl to xu into equal halves, no account is taken of the magnitudes of f(xl) and f(xu). For example, if f(xl) is much closer to zero than f(xu), it is likely that the root is closer to xl than to xu (Fig. 5.12). An alternative method that exploits this graphical insight is to join f(xl) and f(xu) by a straight line. The intersection of this line with the x axis represents an improved estimate of the root. The fact that the replacement of the curve by a straight line gives a “false position” of the root is the origin of the name, method of false position, or in Latin, regula falsi. It is also called the linear interpolation method. Using similar triangles (Fig. 5.12), the intersection of the straight line with the x axis can be estimated as f(xl ) f(xu ) 5 xr 2 xl xr 2 xu

(5.6)

which can be solved for (see Box 5.1 for details). xr 5 xu 2

FIGURE 5.12 A graphical depiction of the method of false position. Similar triangles used to derive the formula for the method are shaded.

f(xu )(xl 2 xu ) f(xl ) 2 f(xu )

(5.7)

f (x) f (xu)

xr xl xu f (xl)

x

136

BRACKETING METHODS

Box 5.1

Derivation of the Method of False Position then adding and subtracting xu on the right-hand side:

Cross-multiply Eq. (5.6) to yield f (xl ) (xr 2 xu ) 5 f (xu ) (xr 2 xl )

xr 5 xu 1

Collect terms and rearrange:

Collecting terms yields

xr [ f(xl ) 2 f (xu ) ] 5 xu f (xl ) 2 xl f (xu ) Divide by f(xl) 2 f(xu): xr 5

xu f (xl ) 2 xl f (xu ) f (xl ) 2 f (xu )

(B5.1.1)

This is one form of the method of false position. Note that it allows the computation of the root xr as a function of the lower and upper guesses xl and xu. It can be put in an alternative form by expanding it: xr 5

xu f (xl ) xl f (xu ) 2 xu 2 f (xl ) 2 f (xu ) f (xl ) 2 f (xu )

xl f (xu ) xu f (xl ) 2 f (xl ) 2 f (xu ) f (xl ) 2 f (xu )

xr 5 xu 1

xl f (xu ) xu f (xu ) 2 f (xl ) 2 f (xu ) f (xl ) 2 f(xu )

xr 5 xu 2

f (xu ) (xl 2 xu ) f (xl ) 2 f (xu )

or

which is the same as Eq. (5.7). We use this form because it involves one less function evaluation and one less multiplication than Eq. (B5.1.1). In addition, it is directly comparable with the secant method, which will be discussed in Chap. 6.

This is the false-position formula. The value of xr computed with Eq. (5.7) then replaces whichever of the two initial guesses, xl or xu, yields a function value with the same sign as f(xr). In this way, the values of xl and xu always bracket the true root. The process is repeated until the root is estimated adequately. The algorithm is identical to the one for bisection (Fig. 5.5) with the exception that Eq. (5.7) is used for step 2. In addition, the same stopping criterion [Eq. (5.2)] is used to terminate the computation. EXAMPLE 5.5

False Position Problem Statement. Use the false-position method to determine the root of the same equation investigated in Example 5.1 [Eq. (E5.1.1)]. Solution. As in Example 5.3, initiate the computation with guesses of xl 5 12 and xu 5 16. First iteration: xl 5 12 f (xl ) 5 6.1139 xu 5 16 f (xu ) 5 22.2303 22.2303(12 2 16) xr 5 16 2 5 14.309 6.1139 2 (22.2303) which has a true relative error of 0.88 percent. Second iteration: f(xl ) f(xr ) 5 21.5376

5.3

THE FALSE-POSITION METHOD

137

Therefore, the root lies in the first subinterval, and xr becomes the upper limit for the next iteration, xu 5 14.9113: xl 5 12 f(xl ) 5 6.1139 xu 5 14.9309 f(xu ) 5 20.2515 20.2515(12 2 14.9309) xr 5 14.9309 2 5 14.8151 6.1139 2 (20.2515) which has true and approximate relative errors of 0.09 and 0.78 percent. Additional iterations can be performed to refine the estimate of the roots.

A feeling for the relative efficiency of the bisection and false-position methods can be appreciated by referring to Fig. 5.13, where we have plotted the true percent relative errors for Examples 5.4 and 5.5. Note how the error for false position decreases much faster than for bisection because of the more efficient scheme for root location in the false-position method. Recall in the bisection method that the interval between xl and xu grew smaller during the course of a computation. The interval, as defined by ¢xy2 5 Z xu 2 xl Z y2 for the first iteration, therefore provided a measure of the error for this approach. This is not the case

FIGURE 5.13 Comparison of the relative errors of the bisection and the false-position methods. 10

True percent relative error

Bisection 1

10– 1

False position

10– 2

10– 3

10– 4 0

3 Iterations

6

138

BRACKETING METHODS

for the method of false position because one of the initial guesses may stay fixed throughout the computation as the other guess converges on the root. For instance, in Example 5.5 the lower guess xl remained at 12 while xu converged on the root. For such cases, the interval does not shrink but rather approaches a constant value. Example 5.5 suggests that Eq. (5.2) represents a very conservative error criterion. In fact, Eq. (5.2) actually constitutes an approximation of the discrepancy of the previous iteration. This is because for a case such as Example 5.5, where the method is converging quickly (for example, the error is being reduced nearly an order of magnitude per iteration), the root for the present iteration xnew is a much better estimate of the true value r than the result of the previous iteration xold r . Thus, the quantity in the numerator of Eq. (5.2) actually represents the discrepancy of the previous iteration. Consequently, we are assured that satisfaction of Eq. (5.2) ensures that the root will be known with greater accuracy than the prescribed tolerance. However, as described in the next section, there are cases where false position converges slowly. For these cases, Eq. (5.2) becomes unreliable, and an alternative stopping criterion must be developed. 5.3.1 Pitfalls of the False-Position Method Although the false-position method would seem to always be the bracketing method of preference, there are cases where it performs poorly. In fact, as in the following example, there are certain cases where bisection yields superior results. EXAMPLE 5.6

A Case Where Bisection Is Preferable to False Position Problem Statement.

Use bisection and false position to locate the root of

10

f(x) 5 x 2 1 between x 5 0 and 1.3. Solution. Using bisection, the results can be summarized as Iteration

xl

xu

xr

a (%)

t (%)

1 2 3 4 5

0 0.65 0.975 0.975 0.975

1.3 1.3 1.3 1.1375 1.05625

0.65 0.975 1.1375 1.05625 1.015625

100.0 33.3 14.3 7.7 4.0

35 2.5 13.8 5.6 1.6

Thus, after five iterations, the true error is reduced to less than 2 percent. For false position, a very different outcome is obtained: Iteration

xl

xu

xr

a (%)

t (%)

1 2 3 4 5

0 0.09430 0.18176 0.26287 0.33811

1.3 1.3 1.3 1.3 1.3

0.09430 0.18176 0.26287 0.33811 0.40788

48.1 30.9 22.3 17.1

90.6 81.8 73.7 66.2 59.2

5.3

THE FALSE-POSITION METHOD

139

f (x)

10

5

0

1.0

x

FIGURE 5.14 Plot of f (x) 5 x10 2 1, illustrating slow convergence of the false-position method.

After five iterations, the true error has only been reduced to about 59 percent. In addition, note that ea , et . Thus, the approximate error is misleading. Insight into these results can be gained by examining a plot of the function. As in Fig. 5.14, the curve violates the premise upon which false position was based—that is, if f(xl) is much closer to zero than f(xu), then the root is closer to xl than to xu (recall Fig. 5.12). Because of the shape of the present function, the opposite is true. The forgoing example illustrates that blanket generalizations regarding root-location methods are usually not possible. Although a method such as false position is often superior to bisection, there are invariably cases that violate this general conclusion. Therefore, in addition to using Eq. (5.2), the results should always be checked by substituting the root estimate into the original equation and determining whether the result is close to zero. Such a check should be incorporated into all computer programs for root location. The example also illustrates a major weakness of the false-position method: its onesidedness. That is, as iterations are proceeding, one of the bracketing points will tend to

140

BRACKETING METHODS

stay fixed. This can lead to poor convergence, particularly for functions with significant curvature. The following section provides a remedy. 5.3.2 Modiﬁed False Position One way to mitigate the “one-sided” nature of false position is to have the algorithm detect when one of the bounds is stuck. If this occurs, the function value at the stagnant bound can be divided in half. This is called the modified false-position method. The algorithm in Fig. 5.15 implements this strategy. Notice how counters are used to determine when one of the bounds stays fixed for two iterations. If this occurs, the function value at this stagnant bound is halved. The effectiveness of this algorithm can be demonstrated by applying it to Example 5.6. If a stopping criterion of 0.01% is used, the bisection and standard false-position FIGURE 5.15 Pseudocode for the modiﬁed false-position method.

FUNCTION ModFalsePos(xl, xu, es, imax, xr, iter, ea) iter 5 0 fl 5 f(xl) fu 5 f(xu) DO xrold 5 xr xr 5 xu 2 fu * (xl 2 xu) / (fl 2 fu) fr 5 f(xr) iter 5 iter 1 1 IF xr , . 0 THEN ea 5 Abs((xr 2 xrold) / xr) * 100 END IF test 5 fl * fr IF test , 0 THEN xu 5 xr fu 5 f(xu) iu 5 0 il 5 il 1 1 If il $ 2 THEN fl 5 fl / 2 ELSE IF test . 0 THEN xl 5 xr fl 5 f(xl) il 5 0 iu 5 iu 1 1 IF iu $ 2 THEN fu 5 fu / 2 ELSE ea 5 0 END IF IF ea , es OR iter $ imax THEN EXIT END DO ModFalsePos 5 xr End MODFALSEPOS

5.4

INCREMENTAL SEARCHES AND DETERMINING INITIAL GUESSES

141

methods would converge in 14 and 39 iterations, respectively. In contrast, the modified false-position method would converge in 12 iterations. Thus, for this example, it is somewhat more efficient than bisection and is vastly superior to the unmodified falseposition method.

5.4

INCREMENTAL SEARCHES AND DETERMINING INITIAL GUESSES Besides checking an individual answer, you must determine whether all possible roots have been located. As mentioned previously, a plot of the function is usually very useful in guiding you in this task. Another option is to incorporate an incremental search at the beginning of the computer program. This consists of starting at one end of the region of interest and then making function evaluations at small increments across the region. When the function changes sign, it is assumed that a root falls within the increment. The x values at the beginning and the end of the increment can then serve as the initial guesses for one of the bracketing techniques described in this chapter. A potential problem with an incremental search is the choice of the increment length. If the length is too small, the search can be very time consuming. On the other hand, if the length is too great, there is a possibility that closely spaced roots might be missed (Fig. 5.16). The problem is compounded by the possible existence of multiple roots. A partial remedy for such cases is to compute the first derivative of the function f'(x) at the beginning and the end of each interval. If the derivative changes sign, it suggests that a minimum or maximum may have occurred and that the interval should be examined more closely for the existence of a possible root. Although such modifications or the employment of a very fine increment can alleviate the problem, it should be clear that brute-force methods such as incremental search are not foolproof. You would be wise to supplement such automatic techniques with any other information that provides insight into the location of the roots. Such information can be found in plotting and in understanding the physical problem from which the equation originated.

FIGURE 5.16 Cases where roots could be missed because the increment length of the search procedure is too large. Note that the last root on the right is multiple and would be missed regardless of increment length.

f (x)

x0

x1

x2

x3

x4

x5

x6

x

142

BRACKETING METHODS

PROBLEMS Determine the real roots of f (x) 5 20.5x2 1 2.5x 1 4.5: Graphically. Using the quadratic formula. Using three iterations of the bisection method to determine the highest root. Employ initial guesses of xl 5 5 and xu 5 10. Compute the estimated error ea and the true error et after each iteration. 5.2 Determine the real root of f (x) 5 5x3 2 5x2 1 6x 2 2: (a) Graphically. (b) Using bisection to locate the root. Employ initial guesses of xl 5 0 and xu 5 1 and iterate until the estimated error ea falls below a level of es 5 10%. 5.3 Determine the real root of f (x) 5 225 1 82x 2 90x2 1 44x3 2 8x4 1 0.7x5 : (a) Graphically. (b) Using bisection to determine the root to es 5 10%. Employ initial guesses of xl 5 0.5 and xu 5 1.0. (c) Perform the same computation as in (b) but use the falseposition method and es 5 0.2%. 5.4 (a) Determine the roots of f (x) 5 212 2 21x 1 18x2 2 2.75x3 graphically. In addition, determine the first root of the function with (b) bisection, and (c) false position. For (b) and (c) use initial guesses of xl 5 21 and xu 5 0, and a stopping criterion of 1%. 5.5 Locate the first nontrivial root of sin x 5 x2 where x is in radians. Use a graphical technique and bisection with the initial interval from 0.5 to 1. Perform the computation until ea is less than es 5 2%. Also perform an error check by substituting your final answer into the original equation. 5.6 Determine the positive real root of ln (x2 ) 5 0.7 (a) graphically, (b) using three iterations of the bisection method, with initial guesses of xl 5 0.5 and xu 5 2, and (c) using three iterations of the false-position method, with the same initial guesses as in (b). 5.7 Determine the real root of f (x) 5 (0.8 2 0.3x)yx: (a) Analytically. (b) Graphically. (c) Using three iterations of the false-position method and initial guesses of 1 and 3. Compute the approximate error ea and the true error et after each iteration. Is there a problem with the result? 5.8 Find the positive square root of 18 using the false-position method to within es 5 0.5%. Employ initial guesses of xl 5 4 and xu 5 5. 5.9 Find the smallest positive root of the function (x is in radians) x2Z cos 1x Z 5 5 using the false-position method. To locate the region in which the root lies, first plot this function for values of x between 0 and 5. Perform the computation until ea falls below es 5 1%. Check your final answer by substituting it into the original function. 5.1 (a) (b) (c)

5.10 Find the positive real root of f (x) 5 x4 2 8x3 2 35x2 1 450x 2 1001 using the false-position method. Use initial guesses of xl 5 4.5 and xu 5 6 and perform five iterations. Compute both the true and approximate errors based on the fact that the root is 5.60979. Use a plot to explain your results and perform the computation to within es 5 1.0%. 5.11 Determine the real root of x3.5 5 80: (a) analytically and (b) with the false-position method to within es 5 2.5%. Use initial guesses of 2.0 and 5.0. 5.12 Given f (x) 5 22x6 2 1.5x4 1 10x 1 2 Use bisection to determine the maximum of this function. Employ initial guesses of xl 5 0 and xu 5 1, and perform iterations until the approximate relative error falls below 5%. 5.13 The velocity y of a falling parachutist is given by y5

gm ( 1 2 e 2 (cym)t ) c

where g 5 9.81 mys2 . For a parachutist with a drag coefficient c 5 15 kg/s, compute the mass m so that the velocity is y 5 36 m/s at t 5 10 s. Use the false-position method to determine m to a level of es 5 0.1%. 5.14 Use bisection to determine the drag coefficient needed so that an 82-kg parachutist has a velocity of 36 m/s after 4 s of free fall. Note: The acceleration of gravity is 9.81 m/s2. Start with initial guesses of xl 5 3 and xu 5 5 and iterate until the approximate relative error falls below 2%. Also perform an error check by substituting your final answer into the original equation. 5.15 As depicted in Fig. P5.15, the velocity of water, y (m/s), discharged from a cylindrical tank through a long pipe can be computed as y 5 12gH tanh a

H

12gH tb 2L

L v

FIGURE P5.15

PROBLEMS

143

where g 5 9.81 m/s2, H 5 initial head (m), L 5 pipe length (m), and t 5 elapsed time (s). Determine the head needed to achieve y 5 5 m/s in 2.5 s for a 4-m-long pipe (a) graphically, (b) by bisection, and (c) with false position. Employ initial guesses of xl 5 0 and xu 5 2 m with a stopping criterion of e s 5 1%. Check you results. 5.16 Water is flowing in a trapezoidal channel at a rate of Q 5 20 m3/s. The critical depth y for such a channel must satisfy the equation 0512

Q2 gA3c

B 2

2

where g 5 9.81 m/s , Ac 5 the cross-sectional area (m ), and B 5 the width of the channel at the surface (m). For this case, the width and the cross-sectional area can be related to depth y by B 5 3 1 y and Ac 5 3y 1

y2 2

Solve for the critical depth using (a) the graphical method, (b) bisection, and (c) false position. For (b) and (c) use initial guesses of xl 5 0.5 and xu 5 2.5, and iterate until the approximate error falls below 1% or the number of iterations exceeds 10. Discuss your results. 5.17 You are designing a spherical tank (Fig. P5.17) to hold water for a small village in a developing country. The volume of liquid it can hold can be computed as V 5 ph2

[3R 2 h] 3

where V 5 volume (m3), h 5 depth of water in tank (m), and R 5 the tank radius (m).

R

your answer. Determine the approximate relative error after each iteration. Employ initial guesses of 0 and R. 5.18 The saturation concentration of dissolved oxygen in freshwater can be calculated with the equation (APHA, 1992) ln osf 5 2139.34411 1 1

1.575701 3 105 6.642308 3 107 2 Ta T 2a

1.243800 3 1010 8.621949 3 1011 2 T 3a T 4a

where osf 5 the saturation concentration of dissolved oxygen in freshwater at 1 atm (mg/L) and Ta 5 absolute temperature (K). Remember that Ta 5 T 1 273.15, where T 5 temperature (°C). According to this equation, saturation decreases with increasing temperature. For typical natural waters in temperate climates, the equation can be used to determine that oxygen concentration ranges from 14.621 mg/L at 0°C to 6.413 mg/L at 40°C. Given a value of oxygen concentration, this formula and the bisection method can be used to solve for temperature in °C. (a) If the initial guesses are set as 0 and 408C, how many bisection iterations would be required to determine temperature to an absolute error of 0.058C? (b) Develop and test a bisection program to determine T as a function of a given oxygen concentration to a prespecified absolute error as in (a). Given initial guesses of 0 and 408C, test your program for an absolute error 5 0.058C and the following cases: osf 5 8, 10, and 12 mg/L. Check your results. 5.19 According to Archimedes principle, the buoyancy force is equal to the weight of fluid displaced by the submerged portion of an object. For the sphere depicted in Fig. P5.19, use bisection to determine the height h of the portion that is above water. Employ the following values for your computation: r 5 1 m, s 5 density of sphere 5 200 kg/m3, and w 5 density of water 5 1000 kg/m3. Note that the volume of the above-water portion of the sphere can be computed with V5

ph2 (3r 2 h) 3

h V

h

r

FIGURE P5.17 If R 5 3 m, to what depth must the tank be filled so that it holds 30 m3? Use three iterations of the false-position method to determine

FIGURE P5.19

144

BRACKETING METHODS

5.20 Perform the same computation as in Prob. 5.19, but for the frustrum of a cone, as depicted in Fig. P5.20. Employ the following values for your computation: r1 5 0.5 m, r2 5 1 m, h 5 1 m, f 5 frustrum density 5 200 kg/m3, and w 5 water density 5 1000 kg/m3. Note that the volume of a frustrum is given by V5

ph 2 ( r1 1 r22 1 r1r2 ) 3 r1 h1 h

r2

FIGURE P5.20 5.21 Integrate the algorithm outlined in Fig. 5.10 into a complete, user-friendly bisection subprogram. Among other things: (a) Place documentation statements throughout the subprogram to identify what each section is intended to accomplish. (b) Label the input and output.

(c) Add an answer check that substitutes the root estimate into the original function to verify whether the final result is close to zero. (d) Test the subprogram by duplicating the computations from Examples 5.3 and 5.4. 5.22 Develop a subprogram for the bisection method that minimizes function evaluations based on the pseudocode from Fig. 5.11. Determine the number of function evaluations (n) per total iterations. Test the program by duplicating Example 5.6. 5.23 Develop a user-friendly program for the false-position method. The structure of your program should be similar to the bisection algorithm outlined in Fig. 5.10. Test the program by duplicating Example 5.5. 5.24 Develop a subprogram for the false-position method that minimizes function evaluations in a fashion similar to Fig. 5.11. Determine the number of function evaluations (n) per total iterations. Test the program by duplicating Example 5.6. 5.25 Develop a user-friendly subprogram for the modified falseposition method based on Fig. 5.15. Test the program by determining the root of the function described in Example 5.6. Perform a number of runs until the true percent relative error falls below 0.01%. Plot the true and approximate percent relative errors versus number of iterations on semilog paper. Interpret your results. 5.26 Develop a function for bisection in a similar fashion to Fig. 5.10. However, rather than using the maximum iterations and Eq. (5.2), employ Eq. (5.5) as your stopping criterion. Make sure to round the result of Eq. (5.5) up to the next highest integer. Test your function by solving Example 5.3 using Ea,d 5 0.0001.

CHAPTER

6 Open Methods

For the bracketing methods in Chap. 5, the root is located within an interval prescribed by a lower and an upper bound. Repeated application of these methods always results in closer estimates of the true value of the root. Such methods are said to be convergent because they move closer to the truth as the computation progresses (Fig. 6.1a). In contrast, the open methods described in this chapter are based on formulas that require only a single starting value of x or two starting values that do not

FIGURE 6.1 Graphical depiction of the fundamental difference between the (a) bracketing and (b) and (c) open methods for root location. In (a), which is the bisection method, the root is constrained within the interval prescribed by xl and xu. In contrast, for the open method depicted in (b) and (c), a formula is used to project from xi to xi11 in an iterative fashion. Thus, the method can either (b) diverge or (c) converge rapidly, depending on the value of the initial guess.

f (x)

f (x)

xi xl

xu

(a)

xl xl

xu xl xl xu

xi + 1

x

x

(b)

xu f (x)

xu

xi xi + 1

x

(c) 145

146

OPEN METHODS

necessarily bracket the root. As such, they sometimes diverge or move away from the true root as the computation progresses (Fig. 6.1b). However, when the open methods converge (Fig. 6.1c), they usually do so much more quickly than the bracketing methods. We will begin our discussion of open techniques with a simple version that is useful for illustrating their general form and also for demonstrating the concept of convergence.

6.1

SIMPLE FIXED-POINT ITERATION As mentioned above, open methods employ a formula to predict the root. Such a formula can be developed for simple fixed-point iteration (or, as it is also called, one-point iteration or successive substitution) by rearranging the function f(x) 5 0 so that x is on the left-hand side of the equation: x 5 g(x)

(6.1)

This transformation can be accomplished either by algebraic manipulation or by simply adding x to both sides of the original equation. For example, x2 2 2x 1 3 5 0 can be simply manipulated to yield x5

x2 1 3 2

whereas sin x 5 0 could be put into the form of Eq. (6.1) by adding x to both sides to yield x 5 sin x 1 x The utility of Eq. (6.1) is that it provides a formula to predict a new value of x as a function of an old value of x. Thus, given an initial guess at the root xi, Eq. (6.1) can be used to compute a new estimate xi11 as expressed by the iterative formula xi11 5 g(xi )

(6.2)

As with other iterative formulas in this book, the approximate error for this equation can be determined using the error estimator [Eq. (3.5)]: ea 5 `

EXAMPLE 6.1

xi11 2 xi ` 100% xi11

Simple Fixed-Point Iteration Problem Statement. Use simple fixed-point iteration to locate the root of f(x) 5 e2x 2 x. Solution. The function can be separated directly and expressed in the form of Eq. (6.2) as xi11 5 e 2xi

6.1

SIMPLE FIXED-POINT ITERATION

147

Starting with an initial guess of x0 5 0, this iterative equation can be applied to compute i

xi

Ea (%)

Et (%)

0 1 2 3 4 5 6 7 8 9 10

0 1.000000 0.367879 0.692201 0.500473 0.606244 0.545396 0.579612 0.560115 0.571143 0.564879

100.0 171.8 46.9 38.3 17.4 11.2 5.90 3.48 1.93 1.11

100.0 76.3 35.1 22.1 11.8 6.89 3.83 2.20 1.24 0.705 0.399

Thus, each iteration brings the estimate closer to the true value of the root: 0.56714329.

6.1.1 Convergence Notice that the true percent relative error for each iteration of Example 6.1 is roughly proportional (by a factor of about 0.5 to 0.6) to the error from the previous iteration. This property, called linear convergence, is characteristic of fixed-point iteration. Aside from the “rate” of convergence, we must comment at this point about the “possibility” of convergence. The concepts of convergence and divergence can be depicted graphically. Recall that in Sec. 5.1, we graphed a function to visualize its structure and behavior (Example 5.1). Such an approach is employed in Fig. 6.2a for the function f(x) 5 e2x 2 x. An alternative graphical approach is to separate the equation into two component parts, as in f1 (x) 5 f2 (x) Then the two equations y1 5 f1 (x)

(6.3)

y2 5 f2 (x)

(6.4)

and

can be plotted separately (Fig. 6.2b). The x values corresponding to the intersections of these functions represent the roots of f(x) 5 0. EXAMPLE 6.2

The Two-Curve Graphical Method Problem Statement. its root graphically.

Separate the equation e2x 2 x 5 0 into two parts and determine

148

OPEN METHODS

Solution. Reformulate the equation as y1 5 x and y2 5 e2x. The following values can be computed: x

y1

y2

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

1.000 0.819 0.670 0.549 0.449 0.368

These points are plotted in Fig. 6.2b. The intersection of the two curves indicates a root estimate of approximately x 5 0.57, which corresponds to the point where the single curve in Fig. 6.2a crosses the x axis.

FIGURE 6.2 Two alternative graphical methods for determining the root of f(x) 5 e2x 2 x. (a) Root at the point where it crosses the x axis; (b) root at the intersection of the component functions.

f (x) f (x) = e– x – x Root x

(a) f (x) f 1(x) = x

f 2(x) = e– x

Root x

(b)

6.1

SIMPLE FIXED-POINT ITERATION

149

The two-curve method can now be used to illustrate the convergence and divergence of fixed-point iteration. First, Eq. (6.1) can be reexpressed as a pair of equations y1 5 x and y2 5 g(x). These two equations can then be plotted separately. As was the case with Eqs. (6.3) and (6.4), the roots of f(x) 5 0 correspond to the abscissa value at the intersection of the two curves. The function y1 5 x and four different shapes for y2 5 g(x) are plotted in Fig. 6.3. For the first case (Fig. 6.3a), the initial guess of x0 is used to determine the corresponding point on the y2 curve [x0, g(x0)]. The point (x1, x1) is located by moving left horizontally to the y1 curve. These movements are equivalent to the first iteration in the fixed-point method: x1 5 g(x0 ) Thus, in both the equation and in the plot, a starting value of x0 is used to obtain an estimate of x1. The next iteration consists of moving to [x1, g(x1)] and then to (x2, x2). This iteration is equivalent to the equation x2 5 g(x1 )

FIGURE 6.3 Iteration cobwebs depicting convergence (a and b) and divergence (c and d) of simple ﬁxed-point iteration. Graphs (a) and (c) are called monotone patterns, whereas (b) and (d) are called oscillating or spiral patterns. Note that convergence occurs when |g9(x)| , 1.

y

y y1 = x

y1 = x y2 = g(x)

y2 = g(x)

x2 x1

x0

x0

x

(a)

x

(b)

y

y y2 = g(x)

y2 = g(x) y1 = x

y1 = x

x0

x

(c)

x0

(d)

x

150

OPEN METHODS

Box 6.1

Convergence of Fixed-Point Iteration

From studying Fig. 6.3, it should be clear that fixed-point iteration converges if, in the region of interest, ug9(x)u , 1. In other words, convergence occurs if the magnitude of the slope of g(x) is less than the slope of the line f(x) 5 x. This observation can be demonstrated theoretically. Recall that the iterative equation is xi11 5 g(xi )

xr 5 g(xr )

(B6.1.3)

Et,i 5 xr 2 xi (B6.1.1)

The derivative mean-value theorem (recall Sec. 4.1.1) states that if a function g(x) and its first derivative are continuous over an interval a # x # b, then there exists at least one value of x 5 j within the interval such that g(b) 2 g(a) b2a

where j is somewhere between xi and xr. This result can then be substituted into Eq. (B6.1.1) to yield

If the true error for iteration i is defined as

Subtracting these equations yields

g¿(j) 5

g(xr ) 2 g(xi ) 5 (xr 2 xi )g¿(j)

xr 2 x i11 5 (xr 2 xi )g¿(j)

Suppose that the true solution is

xr 2 xi11 5 g(xr ) 2 g(xi )

Now, if we let a 5 xi and b 5 xr, the right-hand side of Eq. (B6.1.1) can be expressed as

(B6.1.2)

The right-hand side of this equation is the slope of the line joining g(a) and g(b). Thus, the mean-value theorem states that there is at least one point between a and b that has a slope, designated by g9(j), which is parallel to the line joining g(a) and g(b) (recall Fig. 4.3).

then Eq. (B6.1.3) becomes Et,i11 5 g¿(j)Et,i Consequently, if ug9(x)u , 1, the errors decrease with each iteration. For ug9(x)u . 1, the errors grow. Notice also that if the derivative is positive, the errors will be positive, and hence, the iterative solution will be monotonic (Fig. 6.3a and c). If the derivative is negative, the errors will oscillate (Fig. 6.3b and d). An offshoot of the analysis is that it also demonstrates that when the method converges, the error is roughly proportional to and less than the error of the previous step. For this reason, simple fixedpoint iteration is said to be linearly convergent.

The solution in Fig. 6.3a is convergent because the estimates of x move closer to the root with each iteration. The same is true for Fig. 6.3b. However, this is not the case for Fig. 6.3c and d, where the iterations diverge from the root. Notice that convergence seems to occur only when the absolute value of the slope of y2 5 g(x) is less than the slope of y1 5 x, that is, when ug9(x)u , 1. Box 6.1 provides a theoretical derivation of this result.

6.1.2 Algorithm for Fixed-Point Iteration The computer algorithm for fixed-point iteration is extremely simple. It consists of a loop to iteratively compute new estimates until the termination criterion has been met. Figure 6.4 presents pseudocode for the algorithm. Other open methods can be programmed in a similar way, the major modification being to change the iterative formula that is used to compute the new root estimate.

6.2

FIGURE 6.4 Pseudocode for ﬁxed-point iteration. Note that other open methods can be cast in this general format.

THE NEWTON-RAPHSON METHOD

151

FUNCTION Fixpt(x0, es, imax, iter, ea) xr 5 x0 iter 5 0 DO xrold 5 xr xr 5 g(xrold) iter 5 iter 1 1 IF xr ? O THEN xr 2 xrold ` ? 100 ea 5 ` xr END IF IF ea , es OR iter $ imax EXIT END DO Fixpt 5 xr END Fixpt

f (x) Slope = f ' (x i) f (x i) f (x i) – 0

FIGURE 6.5 Graphical depiction of the Newton-Raphson method. A tangent to the function of xi [that is, f9(xi)] is extrapolated down to the x axis to provide an estimate of the root at xi11.

6.2

0

xi+1

xi

x

xi – xi+1

THE NEWTON-RAPHSON METHOD Perhaps the most widely used of all root-locating formulas is the Newton-Raphson equation (Fig. 6.5). If the initial guess at the root is xi, a tangent can be extended from the point [xi, f(xi)]. The point where this tangent crosses the x axis usually represents an improved estimate of the root.

152

OPEN METHODS

The Newton-Raphson method can be derived on the basis of this geometrical interpretation (an alternative method based on the Taylor series is described in Box 6.2). As in Fig. 6.5, the first derivative at x is equivalent to the slope: f ¿(xi ) 5

f (xi ) 2 0 xi 2 xi11

(6.5)

which can be rearranged to yield xi11 5 xi 2

f(xi ) f ¿(xi )

(6.6)

which is called the Newton-Raphson formula. EXAMPLE 6.3

Newton-Raphson Method Problem Statement. Use the Newton-Raphson method to estimate the root of f(x) 5 e2x 2 x, employing an initial guess of x0 5 0. Solution. The first derivative of the function can be evaluated as f ¿(x) 5 2e2x 2 1 which can be substituted along with the original function into Eq. (6.6) to give xi11 5 xi 2

e2xi 2 xi 2e2xi 2 1

Starting with an initial guess of x0 5 0, this iterative equation can be applied to compute i

xi

Et (%)

0 1 2 3 4

0 0.500000000 0.566311003 0.567143165 0.567143290

100 11.8 0.147 0.0000220 , 1028

Thus, the approach rapidly converges on the true root. Notice that the true percent relative error at each iteration decreases much faster than it does in simple fixed-point iteration (compare with Example 6.1).

6.2.1 Termination Criteria and Error Estimates As with other root-location methods, Eq. (3.5) can be used as a termination criterion. In addition, however, the Taylor series derivation of the method (Box 6.2) provides theoretical insight regarding the rate of convergence as expressed by Ei11 5 O(E 2i ). Thus the error should be roughly proportional to the square of the previous error. In other words,

6.2

Box 6.2

THE NEWTON-RAPHSON METHOD

Derivation and Error Analysis of the Newton-Raphson Method

Aside from the geometric derivation [Eqs. (6.5) and (6.6)], the Newton-Raphson method may also be developed from the Taylor series expansion. This alternative derivation is useful in that it also provides insight into the rate of convergence of the method. Recall from Chap. 4 that the Taylor series expansion can be represented as

be obtained. For this situation xi11 5 xr, where x is the true value of the root. Substituting this value along with f(xr) 5 0 into Eq. (B6.2.1) yields 0 5 f (xi ) 1 f ¿(xi ) (xr 2 xi ) 1

f –(j) (xr 2 xi ) 2 2!

(B6.2.3)

Equation (B6.2.2) can be subtracted from Eq. (B6.2.3) to give

f (xi11 ) 5 f (xi ) 1 f ¿(xi ) (xi11 2 xi ) f –(j) (xi11 2 xi ) 2 1 2!

(B6.2.1)

where j lies somewhere in the interval from xi to xi11. An approximate version is obtainable by truncating the series after the first derivative term: f (xi11 ) > f (xi ) 1 f ¿(xi ) (xi11 2 xi )

0 5 f ¿(xi ) (xr 2 xi11 ) 1

f –(j) (xr 2 xi ) 2 2!

(B6.2.4)

Now, realize that the error is equal to the discrepancy between xi11 and the true value xr, as in Et,i11 5 xr 2 xi11 and Eq. (B6.2.4) can be expressed as

At the intersection with the x axis, f(xi11) would be equal to zero, or 0 5 f (xi ) 1 f ¿(xi ) (xi11 2 xi )

(B6.2.2)

which can be solved for xi11 5 xi 2

153

f (xi ) f ¿(xi )

0 5 f ¿(xi )Et,i11 1

(B6.2.5)

If we assume convergence, both xi and j should eventually be approximated by the root xr, and Eq. (B6.2.5) can be rearranged to yield Et,i11 5

which is identical to Eq. (6.6). Thus, we have derived the NewtonRaphson formula using a Taylor series. Aside from the derivation, the Taylor series can also be used to estimate the error of the formula. This can be done by realizing that if the complete Taylor series were employed, an exact result would

f –(j) 2 E t,i 2!

2f –(xr ) 2 E t,i 2 f ¿(xr )

(B6.2.6)

According to Eq. (B6.2.6), the error is roughly proportional to the square of the previous error. This means that the number of correct decimal places approximately doubles with each iteration. Such behavior is referred to as quadratic convergence. Example 6.4 manifests this property.

the number of significant figures of accuracy approximately doubles with each iteration. This behavior is examined in the following example. EXAMPLE 6.4

Error Analysis of Newton-Raphson Method Problem Statement. As derived in Box 6.2, the Newton-Raphson method is quadratically convergent. That is, the error is roughly proportional to the square of the previous error, as in Et,i11 >

2f –(xr ) 2 E t,i 2f ¿(xr )

Examine this formula and see if it applies to the results of Example 6.3. Solution. The first derivative of f(x) 5 e2x 2 x is f ¿(x) 5 2e2x 2 1

(E6.4.1)

154

OPEN METHODS

which can be evaluated at xr 5 0.56714329 as f 9(0.56714329) 5 21.56714329. The second derivative is f –(x) 5 e2x which can be evaluated as f 0(0.56714329) 5 0.56714329. These results can be substituted into Eq. (E6.4.1) to yield Et,i11 > 2

0.56714329 E 2t,i 5 0.18095E 2t,i 2(21.56714329)

From Example 6.3, the initial error was Et,0 5 0.56714329, which can be substituted into the error equation to predict Et,1 > 0.18095(0.56714329) 2 5 0.0582 which is close to the true error of 0.06714329. For the next iteration, Et,2 > 0.18095(0.06714329) 2 5 0.0008158 which also compares favorably with the true error of 0.0008323. For the third iteration, Et,3 > 0.18095(0.0008323) 2 5 0.000000125 which is the error obtained in Example 6.3. The error estimate improves in this manner because, as we come closer to the root, x and j are better approximated by xr [recall our assumption in going from Eq. (B6.2.5) to Eq. (B6.2.6) in Box 6.2]. Finally, Et,4 > 0.18095(0.000000125) 2 5 2.83 3 10 215 Thus, this example illustrates that the error of the Newton-Raphson method for this case is, in fact, roughly proportional (by a factor of 0.18095) to the square of the error of the previous iteration.

6.2.2 Pitfalls of the Newton-Raphson Method Although the Newton-Raphson method is often very efficient, there are situations where it performs poorly. A special case—multiple roots—will be addressed later in this chapter. However, even when dealing with simple roots, difficulties can also arise, as in the following example. EXAMPLE 6.5

Example of a Slowly Converging Function with Newton-Raphson Problem Statement. Determine the positive root of f(x) 5 x10 2 1 using the NewtonRaphson method and an initial guess of x 5 0.5. Solution. The Newton-Raphson formula for this case is xi11 5 xi 2

x 10 i 2 1 10x 9i

which can be used to compute

6.2

THE NEWTON-RAPHSON METHOD

Iteration

x

0 1 2 3 4 5 . . . `

0.5 51.65 46.485 41.8365 37.65285 33.887565

155

1.0000000

Thus, after the first poor prediction, the technique is converging on the true root of 1, but at a very slow rate.

Aside from slow convergence due to the nature of the function, other difficulties can arise, as illustrated in Fig. 6.6. For example, Fig. 6.6a depicts the case where an inflection point [that is, f 0(x) 5 0] occurs in the vicinity of a root. Notice that iterations beginning at x0 progressively diverge from the root. Figure 6.6b illustrates the tendency of the Newton-Raphson technique to oscillate around a local maximum or minimum. Such oscillations may persist, or as in Fig. 6.6b, a near-zero slope is reached, whereupon the solution is sent far from the area of interest. Figure 6.6c shows how an initial guess that is close to one root can jump to a location several roots away. This tendency to move away from the area of interest is because nearzero slopes are encountered. Obviously, a zero slope [ f 9(x) 5 0] is truly a disaster because it causes division by zero in the Newton-Raphson formula [Eq. (6.6)]. Graphically (see Fig 6.6d), it means that the solution shoots off horizontally and never hits the x axis. Thus, there is no general convergence criterion for Newton-Raphson. Its convergence depends on the nature of the function and on the accuracy of the initial guess. The only remedy is to have an initial guess that is “sufficiently” close to the root. And for some functions, no guess will work! Good guesses are usually predicated on knowledge of the physical problem setting or on devices such as graphs that provide insight into the behavior of the solution. The lack of a general convergence criterion also suggests that good computer software should be designed to recognize slow convergence or divergence. The next section addresses some of these issues. 6.2.3 Algorithm for Newton-Raphson An algorithm for the Newton-Raphson method is readily obtained by substituting Eq. (6.6) for the predictive formula [Eq. (6.2)] in Fig. 6.4. Note, however, that the program must also be modified to compute the first derivative. This can be simply accomplished by the inclusion of a user-defined function.

156

OPEN METHODS

Additionally, in light of the foregoing discussion of potential problems of the NewtonRaphson method, the program would be improved by incorporating several additional features:

f (x)

x1

x2

x0

x

(a) f (x)

x0 x2 x4

x3

x1

x

(b) f (x) x2

x1 x0

x

(c) f (x)

x0

x1

(d) FIGURE 6.6 Four cases where the Newton-Raphson method exhibits poor convergence.

x

6.3

THE SECANT METHOD

157

1. A plotting routine should be included in the program. 2. At the end of the computation, the final root estimate should always be substituted into the original function to compute whether the result is close to zero. This check partially guards against those cases where slow or oscillating convergence may lead to a small value of ea while the solution is still far from a root. 3. The program should always include an upper limit on the number of iterations to guard against oscillating, slowly convergent, or divergent solutions that could persist interminably. 4. The program should alert the user and take account of the possibility that f 9(x) might equal zero at any time during the computation.

6.3

THE SECANT METHOD A potential problem in implementing the Newton-Raphson method is the evaluation of the derivative. Although this is not inconvenient for polynomials and many other functions, there are certain functions whose derivatives may be extremely difficult or inconvenient to evaluate. For these cases, the derivative can be approximated by a backward finite divided difference, as in (Fig. 6.7) f ¿(xi ) >

f (xi21 ) 2 f (xi ) xi21 2 xi

This approximation can be substituted into Eq. (6.6) to yield the following iterative equation: f (xi )(xi21 2 xi ) f (xi21 ) 2 f (xi )

xi11 5 xi 2

FIGURE 6.7 Graphical depiction of the secant method. This technique is similar to the Newton-Raphson technique (Fig. 6.5) in the sense that an estimate of the root is predicted by extrapolating a tangent of the function to the x axis. However, the secant method uses a difference rather than a derivative to estimate the slope.

(6.7)

f (x ) f (x i )

f (x i – 1)

xi – 1

xi

x

158

OPEN METHODS

Equation (6.7) is the formula for the secant method. Notice that the approach requires two initial estimates of x. However, because f(x) is not required to change signs between the estimates, it is not classified as a bracketing method. EXAMPLE 6.6

The Secant Method Problem Statement. Use the secant method to estimate the root of f(x) 5 e2x 2 x. Start with initial estimates of x21 5 0 and x0 5 1.0. Solution. Recall that the true root is 0.56714329. . . . First iteration: x21 5 0 f (x21 ) 5 1.00000

x0 5 1 f(x0 ) 5 20.63212 x1 5 1 2

20.63212(0 2 1) 5 0.61270 et 5 8.0% 1 2 (20.63212)

Second iteration: x0 5 1 x1 5 0.61270

f (x0 ) 5 20.63212 f (x1 ) 5 20.07081

(Note that both estimates are now on the same side of the root.) x2 5 0.61270 2

20.07081(1 2 0.61270) 5 0.56384 et 5 0.58% 20.63212 2 (20.07081)

Third iteration: x1 5 0.61270 f (x1 ) 5 20.07081

x2 5 0.56384 f (x2 ) 5 0.00518

x3 5 0.56384 2

0.00518(0.61270 2 0.56384) 5 0.56717 et 5 0.0048% 20.07081 2 (20.00518)

6.3.1 The Difference Between the Secant and False-Position Methods Note the similarity between the secant method and the false-position method. For example, Eqs. (6.7) and (5.7) are identical on a term-by-term basis. Both use two initial estimates to compute an approximation of the slope of the function that is used to project to the x axis for a new estimate of the root. However, a critical difference between the methods is how one of the initial values is replaced by the new estimate. Recall that in the false-position method the latest estimate of the root replaces whichever of the original values yielded a function value with the same sign as f(xr). Consequently, the two estimates always bracket the root. Therefore, for all practical purposes, the method always converges because the root is kept within the bracket. In contrast, the secant method replaces the values in strict sequence, with the new value xi11 replacing xi and xi replacing xi21. As a result, the two values can sometimes lie on the same side of the root. For certain cases, this can lead to divergence.

6.3

EXAMPLE 6.7

THE SECANT METHOD

159

Comparison of Convergence of the Secant and False-Position Techniques Problem Statement. Use the false-position and secant methods to estimate the root of f(x) 5 ln x. Start the computation with values of xl 5 xi21 5 0.5 and xu 5 xi 5 5.0. Solution. For the false-position method, the use of Eq. (5.7) and the bracketing criterion for replacing estimates results in the following iterations: Iteration

xl

xu

xr

1 2 3

0.5 0.5 0.5

5.0 1.8546 1.2163

1.8546 1.2163 1.0585

As can be seen (Fig. 6.8a and c), the estimates are converging on the true root which is equal to 1.

FIGURE 6.8 Comparison of the false-position and the secant methods. The ﬁrst iterations (a) and (b) for both techniques are identical. However, for the second iterations (c) and (d), the points used differ. As a consequence, the secant method can diverge, as indicated in (d). Secant

False position f (x)

f (x u )

f (x)

f (x i )

x

xr

x

xr

f (x l )

f (x i – 1)

(a)

(b)

f (x)

f (x)

f (x i – 1)

f (x i )

f (x u ) xr x

xr

x

f (x l )

(c)

(d)

160

OPEN METHODS

For the secant method, using Eq. (6.7) and the sequential criterion for replacing estimates results in Iteration

xi⫺1

xi

xi⫹1

1 2

0.5 5.0

5.0 1.8546

1.8546 0.10438

As in Fig. 6.8d, the approach is divergent.

Although the secant method may be divergent, when it converges it usually does so at a quicker rate than the false-position method. For instance, Fig. 6.9 demonstrates the superiority of the secant method in this regard. The inferiority of the false-position method is due to one end staying fixed to maintain the bracketing of the root. This property, which is an advantage in that it prevents divergence, is a shortcoming with regard to the rate of convergence; it makes the finite-difference estimate a less-accurate approximation of the derivative.

FIGURE 6.9 Comparison of the true percent relative errors et for the methods to determine the roots of f(x) 5 e2x 2 x.

10

10– 1

10– 2

t ec is

n io

Secant

10– 4

ion posit False

10– 3

B

Newton-Raphson

True percent relative error

1

10– 5

10– 6

20 Iterations

6.3

THE SECANT METHOD

161

6.3.2 Algorithm for the Secant Method As with the other open methods, an algorithm for the secant method is obtained simply by modifying Fig. 6.4 so that two initial guesses are input and by using Eq. (6.7) to calculate the root. In addition, the options suggested in Sec. 6.2.3 for the Newton-Raphson method can also be applied to good advantage for the secant program. 6.3.3 Modiﬁed Secant Method Rather than using two arbitrary values to estimate the derivative, an alternative approach involves a fractional perturbation of the independent variable to estimate f9(x), f ¿(xi ) >

f(xi 1 dxi ) 2 f(xi ) dxi

where d 5 a small perturbation fraction. This approximation can be substituted into Eq. (6.6) to yield the following iterative equation: xi11 5 xi 2

dxi f (xi ) f (xi 1 dxi ) 2 f (xi )

EXAMPLE 6.8

(6.8)

Modiﬁed Secant Method Problem Statement. Use the modified secant method to estimate the root of f(x) 5 e2x 2 x. Use a value of 0.01 for d and start with x0 5 1.0. Recall that the true root is 0.56714329. . . . Solution. First iteration: x0 5 1 f(x0 ) 5 20.63212 x0 1 dx0 5 1.01 f(x0 1 dx0 ) 5 20.64578 0.01(20.63212) x1 5 1 2 5 0.537263 Zet Z 5 5.3% 20.64578 2 (20.63212) Second iteration: x0 5 0.537263 f(x0 ) 5 0.047083 x0 1 dx0 5 0.542635 f(x0 1 dx0 ) 5 0.038579 0.005373(0.047083) x1 5 0.537263 2 5 0.56701 Zet Z 5 0.0236% 0.038579 2 0.047083 Third iteration: x0 5 0.56701 f(x0 ) 5 0.000209 x0 1 dx0 5 0.572680 f(x0 1 dx0 ) 5 20.00867 0.00567(0.000209) x1 5 0.56701 2 5 0.567143 Zet Z 5 2.365 3 1025 % 20.00867 2 0.000209

162

OPEN METHODS

The choice of a proper value for ␦ is not automatic. If ␦ is too small, the method can be swamped by round-off error caused by subtractive cancellation in the denominator of Eq. (6.8). If it is too big, the technique can become inefficient and even divergent. However, if chosen correctly, it provides a nice alternative for cases where evaluating the derivative is difficult and developing two initial guesses is inconvenient.

6.4

BRENT’S METHOD Wouldn’t it be nice to have a hybrid approach that combined the reliability of bracketing with the speed of the open methods? Brent’s root-location method is a clever algorithm that does just that by applying a speedy open method wherever possible, but reverting to a reliable bracketing method if necessary. The approach was developed by Richard Brent (1973) based on an earlier algorithm of Theodorus Dekker (1969). The bracketing technique is the trusty bisection method (Sec. 5.2) whereas two different open methods are employed. The first is the secant method described in Sec. 6.3. As explained next, the second is inverse quadratic interpolation. 6.4.1 Inverse Quadratic Interpolation Inverse quadratic interpolation is similar in spirit to the secant method. As in Fig. 6.10a, the secant method is based on computing a straight line that goes through two guesses. The intersection of this straight line with the x axis represents the new root estimate. For this reason, it is sometimes referred to as a linear interpolation method. Now suppose that we had three points. In that case, we could determine a quadratic function of x that goes through the three points (Fig. 6.10b). Just as with the linear secant method, the intersection of this parabola with the x axis would represent the new root estimate. And as illustrated in Fig. 6.10b, using a curve rather than a straight line often yields a better estimate. Although this would seem to represent a great improvement, the approach has a fundamental flaw: It is possible that the parabola might not intersect the x axis! Such would be the case when the resulting parabola had complex roots. This is illustrated by the parabola, y 5 f(x), in Fig. 6.11.

FIGURE 6.10 Comparison of (a) the secant method and (b) inverse quadratic interpolation. Note that the dark parabola passing through the three points in (b) is called “inverse” because it is written in y rather than in x.

f (x)

f (x)

x

(a)

x

(b)

6.4

BRENT’S METHOD

163

y 6

x = f (y) 4

FIGURE 6.11 Two parabolas ﬁt to three points. The parabola written as a function of x, y 5 f (x), has complex roots and hence does not intersect the x axis. In contrast, if the variables are reversed, and the parabola developed as x 5 f (y), the function does intersect the x axis.

y = f (x) 2

Root 0

1

2

3

5

x

The difficulty can be rectified by employing inverse quadratic interpolation. That is, rather than using a parabola in x, we can fit the points with a parabola in y. This amounts to reversing the axes and creating a “sideways” parabola [the curve, x 5 f(y), in Fig. 6.11]. If the three points are designated as (xi22, yi22), (xi21, yi21), and (xi, yi), a quadratic function of y that passes through the points can be generated as g(y) 5

(y 2 yi21 )(y 2 yi ) (y 2 yi22 )(y 2 yi ) xi22 1 xi21 (yi22 2 yi21 )(yi22 2 yi ) (yi21 2 yi22 )(yi21 2 yi ) 1

(y 2 yi22 )(y 2 yi21 ) xi (yi 2 yi22 )(yi 2 yi21 )

(6.9)

As we will learn in Sec. 18.2, this form is called a Lagrange polynomial. The root, xi11, corresponds to y 5 0, which when substituted into Eq. (6.9) yields xi11 5

(yi22

yi21 yi yi22 yi xi22 1 xi21 2 yi21 )(yi22 2 yi ) (yi21 2 yi21 2 yi ) 1

yi22 yi21 xi (yi 2 yi22 )(yi 2 yi21 )

(6.10)

As shown in Fig. 6.11, such a “sideways” parabola always intersects the x axis. EXAMPLE 6.9

Inverse Quadratic Interpolation Problem Statement. Develop quadratic equations in both x and y for the data points depicted in Fig. 6.11: (1, 2), (2, 1), and (4, 5). For the first, y 5 f(x), employ the quadratic formula to illustrate that the roots are complex. For the latter, x 5 g(y), use inverse quadratic interpolation (Eq. 6.10) to determine the root estimate.

164

OPEN METHODS

Solution. By reversing the x’s and y’s, Eq. (6.9) can be used to generate a quadratic in x as f(x) 5

(x 2 1) (x 2 4) (x 2 1) (x 2 2) (x 2 2) (x 2 4) 21 11 5 (1 2 2)(1 2 4) (2 2 1)(2 2 4) (4 2 1)(4 2 2)

or collecting terms f(x) 5 x2 2 4x 1 5 This equation was used to generate the parabola, y 5 f(x), in Fig. 6.11. The quadratic formula can be used to determine that the roots for this case are complex, x5

4 6 2(24) 2 2 4(1)(5) 52 6 i 2

Equation (6.9) can be used to generate the quadratic in y as g(y) 5

(y 2 1) (y 2 5) (y 2 2) (y 2 5) (y 2 2) (y 2 1) 11 21 4 (2 2 1)(2 2 5) (1 2 2)(1 2 5) (5 2 2)(5 2 1)

or collecting terms g(y) 5 0.5x2 2 2.5x 1 4 Finally, Eq. (6.10) can be used to determine the root as xi11 5

21(25) 22(25) 22(21) 11 21 454 (2 2 1)(2 2 5) (1 2 2)(1 2 5) (5 2 2)(5 2 1)

Before proceeding to Brent’s algorithm, we need to mention one more case where inverse quadratic interpolation does not work. If the three y values are not distinct (that is, yi22 5 yi21 or yi21 5 yi), an inverse quadratic function does not exist. So this is where the secant method comes into play. If we arrive at a situation where the y values are not distinct, we can always revert to the less efficient secant method to generate a root using two of the points. If yi22 5 yi21, we use the secant method with xi21 and xi. If yi21 5 yi, we use xi22 and xi21. 6.4.2 Brent’s Method Algorithm The general idea behind the Brent’s root finding method is whenever possible to use one of the quick open methods. In the event that these generate an unacceptable result (i.e., a root estimate that falls outside the bracket), the algorithm reverts to the more conservative bisection method. Although bisection may be slower, it generates an estimate guaranteed to fall within the bracket. This process is then repeated until the root is located to within an acceptable tolerance. As might be expected, bisection typically dominates at first but as the root is approached, the technique shifts to the faster open methods. Figure 6.12 presents pseudocode for the algorithm based on a MATLAB software M-file developed by Cleve Moler (2005). It represents a stripped down version of

6.4

FIGURE 6.12 Pseudocode for Brent’s root ﬁnding algorithm based on a MATLAB m-ﬁle developed by Cleve Moler (2005).

BRENT’S METHOD

Function fzerosimp(xl, xu) eps 5 2.22044604925031E-16 tol 5 0.000001 a 5 xl: b 5 xu: fa 5 f(a): fb 5 f(b) c 5 a: fc 5 fa: d 5 b 2 c: e 5 d DO IF fb 5 0 EXIT IF Sgn(fa) 5 Sgn(fb) THEN (If necessary, rearrange points) a 5 c: fa 5 fc: d 5 b 2 c: e 5 d ENDIF IF |fa| , |fb| THEN c 5 b: b 5 a: a 5 c fc 5 fb: fb 5 fa: fa 5 fc ENDIF m 5 0.5 * (a 2 b) (Termination test and possible exit) tol 5 2 * eps * max(|b|, 1) IF |m| # tol Or fb 5 0. THEN EXIT ENDIF (Choose open methods or bisection) IF |e| $ tol And |fc| . |fb| THEN s 5 fb / fc IF a 5 c THEN (Secant method) p 52 * m *s q 51 2 s ELSE (Inverse quadratic interpolation) q 5 fc / fa: r 5 fb / fa p 5 s * (2 * m * q * (q 2 r) 2 (b 2 c) * (r 2 1)) q 5 (q 2 1) * (r 2 1) * (s 2 1) ENDIF IF p . 0 THEN q 5 2q ELSE p 5 2p IF 2 * p , 3 * m * q 2 |tol * q| AND p , |0.5 * e * q| THEN e 5 d: d 5 p / q ELSE d 5 m: e 5 m ENDIF ELSE (Bisection) d 5 m: e 5 m ENDIF c 5 b: fc 5 fb IF |d| . tol THEN b 5 b 1 d Else b 5 b 2 Sgn(b 2 a) * tol fb 5 f(b) ENDDO fzerosimp 5 b END fzerosimp

165

166

OPEN METHODS

the fzero function which is the professional root location function employed in MATLAB. For that reason, we call the simplified version: fzerosimp. Note that it requires another function, f, that holds the equation for which the root is being evaluated. The fzerosimp function is passed two initial guesses that must bracket the root. After assigning values for machine epsilon and a tolerance, the three variables defining the search interval (a, b, c) are initialized, and f is evaluated at the endpoints. A main loop is then implemented. If necessary, the three points are rearranged to satisfy the conditions required for the algorithm to work effectively. At this point, if the stopping criteria are met, the loop is terminated. Otherwise, a decision structure chooses among the three methods and checks whether the outcome is acceptable. A final section then evaluates f at the new point and the loop is repeated. Once the stopping criteria are met, the loop terminates and the final root estimate is returned. Note that Sec. 7.7.2 presents an application of Brent’s method where we illustrate how the MATLAB’s fzero function works. In addition, it is employed in Case Study 8.4 to determine the friction factor for air flow through a tube.

6.5

MULTIPLE ROOTS A multiple root corresponds to a point where a function is tangent to the x axis. For example, a double root results from f(x) 5 (x 2 3) (x 2 1) (x 2 1) 3

(6.11) 2

or, multiplying terms, f(x) 5 x 2 5x 1 7x 2 3. The equation has a double root because one value of x makes two terms in Eq. (6.11) equal to zero. Graphically, this corresponds to the curve touching the x axis tangentially at the double root. Examine Fig. 6.13a at x 5 1. Notice that the function touches the axis but does not cross it at the root. A triple root corresponds to the case where one x value makes three terms in an equation equal to zero, as in f(x) 5 (x 2 3) (x 2 1) (x 2 1) (x 2 1) or, multiplying terms, f(x) 5 x4 2 6x3 1 12x2 2 10x 1 3. Notice that the graphical depiction (Fig. 6.13b) again indicates that the function is tangent to the axis at the root, but that for this case the axis is crossed. In general, odd multiple roots cross the axis, whereas even ones do not. For example, the quadruple root in Fig. 6.13c does not cross the axis. Multiple roots pose some difficulties for many of the numerical methods described in Part Two: 1. The fact that the function does not change sign at even multiple roots precludes the use of the reliable bracketing methods that were discussed in Chap. 5. Thus, of the methods covered in this book, you are limited to the open methods that may diverge. 2. Another possible problem is related to the fact that not only f(x) but also f9(x) goes to zero at the root. This poses problems for both the Newton-Raphson and secant methods, which both contain the derivative (or its estimate) in the denominator of

6.5

f (x) 4

Double root

0 1

3

x

–4

(a) f (x) 4

Triple root

3

x

–4

(b) Quadruple root

(6.12)

where m is the multiplicity of the root (that is, m 5 2 for a double root, m 5 3 for a triple root, etc.). Of course, this may be an unsatisfactory alternative because it hinges on foreknowledge of the multiplicity of the root.

f (x) f ¿(x)

u(x) 5

0 1

f(xi ) f ¿(xi )

Another alternative, also suggested by Ralston and Rabinowitz (1978), is to define a new function u(x), that is, the ratio of the function to its derivative, as in

f (x) 4

167

their respective formulas. This could result in division by zero when the solution converges very close to the root. A simple way to circumvent these problems is based on the fact that it can be demonstrated theoretically (Ralston and Rabinowitz, 1978) that f(x) will always reach zero before f9(x). Therefore, if a zero check for f(x) is incorporated into the computer program, the computation can be terminated before f9(x) reaches zero. 3. It can be demonstrated that the Newton-Raphson and secant methods are linearly, rather than quadratically, convergent for multiple roots (Ralston and Rabinowitz, 1978). Modifications have been proposed to alleviate this problem. Ralston and Rabinowitz (1978) have indicated that a slight change in the formulation returns it to quadratic convergence, as in xi11 5 xi 2 m

0 1

MULTIPLE ROOTS

3

x

–4

(c) FIGURE 6.13 Examples of multiple roots that are tangential to the x axis. Notice that the function does not cross the axis on either side of even multiple roots (a) and (c), whereas it crosses the axis for odd cases (b).

It can be shown that this function has roots at all the same locations as the original function. Therefore, Eq. (6.13) can be substituted into Eq. (6.6) to develop an alternative form of the Newton-Raphson method: xi11 5 xi 2

u(xi ) u¿(xi )

(6.14)

Equation (6.13) can be differentiated to give u¿(x) 5

f ¿(x) f ¿(x) 2 f(x) f –(x) [ f ¿(x)] 2

(6.15)

Equations (6.13) and (6.15) can be substituted into Eq. (6.14) and the result simplified to yield xi11 5 xi 2

EXAMPLE 6.10

(6.13)

f(xi ) f ¿(xi ) [ f ¿(xi )] 2 2 f(xi ) f –(xi )

(6.16)

Modiﬁed Newton-Raphson Method for Multiple Roots Problem Statement. Use both the standard and modified Newton-Raphson methods to evaluate the multiple root of Eq. (6.11), with an initial guess of x0 5 0.

168

OPEN METHODS

Solution. The first derivative of Eq. (6.11) is f ¿(x) 5 3x2 2 10x 1 7, and therefore, the standard Newton-Raphson method for this problem is [Eq. (6.6)] xi11 5 xi 2

x3i 2 5x2i 1 7xi 2 3 3x2i 2 10xi 1 7

which can be solved iteratively for i

xi

et (%)

0 1 2 3 4 5 6

0 0.4285714 0.6857143 0.8328654 0.9133290 0.9557833 0.9776551

100 57 31 17 8.7 4.4 2.2

As anticipated, the method is linearly convergent toward the true value of 1.0. For the modified method, the second derivative is f 0(x) 5 6x 2 10, and the iterative relationship is [Eq. (6.16)] xi11 5 xi 2

(x3i 2 5x2i 1 7xi 2 3)(3x2i 2 10xi 1 7) (3x2i 2 10xi 1 7) 2 2 (x3i 2 5x2i 1 7xi 2 3)(6xi 2 10)

which can be solved for i

xi

et (%)

0 1 2 3

0 1.105263 1.003082 1.000002

100 11 0.31 0.00024

Thus, the modified formula is quadratically convergent. We can also use both methods to search for the single root at x 5 3. Using an initial guess of x0 5 4 gives the following results: i

Standard

et (%)

Modiﬁed

et (%)

0 1 2 3 4 5

4 3.4 3.1 3.008696 3.000075 3.000000

33 13 3.3 0.29 0.0025 2 3 1027

4 2.636364 2.820225 2.961728 2.998479 2.999998

33 12 6.0 1.3 0.051 7.7 3 1025

Thus, both methods converge quickly, with the standard method being somewhat more efficient.

6.6

SYSTEMS OF NONLINEAR EQUATIONS

169

The preceding example illustrates the trade-offs involved in opting for the modified Newton-Raphson method. Although it is preferable for multiple roots, it is somewhat less efficient and requires more computational effort than the standard method for simple roots. It should be noted that a modified version of the secant method suited for multiple roots can also be developed by substituting Eq. (6.13) into Eq. (6.7). The resulting formula is (Ralston and Rabinowitz, 1978) xi11 5 xi 2

6.6

u(xi )(xi21 2 xi ) u(xi21 ) 2 u(xi )

SYSTEMS OF NONLINEAR EQUATIONS To this point, we have focused on the determination of the roots of a single equation. A related problem is to locate the roots of a set of simultaneous equations, f1 (x1, x2, p , xn ) 5 0 f2 (x1, x2, p , xn ) 5 0 fn (x1, x2, p , xn ) 5 0

(6.17)

The solution of this system consists of a set of x values that simultaneously result in all the equations equaling zero. In Part Three, we will present methods for the case where the simultaneous equations are linear—that is, they can be expressed in the general form f(x) 5 a1 x1 1 a2 x2 1 p 1 an xn 2 b 5 0

(6.18)

where the b and the a’s are constants. Algebraic and transcendental equations that do not fit this format are called nonlinear equations. For example, x2 1 xy 5 10 and y 1 3xy2 5 57 are two simultaneous nonlinear equations with two unknowns, x and y. They can be expressed in the form of Eq. (6.17) as u(x, y) 5 x2 1 xy 2 10 5 0 2

y(x, y) 5 y 1 3xy 2 57 5 0

(6.19a) (6.19b)

Thus, the solution would be the values of x and y that make the functions u(x, y) and y(x, y) equal to zero. Most approaches for determining such solutions are extensions of the open methods for solving single equations. In this section, we will investigate two of these: fixed-point iteration and Newton-Raphson.

170

OPEN METHODS

6.6.1 Fixed-Point Iteration The fixed-point-iteration approach (Sec. 6.1) can be modified to solve two simultaneous, nonlinear equations. This approach will be illustrated in the following example.

EXAMPLE 6.11

Fixed-Point Iteration for a Nonlinear System Problem Statement. Use fixed-point iteration to determine the roots of Eq. (6.19). Note that a correct pair of roots is x 5 2 and y 5 3. Initiate the computation with guesses of x 5 1.5 and y 5 3.5. Solution. Equation (6.19a) can be solved for xi11 5

10 2 x2i yi

(E6.11.1)

and Eq. (6.19b) can be solved for yi11 5 57 2 3xi y2i

(E6.11.2)

Note that we will drop the subscripts for the remainder of the example. On the basis of the initial guesses, Eq. (E6.11.1) can be used to determine a new value of x: x5

10 2 (1.5) 2 5 2.21429 3.5

This result and the initial value of y 5 3.5 can be substituted into Eq. (E6.11.2) to determine a new value of y: y 5 57 2 3(2.21429)(3.5) 2 5 224.37516 Thus, the approach seems to be diverging. This behavior is even more pronounced on the second iteration: 10 2 (2.21429) 2 5 20.20910 224.37516 y 5 57 2 3(20.20910)(224.37516) 2 5 429.709

x5

Obviously, the approach is deteriorating. Now we will repeat the computation but with the original equations set up in a different format. For example, an alternative formulation of Eq. (6.19a) is x 5 210 2 xy and of Eq. (6.19b) is y5

57 2 y B 3x

Now the results are more satisfactory: x 5 210 2 1.5(3.5) 5 2.17945

6.6

SYSTEMS OF NONLINEAR EQUATIONS

y5

171

57 2 3.5 5 2.86051 B 3(2.17945)

x 5 210 2 2.17945(2.86051) 5 1.94053 y5

57 2 2.86051 5 3.04955 B 3(1.94053)

Thus, the approach is converging on the true values of x 5 2 and y 5 3.

The previous example illustrates the most serious shortcoming of simple fixed-point iteration—that is, convergence often depends on the manner in which the equations are formulated. Additionally, even in those instances where convergence is possible, divergence can occur if the initial guesses are insufficiently close to the true solution. Using reasoning similar to that in Box 6.1, it can be demonstrated that sufficient conditions for convergence for the two-equation case are `

0u 0u ` 1 ` ` , 1 0x 0y

`

0y 0y ` 1 ` ` , 1 0x 0y

and

These criteria are so restrictive that fixed-point iteration has limited utility for solving nonlinear systems. However, as we will describe later in the book, it can be very useful for solving linear systems. 6.6.2 Newton-Raphson Recall that the Newton-Raphson method was predicated on employing the derivative (that is, the slope) of a function to estimate its intercept with the axis of the independent variable—that is, the root (Fig. 6.5). This estimate was based on a first-order Taylor series expansion (recall Box 6.2), f(xi11 ) 5 f(xi ) 1 (xi11 2 xi ) f ¿(xi )

(6.20)

where xi is the initial guess at the root and xi11 is the point at which the slope intercepts the x axis. At this intercept, f(xi11) by definition equals zero and Eq. (6.20) can be rearranged to yield xi11 5 xi 2

f(xi ) f ¿(xi )

(6.21)

which is the single-equation form of the Newton-Raphson method. The multiequation form is derived in an identical fashion. However, a multivariable Taylor series must be used to account for the fact that more than one independent

172

OPEN METHODS

variable contributes to the determination of the root. For the two-variable case, a firstorder Taylor series can be written [recall Eq. (4.26)] for each nonlinear equation as ui11 5 ui 1 (xi11 2 xi )

0ui 0ui 1 (yi11 2 yi ) 0x 0y

(6.22a)

yi11 5 yi 1 (xi11 2 xi )

0yi 0yi 1 (yi11 2 yi ) 0x 0y

(6.22b)

and

Just as for the single-equation version, the root estimate corresponds to the values of x and y, where ui11 and yi11 equal zero. For this situation, Eq. (6.22) can be rearranged to give 0ui xi11 1 0x 0yi xi11 1 0x

0ui 0ui 0ui yi11 5 2ui 1 xi 1 yi 0y 0x 0y 0yi 0yi 0yi yi11 5 2yi 1 xi 1 yi 0y 0x 0y

(6.23a) (6.23b)

Because all values subscripted with i’s are known (they correspond to the latest guess or approximation), the only unknowns are xi11 and yi11. Thus, Eq. (6.23) is a set of two linear equations with two unknowns [compare with Eq. (6.18)]. Consequently, algebraic manipulations (for example, Cramer’s rule) can be employed to solve for

xi11

0yi 0ui 2 yi 0y 0y 5 xi 2 0ui 0yi 0ui 0yi 2 0x 0y 0y 0x

(6.24a)

yi11

0ui 0yi 2 ui 0x 0x 5 yi 2 0ui 0yi 0ui 0yi 2 0x 0y 0y 0x

(6.24b)

ui

yi

The denominator of each of these equations is formally referred to as the determinant of the Jacobian of the system. Equation (6.24) is the two-equation version of the Newton-Raphson method. As in the following example, it can be employed iteratively to home in on the roots of two simultaneous equations. EXAMPLE 6.12

Newton-Raphson for a Nonlinear System Problem Statement. Use the multiple-equation Newton-Raphson method to determine roots of Eq. (6.19). Note that a correct pair of roots is x 5 2 and y 5 3. Initiate the computation with guesses of x 5 1.5 and y 5 3.5. Solution. First compute the partial derivatives and evaluate them at the initial guesses of x and y: 0u0 0u0 5 2x 1 y 5 2(1.5) 1 3.5 5 6.5 5 x 5 1.5 0x 0y

PROBLEMS

173

0y0 0y0 5 3y2 5 3(3.5) 2 5 36.75 5 1 1 6xy 5 1 1 6(1.5)(3.5) 5 32.5 0x 0y Thus, the determinant of the Jacobian for the first iteration is 6.5(32.5) 2 1.5(36.75) 5 156.125 The values of the functions can be evaluated at the initial guesses as u0 5 (1.5) 2 1 1.5(3.5) 2 10 5 22.5 y0 5 3.5 1 3(1.5)(3.5) 2 2 57 5 1.625 These values can be substituted into Eq. (6.24) to give x 5 1.5 2

22.5(32.5) 2 1.625(1.5) 5 2.03603 156.125

y 5 3.5 2

1.625(6.5) 2 (22.5)(36.75) 5 2.84388 156.125

Thus, the results are converging to the true values of x 5 2 and y 5 3. The computation can be repeated until an acceptable accuracy is obtained.

Just as with fixed-point iteration, the Newton-Raphson approach will often diverge if the initial guesses are not sufficiently close to the true roots. Whereas graphical methods could be employed to derive good guesses for the single-equation case, no such simple procedure is available for the multiequation version. Although there are some advanced approaches for obtaining acceptable first estimates, often the initial guesses must be obtained on the basis of trial and error and knowledge of the physical system being modeled. The two-equation Newton-Raphson approach can be generalized to solve n simultaneous equations. Because the most efficient way to do this involves matrix algebra and the solution of simultaneous linear equations, we will defer discussion of the general approach to Part Three.

PROBLEMS 6.1 Use simple fixed-point iteration to locate the root of f (x) 5 sin ( 1x) 2 x Use an initial guess of x0 5 0.5 and iterate until ea # 0.01%. Verify that the process is linearly convergent as described in Box 6.1. 6.2 Determine the highest real root of f (x) 5 2x3 2 11.7x2 1 17.7x 2 5 (a) Graphically. (b) Fixed-point iteration method (three iterations, x0 5 3). Note: Make certain that you develop a solution that converges on the root. (c) Newton-Raphson method (three iterations, x0 5 3).

(d) Secant method (three iterations, x21 5 3, x0 5 4). (e) Modified secant method (three iterations, x0 5 3, d 5 0.01). Compute the approximate percent relative errors for your solutions. 6.3 Use (a) fixed-point iteration and (b) the Newton-Raphson method to determine a root of f(x) 5 20.9x2 1 1.7x 1 2.5 using x0 5 5. Perform the computation until ea is less than es 5 0.01%. Also perform an error check of your final answer. 6.4 Determine the real roots of f (x) 5 21 1 5.5x 2 4x2 1 0.5x3: (a) graphically and (b) using the Newton-Raphson method to within es 5 0.01%. 6.5 Employ the Newton-Raphson method to determine a real root for f (x) 5 21 1 5.5x 2 4x2 1 0.5x3 using initial guesses of (a) 4.52

174

OPEN METHODS

and (b) 4.54. Discuss and use graphical and analytical methods to explain any peculiarities in your results. 6.6 Determine the lowest real root of f (x) 5 212 2 21x 1 18x2 2 2.4x3: (a) graphically and (b) using the secant method to a value of es corresponding to three significant figures. 6.7 Locate the first positive root of f (x) 5 sin x 1 cos (1 1 x2 ) 2 1 where x is in radians. Use four iterations of the secant method with initial guesses of (a) xi21 5 1.0 and xi 5 3.0; (b) xi21 5 1.5 and xi 5 2.5, and (c) xi21 5 1.5 and xi 5 2.25 to locate the root. (d) Use the graphical method to explain your results. 6.8 Determine the real root of x3.5 5 80, with the modified secant method to within es 5 0.1% using an initial guess of x0 5 3.5 and d 5 0.01. 6.9 Determine the highest real root of f(x) 5 x3 2 6x2 1 11x 2 6.1: (a) Graphically. (b) Using the Newton-Raphson method (three iterations, xi 5 3.5). (c) Using the secant method (three iterations, xi11 5 2.5 and xi 5 3.5). (d) Using the modified secant method (three iterations, xi 5 3.5, d 5 0.01). 6.10 Determine the lowest positive root of f (x) 5 7 sin (x)e 2x 2 1: (a) Graphically. (b) Using the Newton-Raphson method (three iterations, xi 5 0.3). (c) Using the secant method (five iterations, xi21 5 0.5 and xi 5 0.4). (d) Using the modified secant method (three iterations, xi 5 0.3, d 5 0.01). 6.11 Use the Newton-Raphson method to find the root of f (x) 5 e 20.5x (4 2 x) 2 2 Employ initial guesses of (a) 2, (b) 6, and (c) 8. Explain your results. 6.12 Given f (x) 5 22x6 2 1.5x4 1 10x 1 2 Use a root location technique to determine the maximum of this function. Perform iterations until the approximate relative error falls below 5%. If you use a bracketing method, use initial guesses of xl 5 0 and xu 5 1. If you use the Newton-Raphson or the modified secant method, use an initial guess of xi 5 1. If you use the secant method, use initial guesses of xi21 5 0 and xi 5 1. Assuming that convergence is not an issue, choose the technique that is best suited to this problem. Justify your choice. 6.13 You must determine the root of the following easily differentiable function, e0.5x 5 5 2 5x

Pick the best numerical technique, justify your choice and then use that technique to determine the root. Note that it is known that for positive initial guesses, all techniques except fixed-point iteration will eventually converge. Perform iterations until the approximate relative error falls below 2%. If you use a bracketing method, use initial guesses of xl 5 0 and xu 5 2. If you use the Newton-Raphson or the modified secant method, use an initial guess of xi 5 0.7. If you use the secant method, use initial guesses of xi21 5 0 and xi 5 2. 6.14 Use (a) the Newton-Raphson method and (b) the modified secant method (d 5 0.05) to determine a root of f(x) 5 x5 2 16.05x4 1 88.75x3 2 192.0375x2 1 116.35x 1 31.6875 using an initial guess of x 5 0.5825 and es 5 0.01%. Explain your results. 6.15 The “divide and average” method, an old-time method for approximating the square root of any positive number a, can be formulated as x5

x 1 ayx 2

Prove that this is equivalent to the Newton-Raphson algorithm. 6.16 (a) Apply the Newton-Raphson method to the function f(x) 5 tanh(x2 2 9) to evaluate its known real root at x 5 3. Use an initial guess of x0 5 3.2 and take a minimum of four iterations. (b) Did the method exhibit convergence onto its real root? Sketch the plot with the results for each iteration shown. 6.17 The polynomial f(x) 5 0.0074x4 2 0.284x3 1 3.355x2 2 12.183x 1 5 has a real root between 15 and 20. Apply the NewtonRaphson method to this function using an initial guess of x0 5 16.15. Explain your results. 6.18 Use the secant method on the circle function (x 1 1)2 1 (y 2 2)2 5 16 to find a positive real root. Set your initial guess to xi 5 3 and xi21 5 0.5. Approach the solution from the first and fourth quadrants. When solving for f(x) in the fourth quadrant, be sure to take the negative value of the square root. Why does your solution diverge? 6.19 You are designing a spherical tank (Fig. P6.19) to hold water for a small village in a developing country. The volume of liquid it can hold can be computed as V 5 ph2

[3R 2 h] 3

where V 5 volume (m3), h 5 depth of water in tank (m), and R 5 the tank radius (m). If R 5 3 m, what depth must the tank be filled to so that it holds 30 m3? Use three iterations of the NewtonRaphson method to determine your answer. Determine the approximate relative error after each iteration. Note that an initial guess of R will always converge.

PROBLEMS

175 Use a graphical approach to obtain your initial guesses. Determine refined estimates with the two-equation Newton-Raphson method described in Sec. 6.6.2. 6.24 Repeat Prob. 6.23 except determine the positive root of

R

y 5 x2 1 1 y 5 2 cos x V

h

6.25 A mass balance for a pollutant in a well-mixed lake can be written as V

FIGURE P6.19

6.20 The Manning equation can be written for a rectangular open channel as Q5

1S(BH) 5y3 n(B 1 2H) 2y3

where Q 5 flow [m3/s], S 5 slope [m/m], H 5 depth [m], and n 5 the Manning roughness coefficient. Develop a fixed-point iteration scheme to solve this equation for H given Q 5 5, S 5 0.0002, B 5 20, and n 5 0.03. Prove that your scheme converges for all initial guesses greater than or equal to zero. 6.21 The function x3 2 2x2 2 4x 1 8 has a double root at x 5 2. Use (a) the standard Newton-Raphson [Eq. (6.6)], (b) the modified Newton-Raphson [Eq. (6.12)], and (c) the modified NewtonRaphson [Eq. (6.16)] to solve for the root at x 5 2. Compare and discuss the rate of convergence using an initial guess of x0 5 1.2. 6.22 Determine the roots of the following simultaneous nonlinear equations using (a) fixed-point iteration and (b) the Newton-Raphson method: y 5 2x2 1 x 1 0.75 y 1 5xy 5 x2 Employ initial guesses of x 5 y 5 1.2 and discuss the results. 6.23 Determine the roots of the simultaneous nonlinear equations (x 2 4) 2 1 (y 2 4) 2 5 5 x2 1 y2 5 16

dc 5 W 2 Qc 2 kV 1c dt

Given the parameter values V 5 1 3 106m3, Q 5 1 3 105 m3/yr, W 5 1 3 106 g/yr, and k 5 0.25 m0.5/g0.5/yr, use the modified secant method to solve for the steady-state concentration. Employ an initial guess of c 5 4 g/m3 and d 5 0.5. Perform three iterations and determine the percent relative error after the third iteration. 6.26 For Prob. 6.25, the root can be located with fixed-point iteration as c5a

W 2 Qc 2 b kV

or as c5

W 2 kV 1c Q

Only one will converge for initial guesses of 2 , c , 6. Select the correct one and demonstrate why it will always work. 6.27 Develop a user-friendly program for the Newton-Raphson method based on Fig. 6.4 and Sec. 6.2.3. Test it by duplicating the computation from Example 6.3. 6.28 Develop a user-friendly program for the secant method based on Fig. 6.4 and Sec. 6.3.2. Test it by duplicating the computation from Example 6.6. 6.29 Develop a user-friendly program for the modified secant method based on Fig. 6.4 and Sec. 6.3.2. Test it by duplicating the computation from Example 6.8. 6.30 Develop a user-friendly program for Brent’s root location method based on Fig. 6.12. Test it by solving Prob. 6.6. 6.31 Develop a user-friendly program for the two-equation Newton-Raphson method based on Sec. 6.6.2. Test it by solving Example 6.12. 6.32 Use the program you developed in Prob. 6.31 to solve Probs. 6.22 and 6.23 to within a tolerance of es 5 0.01%.

7

CHAPTER

Roots of Polynomials

In this chapter, we will discuss methods to find the roots of polynomial equations of the general form fn (x) 5 a0 1 a1x 1 a2 x2 1 p 1 an x n

(7.1)

where n 5 the order of the polynomial and the a’s 5 constant coefficients. Although the coefficients can be complex numbers, we will limit our discussion to cases where they are real. For such cases, the roots can be real and/or complex. The roots of such polynomials follow these rules: 1. For an nth-order equation, there are n real or complex roots. It should be noted that these roots will not necessarily be distinct. 2. If n is odd, there is at least one real root. 3. If complex roots exist, they exist in conjugate pairs (that is, l 1 mi and l 2 mi), where i 5 121. Before describing the techniques for locating the roots of polynomials, we will provide some background. The first section offers some motivation for studying the techniques; the second deals with some fundamental computer manipulations involving polynomials.

7.1

POLYNOMIALS IN ENGINEERING AND SCIENCE Polynomials have many applications in engineering and science. For example, they are used extensively in curve-fitting. However, we believe that one of their most interesting and powerful applications is in characterizing dynamic systems and, in particular, linear systems. Examples include mechanical devices, structures, and electrical circuits. We will be exploring specific examples throughout the remainder of this text. In particular, they will be the focus of several of the engineering applications throughout the remainder of this text. For the time being, we will keep the discussion simple and general by focusing on a simple second-order system defined by the following linear ordinary differential equation (or ODE): a2

176

d2y dt 2

1 a1

dy 1 a0 y 5 F(t) dt

(7.2)

7.1

POLYNOMIALS IN ENGINEERING AND SCIENCE

177

where y and t are the dependent and independent variables, respectively, the a’s are constant coefficients, and F(t) is the forcing function. In addition, it should be noted that Eq. (7.2) can be alternatively expressed as a pair of first-order ODEs by defining a new variable z, z5

dy dt

(7.3)

Equation (7.3) can be substituted along with its derivative into Eq. (7.2) to remove the second-derivative term. This reduces the problem to solving F(t) 2 a1z 2 a0y dz 5 a2 dt

(7.4)

dz 5z dt

(7.5)

In a similar fashion, an nth-order linear ODE can always be expressed as a system of n first-order ODEs. Now let’s look at the solution. The forcing function represents the effect of the external world on the system. The homogeneous or general solution of the equation deals with the case when the forcing function is set to zero, a2

d2y dt

2

1 a1

dy 1 a0 y 5 0 dt

(7.6)

Thus, as the name implies, the general solution should tell us something very fundamental about the system being simulated—that is, how the system responds in the absence of external stimuli. Now, the general solution to all unforced linear systems is of the form y 5 ert. If this function is differentiated and substituted into Eq. (7.6), the result is a 2 r 2e rt 1 a1r e rt 1 a 0 ert 5 0 or canceling the exponential terms, a2r 2 1 a1r 1 a0 5 0

(7.7)

Notice that the result is a polynomial called the characteristic equation. The roots of this polynomial are the values of r that satisfy Eq. (7.7). These r’s are referred to as the system’s characteristic values, or eigenvalues. So, here is the connection between roots of polynomials and engineering and science. The eigenvalue tells us something fundamental about the system we are modeling, and finding the eigenvalues involves finding the roots of polynomials. And, whereas finding the root of a second-order equation is easy with the quadratic formula, finding roots of higher-order systems (and hence, higher-order polynomials) is arduous analytically. Thus, the best general approach requires numerical methods of the type described in this chapter. Before proceeding to these methods, let us take our analysis a bit farther by investigating what specific values of the eigenvalues might imply about the behavior of

178

ROOTS OF POLYNOMIALS

physical systems. First, let us evaluate the roots of Eq. (7.7) with the quadratic formula, r1 2a1 6 2a21 2 4a2 a 0 5 a0 r2 Thus, we get two roots. If the discriminant (a21 2 4a2 a0 ) is positive, the roots are real and the general solution can be represented as y 5 c1er1 t 1 c2er2 t

(7.8)

where the c’s 5 constants that can be determined from the initial conditions. This is called the overdamped case. If the discriminant is zero, a single real root results, and the general solution can be formulated as y 5 (c1 1 c2 t)elt

(7.9)

This is called the critically damped case. If the discriminant is negative, the roots will be complex conjugate numbers, r1 5 l 6 mi r2 and the general solution can be formulated as y 5 c1 e(l1mi)t 1 c2 e(l2mi)t

FIGURE 7.1 The general solution for linear ODEs can be composed of (a) exponential and (b) sinusoidal components. The combination of the two shapes results in the damped sinusoid shown in (c).

y

y

t

(a)

t

(b)

y

t

(c)

7.2

COMPUTING WITH POLYNOMIALS

179

The physical behavior of this solution can be elucidated by using Euler’s formula e mit 5 cos mt 1 i sin mt to reformulate the general solution as (see Boyce and DiPrima, 1992, for details of the derivation) y 5 c1elt cos mt 1 c2elt sin mt

(7.10)

This is called the underdamped case. Equations (7.8), (7.9), and (7.10) express the possible ways that linear systems respond dynamically. The exponential terms mean that the solutions are capable of decaying (negative real part) or growing (positive real part) exponentially with time (Fig. 7.1a). The sinusoidal terms (imaginary part) mean that the solutions can oscillate (Fig. 7.1b). If the eigenvalue has both real and imaginary parts, the exponential and sinusoidal shapes are combined (Fig. 7.1c). Because such knowledge is a key element in understanding, designing, and controlling the behavior of a physical system, characteristic polynomials are very important in engineering and many branches of science. We will explore the dynamics of several engineering systems in the applications covered in Chap. 8.

7.2

COMPUTING WITH POLYNOMIALS Before describing root-location methods, we will discuss some fundamental computer operations involving polynomials. These have utility in their own right as well as providing support for root finding. 7.2.1 Polynomial Evaluation and Differentiation Although it is the most common format, Eq. (7.1) provides a poor means for determining the value of a polynomial for a particular value of x. For example, evaluating a third-order polynomial as f3 (x) 5 a3 x3 1 a2 x2 1 a1x 1 a0

(7.11)

involves six multiplications and three additions. In general, for an nth-order polynomial, this approach requires n(n 1 1)y2 multiplications and n additions. In contrast, a nested format, f3 (x) 5 ((a3 x 1 a2 )x 1 a1 )x 1 a0

(7.12)

involves three multiplications and three additions. For an nth-order polynomial, this approach requires n multiplications and n additions. Because the nested format minimizes the number of operations, it also tends to minimize round-off errors. Note that, depending on your preference, the order of nesting can be reversed: f3 (x) 5 a0 1 x(a1 1 x(a2 1 xa3 )) Succinct pseudocode to implement the nested form can be written simply as DOFOR j 5 n, 0, 21 p 5 p * x1a(j) END DO

(7.13)

180

ROOTS OF POLYNOMIALS

where p holds the value of the polynomial (defined by its coefficients, the a’s) evaluated at x. There are cases (such as in the Newton-Raphson method) where you might want to evaluate both the function and its derivative. This evaluation can also be neatly included by adding a single line to the preceding pseudocode, D0FOR j 5 n, 0, 21 df 5 df * x1p p 5 p * x1a(j) END DO

where df holds the first derivative of the polynomial. 7.2.2 Polynomial Deﬂation Suppose that you determine a single root of an nth-order polynomial. If you repeat your root location procedure, you might find the same root. Therefore, it would be nice to remove the found root before proceeding. This removal process is referred to as polynomial deflation. Before we show how this is done, some orientation might be useful. Polynomials are typically represented in the format of Eq. (7.1). For example, a fifth-order polynomial could be written as f5 (x) 5 2120 2 46x 1 79x2 2 3x3 2 7x4 1 x5

(7.14)

Although this is a familiar format, it is not necessarily the best expression to understand the polynomial’s mathematical behavior. For example, this fifth-order polynomial might be expressed alternatively as f5 (x) 5 (x 1 1) (x 2 4) (x 2 5) (x 1 3) (x 2 2)

(7.15)

This is called the factored form of the polynomial. If multiplication is completed and like terms collected, Eq. (7.14) would be obtained. However, the format of Eq. (7.15) has the advantage that it clearly indicates the function’s roots. Thus, it is apparent that x 5 21, 4, 5, 23, and 2 are all roots because each causes an individual term in Eq. (7.15) to become zero. Now, suppose that we divide this fifth-order polynomial by any of its factors, for example, x 1 3. For this case, the result would be a fourth-order polynomial f4 (x) 5 (x 1 1) (x 2 4) (x 2 5) (x 2 2) 5 240 2 2x 1 27x2 2 10x3 1 x4

(7.16)

with a remainder of zero. In the distant past, you probably learned to divide polynomials using the approach called synthetic division. Several computer algorithms (based on both synthetic division and other methods) are available for performing the operation. One simple scheme is provided by the following pseudocode, which divides an nth-order polynomial by a

7.2

COMPUTING WITH POLYNOMIALS

181

monomial factor x 2 t: r 5 a(n) a(n) 5 0 DOFOR i 5 n21, 0, 21 s 5 a(i) a(i) 5 r r 5 s 1r *t END DO

If the monomial is a root of the polynomial, the remainder r will be zero, and the coefficients of the quotient stored in a, at the end of the loop. EXAMPLE 7.1

Polynomial Deﬂation Problem Statement. Divide the second-order polynomial, f(x) 5 (x 2 4) (x 1 6) 5 x2 1 2x 2 24 by the factor x 2 4. Solution. Using the approach outlined in the above pseudocode, the parameters are n 5 2, a0 5 224, a1 5 2, a2 5 1, and t 5 4. These can be used to compute r 5 a2 5 1 a2 5 0 The loop is then iterated from i 5 2 2 1 5 1 to 0. For i 5 1, s 5 a1 5 2 a1 5 r 5 1 r 5 s 1 rt 5 2 1 1(4) 5 6 For i 5 0, s 5 a0 5 224 a0 5 r 5 6 r 5 224 1 6(4) 5 0 Thus, the result is as expected—the quotient is a0 1 a1x 5 6 1 x, with a remainder of zero. It is also possible to divide by polynomials of higher order. As we will see later in this chapter, the most common task involves dividing by a second-order polynomial or parabola. The subroutine in Fig. 7.2 addresses the more general problem of dividing an nth-order polynomial a by an mth-order polynomial d. The result is an (n 2 m)th-order polynomial q, with an (m 2 1)th-order polynomial as the remainder. Because each calculated root is known only approximately, it should be noted that deflation is sensitive to round-off errors. In some cases, round-off error can grow to the point that the results can become meaningless. Some general strategies can be applied to minimize this problem. For example, round-off error is affected by the order in which the terms are evaluated. Forward deflation refers to the

182

ROOTS OF POLYNOMIALS SUB poldiv(a, n, d, m, q, r) DOFOR j 5 0, n r(j) 5 a(j) q(j) 5 0 END DO DOFOR k 5 n2m, 0, 21 q(k11) 5 r(m1k) y d(m) DOFOR j 5 m1k21, k, 21 r(j) 5 r(j)2q(k11) * b(j2k) END DO END DO DOFOR j 5 m, n r(j) 5 0 END DO n 5 n2m DOFOR i 5 0, n a(i) 5 q(i11) END DO END SUB

FIGURE 7.2 Algorithm to divide a polynomial (deﬁned by its coefﬁcients a) by a lower-order polynomial d.

case where new polynomial coefficients are in order of descending powers of x (that is, from the highest-order to the zero-order term). For this case, it is preferable to divide by the roots of smallest absolute value first. Conversely, for backward deflation (that is, from the zero-order to the highest-order term), it is preferable to divide by the roots of largest absolute value first. Another way to reduce round-off errors is to consider each successive root estimate obtained during deflation as a good first guess. These can then be used as a starting guess, and the root determined again with the original nondeflated polynomial. This is referred to as root polishing. Finally, a problem arises when two deflated roots are inaccurate enough that they both converge on the same undeflated root. In that case, you might be erroneously led to believe that the polynomial has a multiple root (recall Sec. 6.5). One way to detect this problem is to compare each polished root with those that were located previously. Press et al. (2007) discuss this problem in more detail.

7.3

CONVENTIONAL METHODS Now that we have covered some background material on polynomials, we can begin to describe methods to locate their roots. The obvious first step would be to investigate the viability of the bracketing and open approaches described in Chaps. 5 and 6. The efficacy of these approaches depends on whether the problem being solved involves complex roots. If only real roots exist, any of the previously described methods could have utility. However, the problem of finding good initial guesses complicates both the bracketing and the open methods, whereas the open methods could be susceptible to divergence.

7.4

MÜLLER’S METHOD

183

When complex roots are possible, the bracketing methods cannot be used because of the obvious problem that the criterion for defining a bracket (that is, sign change) does not translate to complex guesses. Of the open methods, the conventional Newton-Raphson method would provide a viable approach. In particular, concise code including deflation can be developed. If a language that accommodates complex variables (like Fortran) is used, such an algorithm will locate both real and complex roots. However, as might be expected, it would be susceptible to convergence problems. For this reason, special methods have been developed to find the real and complex roots of polynomials. We describe two—the Müller and Bairstow methods—in the following sections. As you will see, both are related to the more conventional open approaches described in Chap. 6.

7.4

MÜLLER’S METHOD Recall that the secant method obtains a root estimate by projecting a straight line to the x axis through two function values (Fig. 7.3a). Müller’s method takes a similar approach, but projects a parabola through three points (Fig. 7.3b). The method consists of deriving the coefficients of the parabola that goes through the three points. These coefficients can then be substituted into the quadratic formula to obtain the point where the parabola intercepts the x axis—that is, the root estimate. The approach is facilitated by writing the parabolic equation in a convenient form, f2 (x) 5 a(x 2 x2 ) 2 1 b(x 2 x2 ) 1 c

(7.17)

We want this parabola to intersect the three points [x0, f(x0)], [x1, f(x1)], and [x2, f(x2)]. The coefficients of Eq. (7.17) can be evaluated by substituting each of the three points to give f(x0 ) 5 a(x0 2 x2 ) 2 1 b(x0 2 x2 ) 1 c f(x1 ) 5 a(x1 2 x2 ) 2 1 b(x1 2 x2 ) 1 c f(x2 ) 5 a(x2 2 x2 ) 2 1 b(x2 2 x2 ) 1 c FIGURE 7.3 A comparison of two related approaches for locating roots: (a) the secant method and (b) Müller’s method.

Straight line

f (x)

(7.18) (7.19) (7.20)

f (x)

Root estimate

Parabola

x1 Root

x0

x

x2

x0

Root estimate

Root

(a)

x1

(b)

x

184

ROOTS OF POLYNOMIALS

Note that we have dropped the subscript “2” from the function for conciseness. Because we have three equations, we can solve for the three unknown coefficients, a, b, and c. Because two of the terms in Eq. (7.20) are zero, it can be immediately solved for c 5 f(x2). Thus, the coefficient c is merely equal to the function value evaluated at the third guess, x2. This result can then be substituted into Eqs. (7.18) and (7.19) to yield two equations with two unknowns: f(x0 ) 2 f(x2 ) 5 a(x0 2 x2 ) 2 1 b(x0 2 x2 )

(7.21)

f(x1 ) 2 f(x2 ) 5 a(x1 2 x2 ) 2 1 b(x1 2 x2 )

(7.22)

Algebraic manipulation can then be used to solve for the remaining coefficients, a and b. One way to do this involves defining a number of differences, h0 5 x1 2 x0 f(x1 ) 2 f(x0 ) d0 5 x1 2 x0

h1 5 x2 2 x1 f(x2 ) 2 f(x1 ) d1 5 x2 2 x1

(7.23)

These can be substituted into Eqs. (7.21) and (7.22) to give (h0 1 h1 )b 2 (h0 1 h1 ) 2a 5 h0 d0 1 h1 d1 h1

b2

h21

a5

h1d1

which can be solved for a and b. The results can be summarized as a5

d1 2 d0 h1 1 h0

(7.24)

b 5 ah1 1 d1

(7.25)

c 5 f(x2 )

(7.26)

To find the root, we apply the quadratic formula to Eq. (7.17). However, because of potential round-off error, rather than using the conventional form, we use the alternative formulation [Eq. (3.13)] to yield x3 2 x2 5

22c b 6 2b2 2 4ac

(7.27a)

or isolating the unknown x3 on the left side of the equal sign, x3 5 x2 1

22c b 6 2b2 2 4ac

(7.27b)

Note that the use of the quadratic formula means that both real and complex roots can be located. This is a major benefit of the method. In addition, Eq. (7.27a) provides a neat means to determine the approximate error. Because the left side represents the difference between the present (x3) and the previous (x2) root estimate, the error can be calculated as ea 5 `

x3 2 x2 ` 100% x3

7.4

MÜLLER’S METHOD

185

Now, a problem with Eq. (7.27a) is that it yields two roots, corresponding to the 6 term in the denominator. In Müller’s method, the sign is chosen to agree with the sign of b. This choice will result in the largest denominator, and hence, will give the root estimate that is closest to x2. Once x3 is determined, the process is repeated. This brings up the issue of which point is discarded. Two general strategies are typically used: 1. If only real roots are being located, we choose the two original points that are nearest the new root estimate, x3. 2. If both real and complex roots are being evaluated, a sequential approach is employed. That is, just like the secant method, x1, x2, and x3 take the place of x0, x1, and x2. EXAMPLE 7.2

Müller’s Method Problem Statement. Use Müller’s method with guesses of x0, x1, and x2 5 4.5, 5.5, and 5, respectively, to determine a root of the equation f(x) 5 x3 2 13x 2 12 Note that the roots of this equation are 23, 21, and 4. Solution. First, we evaluate the function at the guesses f(4.5) 5 20.625

f(5.5) 5 82.875

f(5) 5 48

which can be used to calculate h0 5 5.5 2 4.5 5 1 82.875 2 20.625 5 62.25 d0 5 5.5 2 4.5

h1 5 5 2 5.5 5 20.5 48 2 82.875 5 69.75 d1 5 5 2 5.5

These values in turn can be substituted into Eqs. (7.24) through (7.26) to compute a5

69.75 2 62.25 5 15 20.5 1 1

b 5 15(20.5) 1 69.75 5 62.25

c 5 48

The square root of the discriminant can be evaluated as 262.252 2 4(15)48 5 31.54461 Then, because Z62.25 1 31.54451Z . Z62.25 2 31.54451Z, a positive sign is employed in the denominator of Eq. (7.27b), and the new root estimate can be determined as x3 5 5 1

22(48) 5 3.976487 62.25 1 31.54451

and develop the error estimate ea 5 `

21.023513 ` 100% 5 25.74% 3.976487

Because the error is large, new guesses are assigned; x0 is replaced by x1, x1 is replaced by x2, and x2 is replaced by x3. Therefore, for the new iteration, x0 5 5.5

x1 5 5

x2 5 3.976487

186

ROOTS OF POLYNOMIALS

and the calculation is repeated. The results, tabulated below, show that the method converges rapidly on the root, xr 5 4: i

xr

Ea (%)

0 1 2 3 4

5 3.976487 4.00105 4 4

25.74 0.6139 0.0262 0.0000119

Pseudocode to implement Müller’s method for real roots is presented in Fig. 7.4. Notice that this routine is set up to take a single initial nonzero guess that is then perturbed to develop the other two guesses. Of course, the algorithm can also be

FIGURE 7.4 Pseudocode for Müller’s method. SUB Muller(xr, h, eps, maxit) x2 5 xr x1 5 xr 1 h*xr x0 5 xr 2 h*xr DO iter 5 iter 1 1 h0 5 x1 2 x0 h1 5 x2 2 x1 d0 5 (f(x1) 2 f(x0)) / h0 d1 5 (f(x2) 2 f(x1)) / h1 a 5 (d1 2 d0) / (h1 1 h0) b 5 a*h1 1 d1 c 5 f(x2) rad 5 SQRT(b*b 2 4*a*c) If |b1rad| . |b2rad| THEN den 5 b 1 rad ELSE den 5 b 2 rad END IF dxr 5 22*c y den xr 5 x2 1 dxr PRINT iter, xr IF (|dxr| , eps*xr OR iter .5 maxit) EXIT x0 5 x1 x1 5 x2 x2 5 xr END DO END Müller

7.5

BAIRSTOW’S METHOD

187

programmed to accommodate three guesses. For languages like Fortran, the code will find complex roots if the proper variables are declared as complex.

7.5

BAIRSTOW’S METHOD Bairstow’s method is an iterative approach related loosely to both the Müller and NewtonRaphson methods. Before launching into a mathematical description of the technique, recall the factored form of the polynomial, f5 (x) 5 (x 1 1) (x 2 4) (x 2 5) (x 1 3) (x 2 2)

(7.28)

If we divided by a factor that is not a root (for example, x 1 6), the quotient would be a fourth-order polynomial. However, for this case, a remainder would result. On the basis of the above, we can elaborate on an algorithm for determining a root of a polynomial: (1) guess a value for the root x 5 t, (2) divide the polynomial by the factor x 2 t, and (3) determine whether there is a remainder. If not, the guess was perfect and the root is equal to t. If there is a remainder, the guess can be systematically adjusted and the procedure repeated until the remainder disappears and a root is located. After this is accomplished, the entire procedure can be repeated for the quotient to locate another root. Bairstow’s method is generally based on this approach. Consequently, it hinges on the mathematical process of dividing a polynomial by a factor. Recall from our discussion of polynomial deflation (Sec. 7.2.2) that synthetic division involves dividing a polynomial by a factor x 2 t. For example, the general polynomial [Eq. (7.1)] fn (x) 5 a0 1 a1 x 1 a2 x 2 1 p 1 an x n

(7.29)

can be divided by the factor x 2 t to yield a second polynomial that is one order lower, fn21 (x) 5 b1 1 b2 x 1 b3 x 2 1 p 1 bn x n21

(7.30)

with a remainder R 5 b0, where the coefficients can be calculated by the recurrence relationship bn 5 an bi 5 ai 1 bi11t

for i 5 n 2 1 to 0

Note that if t were a root of the original polynomial, the remainder b0 would equal zero. To permit the evaluation of complex roots, Bairstow’s method divides the polynomial by a quadratic factor x2 2 rx 2 s. If this is done to Eq. (7.29), the result is a new polynomial fn22 (x) 5 b2 1 b3 x 1 p 1 bn21 x n23 1 bn x n22 with a remainder R 5 b1 (x 2 r) 1 b0

(7.31)

As with normal synthetic division, a simple recurrence relationship can be used to perform the division by the quadratic factor: bn 5 an bn21 5 an21 1 rbn bi 5 ai 1 rbi11 1 sbi12

(7.32a) (7.32b)

for i 5 n 2 2 to 0

(7.32c)

188

ROOTS OF POLYNOMIALS

The quadratic factor is introduced to allow the determination of complex roots. This relates to the fact that, if the coefficients of the original polynomial are real, the complex roots occur in conjugate pairs. If x2 2 rx 2 s is an exact divisor of the polynomial, complex roots can be determined by the quadratic formula. Thus, the method reduces to determining the values of r and s that make the quadratic factor an exact divisor. In other words, we seek the values that make the remainder term equal to zero. Inspection of Eq. (7.31) leads us to conclude that for the remainder to be zero, b0 and b1 must be zero. Because it is unlikely that our initial guesses at the values of r and s will lead to this result, we must determine a systematic way to modify our guesses so that b0 and b1 approach zero. To do this, Bairstow’s method uses a strategy similar to the Newton-Raphson approach. Because both b0 and b1 are functions of both r and s, they can be expanded using a Taylor series, as in [recall Eq. (4.26)] 0b1 0b1 ¢r 1 ¢s 0r 0s 0b0 0b0 b0 (r 1 ¢r, s 1 ¢s) 5 b0 1 ¢r 1 ¢s 0r 0s

b1 (r 1 ¢r, s 1 ¢s) 5 b1 1

(7.33)

where the values on the right-hand side are all evaluated at r and s. Notice that secondand higher-order terms have been neglected. This represents an implicit assumption that 2r and 2s are small enough that the higher-order terms are negligible. Another way of expressing this assumption is to say that the initial guesses are adequately close to the values of r and s at the roots. The changes, Dr and Ds, needed to improve our guesses can be estimated by setting Eq. (7.33) equal to zero to give 0b1 0b1 ¢r 1 ¢s 5 2b1 0r 0s 0b0 0b0 ¢r 1 ¢s 5 2b0 0r 0s

(7.34) (7.35)

If the partial derivatives of the b’s can be determined, these are a system of two equations that can be solved simultaneously for the two unknowns, Dr and Ds. Bairstow showed that the partial derivatives can be obtained by a synthetic division of the b’s in a fashion similar to the way in which the b’s themselves were derived: cn 5 bn cn21 5 bn21 1 r cn ci 5 bi 1 r ci11 1 sci12

(7.36a) (7.36b)

for i 5 n 2 2 to 1

(7.36c)

where 0b0y0r 5 c1, 0b0y0s 5 0b1y0r 5 c2, and 0b1y0s 5 c3. Thus, the partial derivatives are obtained by synthetic division of the b’s. Then the partial derivatives can be substituted into Eqs. (7.34) and (7.35) along with the b’s to give c2 ¢r 1 c3 ¢s 5 2b1 c1 ¢r 1 c2 ¢s 5 2b0

7.5

BAIRSTOW’S METHOD

189

These equations can be solved for Dr and Ds, which can in turn be employed to improve the initial guesses of r and s. At each step, an approximate error in r and s can be estimated, as in Zea,r Z 5 `

¢r ` 100% r

(7.37)

Zea,s Z 5 `

¢s ` 100% s

(7.38)

and

When both of these error estimates fall below a prespecified stopping criterion es, the values of the roots can be determined by x5

r 6 2r2 1 4s 2

(7.39)

At this point, three possibilities exist: 1. The quotient is a third-order polynomial or greater. For this case, Bairstow’s method would be applied to the quotient to evaluate new values for r and s. The previous values of r and s can serve as the starting guesses for this application. 2. The quotient is a quadratic. For this case, the remaining two roots could be evaluated directly with Eq. (7.39). 3. The quotient is a first-order polynomial. For this case, the remaining single root can be evaluated simply as x52 EXAMPLE 7.3

s r

(7.40)

Bairstow’s Method Problem Statement. Employ Bairstow’s method to determine the roots of the polynomial f5 (x) 5 x5 2 3.5x4 1 2.75x3 1 2.125x2 2 3.875x 1 1.25 Use initial guesses of r 5 s 5 21 and iterate to a level of es 5 1%. Solution. Equations (7.32) and (7.36) can be applied to compute b5 5 1 b4 5 24.5 b0 5 11.375 c5 5 1 c4 5 25.5

b3 5 6.25 c3 5 10.75

b2 5 0.375 c2 5 24.875

b1 5 210.5 c1 5 216.375

Thus, the simultaneous equations to solve for Dr and Ds are 24.875¢r 1 10.75¢s 5 10.5 216.375¢r 2 4.875¢s 5 211.375 which can be solved for Dr 5 0.3558 and Ds 5 1.1381. Therefore, our original guesses can be corrected to r 5 21 1 0.3558 5 20.6442 s 5 21 1 1.1381 5 0.1381

190

ROOTS OF POLYNOMIALS

and the approximate errors can be evaluated by Eqs. (7.37) and (7.38), 0 ea,r 0 5 `

0.3558 1.1381 ` 100% 5 55.23% 0 ea, s 0 5 ` ` 100% 5 824.1% 20.6442 0.1381

Next, the computation is repeated using the revised values for r and s. Applying Eqs. (7.32) and (7.36) yields b5 5 1 b4 5 24.1442 b0 5 2.1304 c5 5 1 c4 5 24.7884

b3 5 5.5578

b2 5 22.0276

b1 5 21.8013

c3 5 8.7806

c2 5 28.3454

c1 5 4.7874

Therefore, we must solve 28.3454¢r 1 8.7806¢s 5 1.8013 4.7874¢r 2 8.3454¢s 5 22.1304 for Dr 5 0.1331 and Ds 5 0.3316, which can be used to correct the root estimates as r 5 20.6442 1 0.1331 5 20.5111 s 5 0.1381 1 0.3316 5 0.4697

Zea,r Z 5 26.0% Zea,s Z 5 70.6%

The computation can be continued, with the result that after four iterations the method converges on values of r 5 20.5 ( Zea,r Z 5 0.063%) and s 5 0.5 (Zea,s Z 5 0.040%). Equation (7.39) can then be employed to evaluate the roots as x5

20.5 6 2(20.5) 2 1 4(0.5) 5 0.5,21.0 2

At this point, the quotient is the cubic equation f(x) 5 x3 2 4x2 1 5.25x 2 2.5 Bairstow’s method can be applied to this polynomial using the results of the previous step, r 5 20.5 and s 5 0.5, as starting guesses. Five iterations yield estimates of r 5 2 and s 5 21.249, which can be used to compute x5

2 6 222 1 4(21.249) 5 1 6 0.499i 2

At this point, the quotient is a first-order polynomial that can be directly evaluated by Eq. (7.40) to determine the fifth root: 2.

Note that the heart of Bairstow’s method is the evaluation of the b’s and c’s via Eqs. (7.32) and (7.36). One of the primary strengths of the method is the concise way in which these recurrence relationships can be programmed. Figure 7.5 lists pseudocode to implement Bairstow’s method. The heart of the algorithm consists of the loop to evaluate the b’s and c’s. Also notice that the code to solve the simultaneous equations checks to prevent division by zero. If this is the case, the values of r and s are perturbed slightly and the procedure is begun again. In addition, the algorithm places a user-defined upper limit on the number of iterations (MAXIT) and should be designed to avoid division by zero while calculating the error estimates. Finally, the algorithm requires initial guesses for r and s (rr and ss in the code). If no prior knowledge of the roots exist, they can be set to zero in the calling program.

7.5

BAIRSTOW’S METHOD

(a) Bairstow Algorithm

SUB Bairstow (a,nn,es,rr,ss,maxit,re,im,ier) DIMENSION b(nn), c(nn) r 5 rr; s 5 ss; n 5 nn ier 5 0; ea1 5 1; ea2 5 1 DO IF n , 3 OR iter $ maxit EXIT iter 5 0 DO iter 5 iter 1 1 b(n) 5 a(n) b(n 2 1) 5 a(n 2 1) 1 r * b(n) c(n) 5 b(n) c(n 2 1) 5 b(n 2 1) 1 r * c(n) DO i 5 n 2 2, 0, 21 b(i) 5 a(i) 1 r * b(i 1 1) 1 s * b(i 1 2) c(i) 5 b(i) 1 r * c(i 1 1) 1 s * c(i 1 2) END DO det 5 c(2) * c(2) 2 c(3) * c(1) IF det ﬁ 0 THEN dr 5 (2b(1) * c(2) 1 b(0) * c(3))ydet ds 5 (2b(0) * c(2) 1 b(1) * c(1))ydet r 5 r 1 dr s 5 s 1 ds IF rﬁ0 THEN ea1 5 ABS(dryr) * 100 IF sﬁ0 THEN ea2 5 ABS(dsys) * 100 ELSE r 5r 1 1 s 5 s 11 iter 5 0 END IF IF ea1 # es AND ea2 # es OR iter $ maxit EXIT END DO CALL Quadroot(r,s,r1,i1,r2,i2) re(n) 5 r1 im(n) 5 i1 re(n 2 1) 5 r2 im(n 2 1) 5 i2 n 5 n 22 DO i 5 0, n a(i) 5 b(i 1 2) END DO END DO

191 IF iter , maxit THEN IF n 5 2 THEN r 5 2a(1)ya(2) s 5 2a(0)ya(2) CALL Quadroot(r,s,r1,i1,r2,i2) re(n) 5 r1 im(n) 5 i1 re(n 2 1) 5 r2 im(n 2 1) 5 i2 ELSE re(n) 5 2a(0)ya(1) im(n) 5 0 END IF ELSE ier 5 1 END IF END Bairstow

(b) Roots of Quadratic Algorithm

SUB Quadroot(r,s,r1,i1,r2,i2) disc 5 r ^ 2 1 4 * s IF disc . 0 THEN r1 5 (r 1 SQRT(disc))y2 r2 5 (r 2 SQRT(disc))y2 i1 5 0 i2 5 0 ELSE r1 5 ry2 r2 5 r1 i1 5 SQRT(ABS(disc))y2 i2 5 2i1 END IF END QuadRoot

FIGURE 7.5 (a) Algorithm for implementing Bairstow’s method, along with (b) an algorithm to determine the roots of a quadratic.

192

ROOTS OF POLYNOMIALS

7.6

OTHER METHODS

SOFTWARE

Other methods are available to locate the roots of polynomials. The Jenkins-Traub method is commonly used in software libraries. It is fairly complicated, and a good starting point to understanding it is found in Ralston and Rabinowitz (1978). Laguerre’s method, which approximates both real and complex roots and has cubic convergence, is among the best approaches. A complete discussion can be found in Householder (1970). In addition, Press et al. (2007) present a nice algorithm to implement the method.

7.7

ROOT LOCATION WITH SOFTWARE PACKAGES Software packages have great capabilities for locating roots. In this section, we will give you a taste of some of the more useful ones. 7.7.1 Excel A spreadsheet like Excel can be used to locate a root by trial and error. For example, if you want to find a root of f(x) 5 x 2 cos x first, you can enter a value for x in a cell. Then set up another cell for f(x) that would obtain its value for x from the first cell. You can then vary the x cell until the f(x) cell approaches zero. This process can be further enhanced by using Excel’s plotting capabilities to obtain a good initial guess (Fig. 7.6). Although Excel does facilitate a trial-and-error approach, it also has two standard tools that can be employed for root location: Goal Seek and Solver. Both these tools can be employed to systematically adjust the initial guesses. Goal Seek is expressly used to drive an equation to a value (in our case, zero) by varying a single parameter.

FIGURE 7.6 A spreadsheet set up to determine the root of f(x) 5 x 2 cos x by trial and error. The plot is used to obtain a good initial guess.

7.7

EXAMPLE 7.4

ROOT LOCATION WITH SOFTWARE PACKAGES

193

Using Excel’s Goal Seek Tool to Locate a Single Root Problem Statement. Employ Goal Seek to determine the root of the transcendental function f(x) 5 x 2 cos x Solution. As in Fig. 7.6, the key to solving a single equation with Excel is creating a cell to hold the value of the function in question and then making the value dependent on another cell. Once this is done, the selection Goal Seek is chosen from the What-If Analysis button on your Data ribbon. At this point a dialogue box will be displayed, asking you to set a cell to a value by changing another cell. For the example, suppose that as in Fig. 7.6 your guess is entered in cell A11 and your function result in cell B11. The Goal Seek dialogue box would be filled out as

When the OK button is selected, a message box displays the results,

The cells on the spreadsheet would also be modified to the new values (as shown in Fig. 7.6). The Solver tool is more sophisticated than Goal Seek in that (1) it can vary several cells simultaneously and (2) along with driving a target cell to a value, it can minimize and maximize its value. The next example illustrates how it can be used to solve a system of nonlinear equations. EXAMPLE 7.5

Using Excel’s Solver for a Nonlinear System Problem Statement. Recall that in Sec. 6.6 we obtained the solution of the following set of simultaneous equations, u(x, y) 5 x2 1 xy 2 10 5 0 y(x, y) 5 y 1 3xy2 2 57 5 0

SOFTWARE

194

ROOTS OF POLYNOMIALS

Note that a correct pair of roots is x 5 2 and y 5 3. Use Solver to determine the roots using initial guesses of x 5 1 and y 5 3.5. Solution. As shown below, two cells (B1 and B2) can be created to hold the guesses for x and y. The function values themselves, u(x, y) and y(x, y) can then be entered into two other cells (B3 and B4). As can be seen, the initial guesses result in function values that are far from zero.

Next, another cell can be created that contains a single value reflecting how close both functions are to zero. One way to do this is to sum the squares of the function values. This is done and the result entered in cell B6. If both functions are at zero, this function should also be at zero. Further, using the squared functions avoids the possibility that both functions could have the same nonzero value, but with opposite signs. For this case, the target cell (B6) would be zero, but the roots would be incorrect. Once the spreadsheet is created, the selection Solver is chosen from the Data ribbon.1 At this point a dialogue box will be displayed, querying you for pertinent information. The pertinent cells of the Solver dialogue box would be filled out as

1

Note that you may have to install Solver by choosing Office, Excel Options, Add-Ins. Select Excel Add-Ins from the Manage drop-down box at the bottom of the Excel options menu and click Go. Then, check the Solver box. The Solver then should be installed and a button to access it should appear on your Data ribbon.

7.7

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195

When the OK button is selected, a dialogue box will open with a report on the success of the operation. For the present case, the Solver obtains the correct solution:

It should be noted that the Solver can fail. Its success depends on (1) the condition of the system of equations and/or (2) the quality of the initial guesses. Thus, the successful outcome of the previous example is not guaranteed. Despite this, we have found Solver useful enough to make it a feasible option for quickly obtaining roots in a wide range of engineering applications. 7.7.2 MATLAB As summarized in Table 7.1, MATLAB software is capable of locating roots of single algebraic and transcendental equations. It is superb at manipulating and locating the roots of polynomials. The fzero function is designed to locate one root of a single function. A simplified representation of its syntax is fzero(f,x0,options)

where f is the function you are analyzing, x0 is the initial guess, and options are the optimization parameters (these are changed using the function optimset). If options are omitted, default values are employed. Note that one or two guesses can be employed. If two guesses are employed, they are assumed to bracket a root. The following example illustrates how fzero can be used. TABLE 7.1 Common functions in MATLAB related to root location and polynomial manipulation. Function

Description

fzero roots poly polyval polyvalm residue polyder conv deconv

Root of single function. Find polynomial roots. Construct polynomial with speciﬁed roots. Evaluate polynomial. Evaluate polynomial with matrix argument. Partial-fraction expansion (residues). Differentiate polynomial. Multiply polynomials. Divide polynomials.

SOFTWARE

196

ROOTS OF POLYNOMIALS

EXAMPLE 7.6

Using MATLAB for Root Location Problem Statement. Use the MATLAB function fzero to find the roots of f(x) 5 x10 2 1 within the interval xl 5 0 and xu 5 4. Obviously two roots occur at 21 and 1. Recall that in Example 5.6, we used the false-position method with initial guesses of 0 and 1.3 to determine the positive root. Solution. Using the same initial conditions as in Example 5.6, we can use MATLAB to determine the positive root as in >> x0=[0 1.3]; >> x=fzero(@(x) x^10–1,x0) x =

1

In a similar fashion, we can use initial guesses of 21.3 and 0 to determine the negative root, >> x0=[21.3 0]; >> x=fzero(@(x) x^10–1,x0) x =

–1

We can also employ a single guess. An interesting case would be to use an initial guess of 0, >> x0=0; >> x=fzero(@(x) x^10–1,x0) x =

–1

Thus, for this guess, the underlying algorithm happens to home in on the negative root. The use of optimset can be illustrated by using it to display the actual iterations as the solution progresses: >> x0=0; >> option=optimset('DISP','ITER'); >> x=fzero(@(x) x^10–1,x0,option) Func–count x 1 0 2 –0.0282843 3 0.0282843 4 –0.04

f(x) –1 –1 –1 –1

Procedure initial search search search

• • •

21 22

0.64 –0.905097

–0.988471 –0.631065

search search

7.7

ROOT LOCATION WITH SOFTWARE PACKAGES

23 24

0.905097 –1.28

–0.631065 10.8059

197

search search

Looking for a zero in the interval [–1.28, 0.9051] 25 26 27 28 29 30 31 32 33 34 35 Zero found

0.784528 –0.911674 –0.247736 –0.999999 –0.763868 –0.932363 –1.02193 0.242305 –0.968701 –0.27239 –0.996873 –0.0308299 –0.999702 –0.00297526 –1 5.53132e–006 –1 –7.41965e–009 –1 –1.88738e–014 –1 0 in the interval: [–1.28, 0.9051].

interpolation bisection bisection bisection interpolation interpolation interpolation interpolation interpolation interpolation interpolation

x 5 21

These results illustrate the strategy used by fzero when it is provided with a single guess. First, it searches in the vicinity of the guess until it detects a sign change. Then it uses a combination of bisection and interpolation to home in on the root. The interpolation involves both the secant method and inverse quadratic interpolation (recall Sec. 7.4). It should be noted that the fzero algorithm has more to it than this basic description might imply. You can consult Press et al. (2007) for additional details.

EXAMPLE 7.7

Using MATLAB to Manipulate and Determine the Roots of Polynomials Problem Statement. Explore how MATLAB can be employed to manipulate and determine the roots of polynomials. Use the following equation from Example 7.3, f5 (x) 5 x5 2 3.5x4 1 2.75x3 1 2.125x2 2 3.875x 1 1.25

(E7.7.1)

which has three real roots: 0.5, 21.0, and 2, and one pair of complex roots: 1 6 0.5i. Solution. Polynomials are entered into MATLAB by storing the coefficients as a vector. For example, at the MATLAB prompt (. .) typing and entering the follow line stores the coefficients in the vector a, >> a=[1 –3.5 2.75 2.125 –3.875 1.25];

We can then proceed to manipulate the polynomial. For example, we can evaluate it at x 5 1 by typing >> polyval(a,1)

with the result 1(1)5 2 3.5(1)4 1 2.75(1)3 1 2.125(1)2 2 3.875(1) 1 1.25 5 20.25, ans = –0.2500

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ROOTS OF POLYNOMIALS

We can evaluate the derivative f9(x) 5 5x4 2 14x3 1 8.25x2 1 4.25x 2 3.875 by >> polyder(a) ans = 5.0000 –14.0000

8.2500

4.2500

–3.8750

Next, let us create a quadratic polynomial that has roots corresponding to two of the original roots of Eq. (E7.7.1): 0.5 and 21. This quadratic is (x 2 0.5)(x 1 1) 5 x2 1 0.5x 2 0.5 and can be entered into MATLAB as the vector b, >> b=[1 0.5 –0.5];

We can divide this polynomial into the original polynomial by >> [d,e]5deconv(a,b)

with the result being a quotient (a third-order polynomial d) and a remainder (e), d = 1.0000 –4.0000 e = 0 0 0

5.2500

–2.5000

0

0

0

Because the polynomial is a perfect divisor, the remainder polynomial has zero coefficients. Now, the roots of the quotient polynomial can be determined as >> roots(d)

with the expected result that the remaining roots of the original polynomial (E7.7.1) are found, ans = 2.0000 1.0000 1.0000

+ 0.5000i – 0.5000i

We can now multiply d by b to come up with the original polynomial, >> conv(d,b) ans = 1.0000 –3.5000

2.7500

2.1250

–3.8750

1.2500

Finally, we can determine all the roots of the original polynomial by >> r5roots(a) r = –1.0000 2.0000 1.0000 + 0.5000i 1.0000 – 0.5000i 0.5000

7.7

ROOT LOCATION WITH SOFTWARE PACKAGES

199

7.7.3 Mathcad Mathcad has a numeric mode function called root that can be used to solve an equation of a single variable. The method requires that you supply a function f(x) and either an initial guess or a bracket. When a single guess value is used, root uses the Secant and Müller methods. In the case where two guesses that bracket a root are supplied, it uses a combination of the Ridder method (a variation of false position) and Brent’s method. It iterates until the magnitude of f(x) at the proposed root is less than the predefined value of TOL. The Mathcad implementation has similar advantages and disadvantages as conventional root location methods such as issues concerning the quality of the initial guess and the rate of convergence. Mathcad can find all the real or complex roots of polynomials with polyroots. This numeric or symbolic mode function is based on the Laguerre method. This function does not require initial guesses, and all the roots are returned at the same time. Mathcad contains a numeric mode function called Find that can be used to solve up to 50 simultaneous nonlinear algebraic equations. The Find function chooses an appropriate method from a group of available methods, depending on whether the problem is linear or nonlinear, and other attributes. Acceptable values for the solution may be unconstrained or constrained to fall within specified limits. If Find fails to locate a solution that satisfies the equations and constraints, it returns the error message “did not find solution.” However, Mathcad also contains a similar function called Minerr. This function gives solution results that minimize the errors in the constraints even when exact solutions cannot be found. Thus, the problem of solving for the roots of nonlinear equations is closely related to both optimization and nonlinear least squares. These areas and Minerr are covered in detail in Parts Four and Five. Figure 7.7 shows a typical Mathcad worksheet. The menus at the top provide quick access to common arithmetic operators and functions, various two- and three-dimensional FIGURE 7.7 Mathcad screen to ﬁnd the root of a single equation.

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ROOTS OF POLYNOMIALS

plot types, and the environment to create subprograms. Equations, text, data, or graphs can be placed anywhere on the screen. You can use a variety of fonts, colors, and styles to construct worksheets with almost any design and format that pleases you. Consult the summary of the Mathcad User’s manual in Appendix C or the full manual available from MathSoft. Note that in all our Mathcad examples, we have tried to fit the entire Mathcad session onto a single screen. You should realize that the graph would have to be placed below the commands to work properly. Let’s start with an example that solves for the root of f(x) 5 x 2 cos x. The first step is to enter the function. This is done by typing f(x): which is automatically converted to f(x):5 by Mathcad. The :5 is called the definition symbol. Next an initial guess is input in a similar manner using the definition symbol. Now, soln is defined as root(f(x), x), which invokes the secant method with a starting value of 1.0. Iteration is continued until f(x) evaluated at the proposed root is less than TOL. The value of TOL is set from the Math/Options pull down menu. Finally the value of soln is displayed using a normal equal sign (5). The number of significant figures is set from the Format/Number pull down menu. The text labels and equation definitions can be placed anywhere on the screen in a number of different fonts, styles, sizes, and colors. The graph can be placed anywhere on the worksheet by clicking to the desired location. This places a red cross hair at that location. Then use the Insert/Graph/X-Y Plot pull down menu to place an empty plot on the worksheet with place-holders for the expressions to be graphed and for the ranges of the x and y axes. Simply type f(z) in the placeholder on the y axis and 210 and 10 for the z-axis range. Mathcad does all the rest to produce the graph shown in Fig. 7.7. Once the graph has been created you can use the Format/Graph/X-Y Plot pull down menu to vary the type of graph; change the color, type, and weight of the trace of the function; and add titles, labels and other features. Figure 7.8 shows how Mathcad can be used to find the roots of a polynomial using the polyroots function. First, p(x) and v are input using the :5 definition symbol. Note that v is a vector that contains the coefficients of the polynomial starting with zero-order term and ending in this case with the third-order term. Next, r is defined (using :5) as polyroots(v), which invokes the Laguerre method. The roots contained in r are displayed as rT using a normal equal sign (5). Next, a plot is constructed in a manner similar to the above, except that now two range variables, x and j, are used to define the range of the x axis and the location of the roots. The range variable for x is constructed by typing x and then “:” (which appears as :5) and then 24, and then “,” and then 23.99, and then “;” (which is transformed into .. by Mathcad), and finally 4. This creates a vector of values of x ranging from 24 to 4 with an increment of 0.01 for the x axis with corresponding values for p(x) on the y axis. The j range variable is used to create three values for r and p(r) that are plotted as individual small circles. Note that again, in our effort to fit the entire Mathcad session onto a single screen, we have placed the graph above the commands. You should realize that the graph would have to be below the commands to work properly. The last example shows the solution of a system of nonlinear equations using a Mathcad Solve Block (Fig. 7.9). The process begins with using the definition symbol to create initial guesses for x and y. The word Given then alerts Mathcad that what follows is a system of equations. Then comes the equations and inequalities (not used here). Note that for this application Mathcad requires the use of a symbolic equal sign typed as [Ctrl]5 or , and . to separate the left and right sides of an equation. Now, the variable vec is defined as Find (x,y) and the value of vec is shown using an equal sign.

7.7

FIGURE 7.8 Mathcad screen to solve for roots of polynomial.

FIGURE 7.9 Mathcad screen to solve a system of nonlinear equations.

ROOT LOCATION WITH SOFTWARE PACKAGES

201

202

ROOTS OF POLYNOMIALS

PROBLEMS 7.1 Divide a polynomial f(x) 5 x4 2 7.5x3 1 14.5x2 1 3x 2 20 by the monomial factor x 2 2. Is x 5 2 a root? 7.2 Divide a polynomial f(x) 5 x5 2 5x4 1 x3 2 6x2 2 7x 1 10 by the monomial factor x 2 2. 7.3 Use Müller’s method to determine the positive real root of (a) f(x) 5 x3 1 x2 2 4x 2 4 (b) f(x) 5 x3 2 0.5x2 1 4x 2 2 7.4 Use Müller’s method or MATLAB to determine the real and complex roots of (a) f(x) 5 x3 2 x2 1 2x 2 2 (b) f(x) 5 2x4 1 6x2 1 8 (c) f(x) 5 x4 2 2x3 1 6x2 2 2x 1 5 7.5 Use Bairstow’s method to determine the roots of (a) f(x) 5 22 1 6.2x 2 4x2 1 0.7x3 (b) f(x) 5 9.34 2 21.97x 1 16.3x2 2 3.704x3 (c) f(x) 5 x4 2 2x3 1 6x2 2 2x 1 5 7.6 Develop a program to implement Müller’s method. Test it by duplicating Example 7.2. 7.7 Use the program developed in Prob. 7.6 to determine the real roots of Prob. 7.4a. Construct a graph (by hand or with a software package) to develop suitable starting guesses. 7.8 Develop a program to implement Bairstow’s method. Test it by duplicating Example 7.3. 7.9 Use the program developed in Prob. 7.8 to determine the roots of the equations in Prob. 7.5. 7.10 Determine the real root of x3.5 5 80 with Excel, MATLAB or Mathcad. 7.11 The velocity of a falling parachutist is given by y5

gm ( 1 2 e 2 (cym)t ) c

where g 5 9.81 m/s2. For a parachutist with a drag coefficient c 5 15 kg/s, compute the mass m so that the velocity is y 5 35 m/s at t 5 8 s. Use Excel, MATLAB or Mathcad to determine m. 7.12 Determine the roots of the simultaneous nonlinear equations y 5 2x2 1 x 1 0.75 y 1 5xy 5 x2 Employ initial guesses of x 5 y 5 1.2 and use the Solver tool from Excel or a software package of your choice. 7.13 Determine the roots of the simultaneous nonlinear equations

7.14 Perform the identical MATLAB operations as those in Example 7.7 or use a software package of your choice to find all the roots of the polynomial f(x) 5 (x 1 2) (x 1 5) (x 2 6) (x 2 4) (x 2 8) Note that the poly function can be used to convert the roots to a polynomial. 7.15 Use MATLAB or Mathcad to determine the roots for the equations in Prob. 7.5. 7.16 A two-dimensional circular cylinder is placed in a high-speed uniform flow. Vortices shed from the cylinder at a constant frequency, and pressure sensors on the rear surface of the cylinder detect this frequency by calculating how often the pressure oscillates. Given three data points, use Müller’s method to find the time where the pressure was zero. Time Pressure

0.60

0.62

0.64

20

50

60

7.17 When trying to find the acidity of a solution of magnesium hydroxide in hydrochloric acid, we obtain the following equation A(x) 5 x3 1 3.5x2 2 40 where x is the hydronium ion concentration. Find the hydronium ion concentration for a saturated solution (acidity equals zero) using two different methods in MATLAB (for example, graphically and the roots function). 7.18 Consider the following system with three unknowns a, u, and y: u2 2 2y2 5 a2 u1y52 a2 2 2a 2 u 5 0 Solve for the real values of the unknowns using: (a) the Excel Solver and (b) a symbolic manipulator software package. 7.19 In control systems analysis, transfer functions are developed that mathematically relate the dynamics of a system’s input to its output. A transfer function for a robotic positioning system is given by

(x 2 4) 2 1 (y 2 4) 4 5 5 x2 1 y2 5 16 Use a graphical approach to obtain your initial guesses. Determine refined estimates with the Solver tool from Excel or a software package of your choice.

G(s) 5

C(s) s3 1 9s2 1 26s 1 24 5 4 N(s) s 1 15s3 1 77s2 1 153s 1 90

where G(s) 5 system gain, C(s) 5 system output, N(s) 5 system input, and s 5 Laplace transform complex frequency. Use a

PROBLEMS numerical technique to find the roots of the numerator and denominator and factor these into the form G(s) 5

(s 1 a1 ) (s 1 a2 ) (s 1 a3 ) (s 1 b1 ) (s 1 b2 ) (s 1 b3 ) (s 1 b4 )

where ai and bi 5 the roots of the numerator and denominator, respectively. 7.20 Develop an M-file function for bisection in a similar fashion to Fig. 5.10. Test the function by duplicating the computations from Examples 5.3 and 5.4. 7.21 Develop an M-file function for the false-position method. The structure of your function should be similar to the bisection algorithm outlined in Fig. 5.10. Test the program by duplicating Example 5.5.

203 7.22 Develop an M-file function for the Newton-Raphson method based on Fig. 6.4 and Sec. 6.2.3. Along with the initial guess, pass the function and its derivative as arguments. Test it by duplicating the computation from Example 6.3. 7.23 Develop an M-file function for the secant method based on Fig. 6.4 and Sec. 6.3.2. Along with the two initial guesses, pass the function as an argument. Test it by duplicating the computation from Example 6.6. 7.24 Develop an M-file function for the modified secant method based on Fig. 6.4 and Sec. 6.3.2. Along with the initial guess and the perturbation fraction, pass the function as an argument. Test it by duplicating the computation from Example 6.8.

8

CHAPTER

Case Studies: Roots of Equations The purpose of this chapter is to use the numerical procedures discussed in Chaps. 5, 6, and 7 to solve actual engineering problems. Numerical techniques are important for practical applications because engineers frequently encounter problems that cannot be approached using analytical techniques. For example, simple mathematical models that can be solved analytically may not be applicable when real problems are involved. Thus, more complicated models must be employed. For these cases, it is appropriate to implement a numerical solution on a computer. In other situations, engineering design problems may require solutions for implicit variables in complicated equations. The following case studies are typical of those that are routinely encountered during upper-class courses and graduate studies. Furthermore, they are representative of problems you will address professionally. The problems are drawn from the four major disciplines of engineering: chemical, civil, electrical, and mechanical. These applications also serve to illustrate the trade-offs among the various numerical techniques. The first application, taken from chemical engineering, provides an excellent example of how root-location methods allow you to use realistic formulas in engineering practice. In addition, it also demonstrates how the efficiency of the Newton-Raphson technique is used to advantage when a large number of root-location computations is required. The following engineering design problems are taken from civil, electrical, and mechanical engineering. Section 8.2 uses bisection to determine changes in rainwater chemistry due to increases in atmospheric carbon dioxide. Section 8.3 shows how the roots of transcendental equations can be used in the design of an electrical circuit. Sections 8.2 and 8.3 also illustrate how graphical methods provide insight into the root-location process. Finally, Sec. 8.4 uses a variety of numerical methods to compute the friction factor for fluid flow in a pipe.

8.1

IDEAL AND NONIDEAL GAS LAWS (CHEMICAL/BIO ENGINEERING) Background. The ideal gas law is given by pV 5 nRT

(8.1)

where p is the absolute pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the absolute temperature. Although this equation is 204

8.1

IDEAL AND NONIDEAL GAS LAWS

205

widely used by engineers and scientists, it is accurate over only a limited range of pressure and temperature. Furthermore, Eq. (8.1) is more appropriate for some gases than for others. An alternative equation of state for gases is given by ap 1

a b(y 2 b) 5 RT y2

(8.2)

known as the van der Waals equation, where y 5 V/n is the molal volume and a and b are empirical constants that depend on the particular gas. A chemical engineering design project requires that you accurately estimate the molal volume (y) of both carbon dioxide and oxygen for a number of different temperature and pressure combinations so that appropriate containment vessels can be selected. It is also of interest to examine how well each gas conforms to the ideal gas law by comparing the molal volume as calculated by Eqs. (8.1) and (8.2). The following data are provided: R 5 0.082054 L atm /(mol K) a 5 3.592 f carbon dioxide b 5 0.04267 a 5 1.360 f oxygen b 5 0.03183 The design pressures of interest are 1, 10, and 100 atm for temperature combinations of 300, 500, and 700 K. Solution. Molal volumes for both gases are calculated using the ideal gas law, with n 5 1. For example, if p 5 1 atm and T 5 300 K, y5

RT L atm 300 K V 5 5 0.082054 5 24.6162 L/mol n p mol K 1 atm

These calculations are repeated for all temperature and pressure combinations and presented in Table 8.1. TABLE 8.1 Computations of molal volume.

Temperature, K

Pressure, atm

Molal Volume (Ideal Gas Law), L/mol

300

1 10 100 1 10 100 1 10 100

24.6162 2.4616 0.2462 41.0270 4.1027 0.4103 57.4378 5.7438 0.5744

500

700

Molal Volume (van der Waals) Carbon Dioxide, L/mol

Molal Volume (van der Waals) Oxygen, L/mol

24.5126 2.3545 0.0795 40.9821 4.0578 0.3663 57.4179 5.7242 0.5575

24.5928 2.4384 0.2264 41.0259 4.1016 0.4116 57.4460 5.7521 0.5842

206

CASE STUDIES: ROOTS OF EQUATIONS

The computation of molal volume from the van der Waals equation can be accomplished using any of the numerical methods for finding roots of equations discussed in Chaps. 5, 6, and 7, with f(y) 5 ap 1

a b(y 2 b) 2 RT y2

(8.3)

In this case, the derivative of f(y) is easy to determine and the Newton-Raphson method is convenient and efficient to implement. The derivative of f(y) with respect to y is given by f ¿(y) 5 p 2

a 2ab 1 3 y y2

(8.4)

The Newton-Raphson method is described by Eq. (6.6): yi11 5 yi 2

f(yi ) f ¿(yi )

which can be used to estimate the root. For example, using the initial guess of 24.6162, the molal volume of carbon dioxide at 300 K and 1 atm is computed as 24.5126 L/mol. This result was obtained after just two iterations and has an ea of less than 0.001 percent. Similar computations for all combinations of pressure and temperature for both gases are presented in Table 8.1. It is seen that the results for the ideal gas law differ from those for the van der Waals equation for both gases, depending on specific values for p and T. Furthermore, because some of these results are significantly different, your design of the containment vessels would be quite different, depending on which equation of state was used. In this case, a complicated equation of state was examined using the Newton-Raphson method. The results varied significantly from the ideal gas law for several cases. From a practical standpoint, the Newton-Raphson method was appropriate for this application because f9(y) was easy to calculate. Thus, the rapid convergence properties of the Newton-Raphson method could be exploited. In addition to demonstrating its power for a single computation, the present design problem also illustrates how the Newton-Raphson method is especially attractive when numerous computations are required. Because of the speed of digital computers, the efficiency of various numerical methods for most roots of equations is indistinguishable for a single computation. Even a 1-s difference between the crude bisection approach and the efficient Newton-Raphson does not amount to a significant time loss when only one computation is performed. However, suppose that millions of root evaluations are required to solve a problem. In this case, the efficiency of the method could be a deciding factor in the choice of a technique. For example, suppose that you are called upon to design an automatic computerized control system for a chemical production process. This system requires accurate estimates of molal volumes on an essentially continuous basis to properly manufacture the final product. Gauges are installed that provide instantaneous readings of pressure and temperature. Evaluations of y must be obtained for a variety of gases that are used in the process. For such an application, bracketing methods such as bisection or false position would probably be too time-consuming. In addition, the two initial guesses that are required for

8.2

GREENHOUSE GASES AND RAINWATER

207

these approaches may also interject a critical delay in the procedure. This shortcoming is relevant to the secant method, which also needs two initial estimates. In contrast, the Newton-Raphson method requires only one guess for the root. The ideal gas law could be employed to obtain this guess at the initiation of the process. Then, assuming that the time frame is short enough so that pressure and temperature do not vary wildly between computations, the previous root solution would provide a good guess for the next application. Thus, the close guess that is often a prerequisite for convergence of the Newton-Raphson method would automatically be available. All the above considerations would greatly favor the Newton-Raphson technique for such problems.

8.2

GREENHOUSE GASES AND RAINWATER (CIVIL/ENVIRONMENTAL ENGINEERING) Background. Civil engineering is a broad field that includes such diverse areas as structural, geotechnical, transportation, water-resources, and environmental engineering. The last area has traditionally dealt with pollution control. However, in recent years, environmental engineers (as well as chemical engineers) have addressed broader problems such as climate change. It is well documented that the atmospheric levels of several greenhouse gases have been increasing over the past 50 years. For example, Fig. 8.1 shows data for the partial pressure of carbon dioxide (CO2) collected at Mauna Loa, Hawaii, from 1958 through 2003. The trend in the data can be nicely fit with a quadratic polynomial (in Part Five, we will learn how to determine such polynomials), pCO2 5 0.011825(t 2 1980.5) 2 1 1.356975(t 2 1980.5) 1 339 where pCO2 5 the partial pressure of CO2 in the atmosphere [ppm]. The data indicate that levels have increased over 19% during the period from 315 to 376 ppm.

FIGURE 8.1 Average annual partial pressures of atmospheric carbon dioxide (ppm) measured at Mauna Loa, Hawaii.

370

pCO 2 (ppm)

350

330

310 1950

1960

1970

1980

1990

2000

2010

208

CASE STUDIES: ROOTS OF EQUATIONS

Aside from global warming, greenhouse gases can also influence atmospheric chemistry. One question that we can address is how the carbon dioxide trend is affecting the pH of rainwater. Outside of urban and industrial areas, it is well documented that carbon dioxide is the primary determinant of the pH of the rain. pH is the measure of the activity of hydrogen ions and, therefore, its acidity. For dilute aqueous solutions, it can be computed as pH 5 2log10 [H 1 ]

(8.5)

where [H1] is the molar concentration of hydrogen ions. The following five nonlinear system of equations govern the chemistry of rainwater, K1 5 106

[H 1 ][HCO2 3] KH pCO2

(8.6)

[H 1 ][CO22 3 ] 2 [HCO3 ]

(8.7)

Kw 5 [H 1 ][OH 2 ]

(8.8)

K2 5

cT 5

KH pCO2 106

1 [HCO32 ] 1 [CO22 3 ]

22 2 1 0 5 [HCO2 3 ] 1 2[CO3 ] 1 [OH ] 2 [H ]

(8.9) (8.10)

where KH 5 Henry’s constant, and K1, K2, and Kw are equilibrium coefficients. The five unknowns in this system of five nonlinear equations are cT 5 total inorganic carbon, 22 1 2 [HCO2 3 ] 5 bicarbonate, [CO3 ] 5 carbonate, [H ] 5 hydrogen ion, and [OH ] 5 hydroxyl ion. Notice how the partial pressure of CO2 shows up in Eqs. (8.6) and (8.9). Use these equations to compute the pH of rainwater given that KH 5 1021.46, K1 5 1026.3, K2 5 10210.3, and Kw 5 10214. Compare the results in 1958 when the pCO2 was 315 and in 2003 when it was 375 ppm. When selecting a numerical method for your computation, consider the following: You know with certainty that the pH of rain in pristine areas always falls between 2 and 12. You also know that your measurement devices can only measure pH to two places of decimal precision. Solution. There are a variety of ways to solve this nonlinear system of five equations. One way is to eliminate unknowns by combining them to produce a single function that only depends on [H1]. To do this, first solve Eqs. (8.6) and (8.7) for [HCO2 3 ] 5

K1

KH pCO2 10 [H 1 ] K2 [HCO2 3] [CO22 ] 5 3 1 [H ] 6

(8.11) (8.12)

Substitute Eq. (8.11) into (8.12) [CO22 3 ] 5

K2K1 106 [H 1 ]2

KH pCO2

(8.13)

8.3

DESIGN OF AN ELECTRIC CIRCUIT

209

Equations (8.11) and (8.13) can be substituted along with Eq. (8.8) into Eq. (8.10) to give 05

K1 6

1

10 [H ]

KH pCO2 1 2

K2K1 6

1 2

10 [H ]

KH pCO2 1

Kw 2 [H 1 ] [H 1 ]

(8.14)

Although it might not be apparent, this result is a third-order polynomial in [H1]. Thus, its root can be used to compute the pH of the rainwater. Now we must decide which numerical method to employ to obtain the solution. There are two reasons why bisection would be a good choice. First, the fact that the pH always falls within the range from 2 to 12 provides us with two good initial guesses. Second, because the pH can only be measured to two decimal places of precision, we will be satisfied with an absolute error of Ea,d 5 0.005. Remember that given an initial bracket and the desired relative error, we can compute the number of iterations a priori. Using Eq. (5.5), the result is n 5 log2(10)0.005 5 10.9658. Thus, eleven iterations of bisection will produce the desired precision. If this is done, the result for 1958 will be a pH of 5.6279 with a relative error of 0.0868%. We can be confident that the rounded result of 5.63 is correct to two decimal places. This can be verified by performing another run with more iterations. For example, if we perform 35 iterations, a result of 5.6304 is obtained with an approximate relative error of εa 5 5.17 3 1029%. The same calculation can be repeated for the 2003 conditions to give pH 5 5.59 with εa 5 0.0874%. Interestingly, these results indicate that the 19% rise in atmospheric CO2 levels has produced only a 0.67% drop in pH. Although this is certainly true, remember that the pH represents a logarithmic scale as defined by Eq. (8.5). Consequently, a unit drop in pH represents a 10-fold increase in hydrogen ion. The concentration can be computed as [H1] 5 102pH and the resulting percent change can be calculated as 9.1%. Therefore, the hydrogen ion concentration has increased about 9%. There is quite a lot of controversy related to the true significance of the greenhouse gas trends. However, regardless of the ultimate implications, it is quite sobering to realize that something as large as our atmosphere has changed so much over a relatively short time period. This case study illustrates how numerical methods can be employed to analyze and interpret such trends. Over the coming years, engineers and scientists can hopefully use such tools to gain increased understanding and help rationalize the debate over their ramifications.

8.3

DESIGN OF AN ELECTRIC CIRCUIT (ELECTRICAL ENGINEERING) Background. Electrical engineers often use Kirchhoff’s laws to study the steady-state (not time-varying) behavior of electric circuits. Such steady-state behavior will be examined in Sec. 12.3. Another important problem involves circuits that are transient in nature where sudden temporal changes take place. Such a situation occurs following the closing of the switch in Fig. 8.2. In this case, there will be a period of adjustment following the closing of the switch as a new steady state is reached. The length of this adjustment period is closely related to the storage properties of the capacitor and the inductor. Energy storage may oscillate between these two elements during a transient period. However, resistance in the circuit will dissipate the magnitude of the oscillations.

210

FIGURE 8.2 An electric circuit. When the switch is closed, the current will undergo a series of oscillations until a new steady state is reached.

CASE STUDIES: ROOTS OF EQUATIONS

Switch –

Battery

+

V0

– +

i Capacitor

Inductor

Resistor

The flow of current through the resistor causes a voltage drop (VR) given by VR 5 iR where i 5 the current and R 5 the resistance of the resistor. When R and i have units of ohms and amperes, respectively, VR has units of volts. Similarly, an inductor resists changes in current, such that the voltage drop VL across it is VL 5 L

di dt

where L 5 the inductance. When L and i have units of henrys and amperes, respectively, VL has units of volts and t has units of seconds. The voltage drop across the capacitor (VC) depends on the charge (q) on it: VC 5

q C

(8.15)

where C 5 the capacitance. When the charge is expressed in units of coulombs, the unit of C is the farad. Kirchhoff’s second law states that the algebraic sum of voltage drops around a closed circuit is zero. After the switch is closed we have L

q di 1 Ri 1 5 0 dt C

(8.16)

However, the current is related to the charge according to i5

dq dt

(8.17)

Therefore, L

d 2q dt

2

1R

dq 1 1 q50 dt C

(8.18)

This is a second-order linear ordinary differential equation that can be solved using the methods of calculus. This solution is given by q(t) 5 q0e2Rty(2L)cos c

R 2 1 2 a b td 2L B LC

(8.19)

8.3

q(t) q0

Time

FIGURE 8.3 The charge on a capacitor as a function of time following the closing of the switch in Fig. 8.2.

DESIGN OF AN ELECTRIC CIRCUIT

211

where at t 5 0, q 5 q0 5 V0C, and V0 5 the voltage from the charging battery. Equation (8.19) describes the time variation of the charge on the capacitor. The solution q(t) is plotted in Fig. 8.3. A typical electrical engineering design problem might involve determining the proper resistor to dissipate energy at a specified rate, with known values for L and C. For this problem, assume the charge must be dissipated to 1 percent of its original value (qq0 5 0.01) in t 5 0.05 s, with L 5 5 H and C 5 1024 F. Solution. It is necessary to solve Eq. (8.19) for R, with known values of q, q0, L, and C. However, a numerical approximation technique must be employed because R is an implicit variable in Eq. (8.19). The bisection method will be used for this purpose. The other methods discussed in Chaps. 5 and 6 are also appropriate, although the NewtonRaphson method might be deemed inconvenient because the derivative of Eq. (8.19) is a little cumbersome. Rearranging Eq. (8.19), f(R) 5 e2Rty(2L)cos c

q 1 R 2 2 a b td 2 q0 B LC 2L

or using the numerical values given, f(R) 5 e20.005Rcos[ 22000 2 0.01R2 (0.05)] 2 0.01

(8.20)

Examination of this equation suggests that a reasonable initial range for R is 0 to 400 V (because 2000 2 0.01R2 must be greater than zero). Figure 8.4, a plot of Eq. (8.20), confirms this. Twenty-one iterations of the bisection method give R 5 328.1515 V, with an error of less than 0.0001 percent. FIGURE 8.4 Plot of Eq. (8.20) used to obtain initial guesses for R that bracket the root.

f (R) 0.0

– 0.2

– 0.4

– 0.6

Root ⯝ 325 200

400

R

212

CASE STUDIES: ROOTS OF EQUATIONS

Thus, you can specify a resistor with this rating for the circuit shown in Fig. 8.2 and expect to achieve a dissipation performance that is consistent with the requirements of the problem. This design problem could not be solved efficiently without using the numerical methods in Chaps. 5 and 6.

8.4

PIPE FRICTION (MECHANICAL/AEROSPACE ENGINEERING) Background. Determining fluid flow through pipes and tubes has great relevance in many areas of engineering and science. In mechanical and aerospace engineering, typical applications include the flow of liquids and gases through cooling systems. The resistance to flow in such conduits is parameterized by a dimensionless number called the friction factor. For turbulent flow, the Colebrook equation provides a means to calculate the friction factor, 05

1 e 2.51 1 2.0 log a 1 b 3.7D 1f Re1f

(8.21)

where 5 the roughness (m), D 5 diameter (m), and Re 5 the Reynolds number, Re 5

rVD m

where 5 the fluid’s density (kg/m3), V 5 its velocity (m/s), and 5 dynamic viscosity (N ? s/m2). In addition to appearing in Eq. (8.21), the Reynolds number also serves as the criterion for whether flow is turbulent (Re . 4000). In the present case study, we will illustrate how the numerical methods covered in this part of the book can be employed to determine f for air flow through a smooth, thin tube. For this case, the parameters are 5 1.23 kg/m3, 5 1.79 3 1025 N ? s/m2, D 5 0.005 m, V 5 40 m/s, and 5 0.0015 mm. Note that friction factors range from about 0.008 to 0.08. In addition, an explicit formulation called the Swamee-Jain equation provides an approximate estimate, 1.325

f5 c ln a

e 5.74 2 1 0.9 b d 3.7D Re

(8.22)

Solution. The Reynolds number can be computed as Re 5

rVD 1.23(40)0.005 5 13,743 5 m 1.79 3 1025

This value along with the other parameters can be substituted into Eq. (8.21) to give g( f ) 5

1 0.0000015 2.51 1 2.0 log a 1 b 3.7(0.005) 1f 13,7431f

Before determining the root, it is advisable to plot the function to estimate initial guesses and to anticipate possible difficulties. This can be done easily with tools such

8.4

PIPE FRICTION

213

6 5 4

g( f )

3 2 1 0 –1 –2 –3

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

f

FIGURE 8.5

as MATLAB software, Excel, or Mathcad. For example, a plot of the function can be generated with the following MATLAB commands >> >> >> >>

rho=1.23;mu=1.79e-5;D=0.005;V=40;e=0.0015/1000; Re=rho*V*D/mu; [email protected](f) 1/sqrt(f)+2*log10(e/(3.7*D)+2.51/(Re*sqrt(f))); fplot(g,[0.008 0.08]),grid,xlabel('f'),ylabel('g(f)')

As in Fig. 8.5, the root is located at about 0.03. Because we are supplied initial guesses (xl 5 0.008 and xu 5 0.08), either of the bracketing methods from Chap. 5 could be used. For example, bisection gives a value of f 5 0.0289678 with a percent relative error of error of 5.926 3 1025 in 22 iterations. False position yields a result of similar precision in 26 iterations. Thus, although they produce the correct result, they are somewhat inefficient. This would not be important for a single application, but could become prohibitive if many evaluations were made. We could try to attain improved performance by turning to an open method. Because Eq. (8.21) is relatively straightforward to differentiate, the Newton-Raphson method is a good candidate. For example, using an initial guess at the lower end of the range (x0 5 0.008), Newton-Raphson converges quickly to 0.0289678 with an approximate error of 6.87 3 1026% in only 6 iterations. However, when the initial guess is set at the upper end of the range (x0 5 0.08), the routine diverges! As can be seen by inspecting Fig. 8.5, this occurs because the function’s slope at the initial guess causes the first iteration to jump to a negative value. Further runs demonstrate that for this case, convergence only occurs when the initial guess is below about 0.066.

214

CASE STUDIES: ROOTS OF EQUATIONS

So we can see that although the Newton-Raphson is very efficient, it requires good initial guesses. For the Colebrook equation, a good strategy might be to employ the Swamee-Jain equation (Eq. 8.22) to provide the initial guess as in f5

1.325 5 0.029031 2 0.0000015 5.74 c ln a 1 bd 3.7(0.005) 137430.9

For this case, Newton-Raphson converges in only 3 iterations quickly to 0.0289678 with an approximate error of 8.51 3 10210%. Aside from our homemade functions, we can also use professional root finders like MATLAB’s built-in fzero function. However, just as with the Newton-Raphson method, divergence also occurs when fzero function is used with a single guess. However, in this case, guesses at the lower end of the range cause problems. For example, >> >> >> >>

rho=1.23;mu=1.79e-5;D=0.005;V=40;e=0.0015/1000; Re=rho*V*D/mu [email protected](f) 1/sqrt(f)+2*log10(e/(3.7*D)+2.51/(Re*sqrt(f))); fzero(g,0.008)

Exiting fzero: aborting search for an interval containing a sign change because complex function value encountered during search. (Function value at -0.0028 is -4.9202820.2423i.) Check function or try again with a different starting value. ans = NaN

If the iterations are displayed using optimset (recall Sec. 7.7.2), it is revealed that a negative value occurs during the search phase before a sign change is detected and the routine aborts. However, for single initial guesses above about 0.016, the routine works nicely. For example, for the guess of 0.08 that caused problems for Newton-Raphson, fzero does just fine, >> fzero(g,0.08) ans = 0.02896781017144

As a final note, let’s see whether convergence is possible for simple fixed-point iteration. The easiest and most straightforward version involves solving for the first f in Eq. (8.21), fi11 5

0.25 2 e 2.51 alog a bb 1 3.7D Re1fi

(8.23)

The cobweb display of this function depicted indicates a surprising result (Fig. 8.6). Recall that fixed-point iteration converges when the y2 curve has a relatively flat slope (i.e., Zg9()Z , 1). As indicated by Fig. 8.6, the fact that the y2 curve is quite flat in the range from f 5 0.008 to 0.08 means that not only does fixed-point iteration converge,

PROBLEMS

215

y 0.05

y1 = x

0.04

0.03

y2 = g(x)

0.02

0.01

0

0

0.02

0.04

0.06

0.08

x

FIGURE 8.6

but it converges fairly rapidly! In fact, for initial guesses anywhere between 0.008 and 0.08, fixed-point iteration yields predictions with percent relative errors less than 0.008% in six or fewer iterations. Thus, this simple approach that requires only one guess and no derivative estimates performs really well for this particular case. The take-home message from this case study is that even great, professionallydeveloped software like MATLAB is not always foolproof. Further, there is usually no single method that works best for all problems. Sophisticated users understand the strengths and weaknesses of the available numerical techniques. In addition, they understand enough of the underlying theory so that they can effectively deal with situations where a method breaks down.

PROBLEMS Chemical/Bio Engineering 8.1 Perform the same computation as in Sec. 8.1, but for ethyl alcohol (a 5 12.02 and b 5 0.08407) at a temperature of 375 K and p of 2.0 atm. Compare your results with the ideal gas law. Use any of the numerical methods discussed in Chaps. 5 and 6 to perform the computation. Justify your choice of technique. 8.2 In chemical engineering, plug flow reactors (that is, those in which fluid flows from one end to the other with minimal mixing along the longitudinal axis) are often used to convert reactants into products. It has been determined that the efficiency of the conversion can sometimes be improved by recycling a portion of the

product stream so that it returns to the entrance for an additional pass through the reactor (Fig. P8.2). The recycle rate is defined as R5

volume of fluid returned to entrance volume leaving the system

Suppose that we are processing a chemical A to generate a product B. For the case where A forms B according to an autocatalytic reaction (that is, in which one of the products acts as a catalyst or stimulus for the reaction), it can be shown that an optimal recycle rate must satisfy

216

CASE STUDIES: ROOTS OF EQUATIONS

Feed

Plug flow reactor

Product

Recycle

1 1 R(1 2 XAf ) R(1 2 XAf )

5

R11 R[1 1 R(1 2 XAf )]

H2 1 12 O2

If it is assumed that this is the only reaction involved, the mole fraction x of H2O that dissociates can be represented by K5

2pt x 1 2 xA2 1 x

(P8.3.1)

where K 5 the reaction equilibrium constant and pt 5 the total pressure of the mixture. If pt 5 3 atm and K 5 0.05, determine the value of x that satisfies Eq. (P8.3.1). 8.4 The following equation pertains to the concentration of a chemical in a completely mixed reactor: c 5 cin ( 1 2 e20.04t ) 1 c0e20.04t If the initial concentration c0 5 4 and the inflow concentration cin 5 10, compute the time required for c to be 93 percent of cin. 8.5 A reversible chemical reaction 2A 1 B

C

can be characterized by the equilibrium relationship K5

(ca,0 2 2x) 2 (cb,0 2 x)

where the subscript 0 designates the initial concentration of each constituent. If K 5 0.015, ca,0 5 42, cb,0 5 30, and cc,0 5 4, determine the value of x. (a) Obtain the solution graphically. (b) On the basis of (a), solve for the root with initial guesses of xl 5 0 and xu 5 20 to s 5 0.5%. Choose either bisection or false position to obtain your solution. Justify your choice. 8.6 The following chemical reactions take place in a closed system 2A 1 B

where XAf 5 the fraction of reactant A that is converted to product B. The optimal recycle rate corresponds to the minimum-sized reactor needed to attain the desired level of conversion. Use a numerical method to determine the recycle ratios needed to minimize reactor size for a fractional conversion of XAf 5 0.9. 8.3 In a chemical engineering process, water vapor (H2O) is heated to sufficiently high temperatures that a significant portion of the water dissociates, or splits apart, to form oxygen (O2) and hydrogen (H2): H2O

(cc,0 1 x)

K5

FIGURE P8.2 Schematic representation of a plug ﬂow reactor with recycle.

ln

moles of C that are produced. Conservation of mass can be used to reformulate the equilibrium relationship as

cc c2a cb

where the nomenclature ci represents the concentration of constituent i. Suppose that we define a variable x as representing the number of

C C

A1D

At equilibrium, they can be characterized by K1 5

cc

c2a cb cc K2 5 ca cd

where the nomenclature represents the concentration of constituent i. If x1 and x2 are the number of moles of C that are produced due to the first and second reactions, respectively, use an approach similar to that of Prob. 8.5 to reformulate the equilibrium relationships in terms of the initial concentrations of the constituents. Then, use the Newton-Raphson method to solve the pair of simultaneous nonlinear equations for x1 and x2 if K1 5 4 3 1024, K2 5 3.7 3 1022, ca,0 5 50, cb,0 5 20, cc,0 5 5, and cd,0 5 10. Use a graphical approach to develop your initial guesses. 8.7 The Redlich-Kwong equation of state is given by p5

RT a 2 y2b y(y 1 b) 1T

where R 5 the universal gas constant [5 0.518 kJ/(kg K)], T 5 absolute temperature (K), p 5 absolute pressure (kPa), and y 5 the volume of a kg of gas (m3/kg). The parameters a and b are calculated by a 5 0.427

R2 T 2.5 Tc c b 5 0.0866R pc pc

where pc 5 critical pressure (kPa) and Tc 5 critical temperature (K). As a chemical engineer, you are asked to determine the amount of methane fuel (pc 5 4600 kPa and Tc 5 191 K) that can be held in a 3-m3 tank at a temperature of 2408C with a pressure of 65,000 kPa. Use a root-locating method of your choice to calculate y and then determine the mass of methane contained in the tank.

PROBLEMS 8.8 The volume V of liquid in a hollow horizontal cylinder of radius r and length L is related to the depth of the liquid h by V 5 c r2cos21 a

r2h b 2 (r 2 h) 22rh 2 h2 d L r

Determine h given r 5 2 m, L 5 5 m, and V 5 8 m3. Note that if you are using a programming language or software tool that is not rich in trigonometric functions, the arc cosine can be computed with cos21x 5

p x b 2 tan21 a 2 21 2 x2

8.9 The volume V of liquid in a spherical tank of radius r is related to the depth h of the liquid by 2

V5

ph (3r 2 h) 3

Determine h given r 5 1 m and V 5 0.5 m3. 8.10 For the spherical tank in Prob. 8.9, it is possible to develop the following two fixed-point formulas: h5

h3 1 (3Vyp) B 3r

217 Given the parameter values listed below, find the void fraction e of the bed. Dp Go m

¢PrDp G2oL

h5

3

B

3 arh2 2

V b p

If r 5 1 m and V 5 0.5 m3, determine whether either of these is stable, and the range of initial guesses for which they are stable. 8.11 The operation of a constant density plug flow reactor for the production of a substance via an enzymatic reaction is described by the equation below, where V is the volume of the reactor, F is the flow rate of reactant C, Cin and Cout are the concentrations of reactant entering and leaving the reactor, respectively, and K and kmax are constants. For a 100-L reactor, with an inlet concentration of Cin 5 0.2 M, an inlet flow rate of 80 L/s, kmax 5 1022 s21, and K 5 0.1 M, find the concentration of C at the outlet of the reactor. V 52 F

#

Cout

Cin

K 1 1 dC kmaxC kmax

8.12 The Ergun equation, shown below, is used to describe the flow of a fluid through a packed bed. DP is the pressure drop, r is the density of the fluid, Go is the mass velocity (mass flow rate divided by cross-sectional area), Dp is the diameter of the particles within the bed, m is the fluid viscosity, L is the length of the bed, and e is the void fraction of the bed. ¢Pr Dp e3 12e 5 150 1 1.75 2 (DpGo ym) Go L 1 2 e

5 10

8.13 The pressure drop in a section of pipe can be calculated as ¢p 5 f

LrV 2 2D

where Dp 5 the pressure drop (Pa), f 5 the friction factor, L 5 the length of pipe [m], 5 density (kg/m3), V 5 velocity (m /s), and D 5 diameter (m). For turbulent flow, the Colebrook equation provides a means to calculate the friction factor, 1 2.51 e 1 5 22.0 log a b 3.7D 1f Re 1f where 5 the roughness (m), and Re 5 the Reynolds number, Re 5

and

5 1000

rVD m

where 5 dynamic viscosity (N ? s/m2). (a) Determine Dp for a 0.2-m-long horizontal stretch of smooth drawn tubing given 5 1.23 kg/m3, m 5 1.79 3 1025 N ? s/m2, D 5 0.005 m, V 5 40 m/s, and e 5 0.0015 mm. Use a numerical method to determine the friction factor. Note that smooth pipes with Re , 105, a good initial guess can be obtained using the Blasius formula, f 5 0.316yRe0.25. (b) Repeat the computation but for a rougher commercial steel pipe (e 5 0.045 mm). Civil and Environmental Engineering 8.14 In structural engineering, the secant formula defines the force per unit area, PyA, that causes a maximum stress m in a column of given slenderness ratio Lyk: sm P 5 2 A 1 1 ( ecyk ) sec[0.5 1Py(EA) (Lyk)] where ecyk2 5 the eccentricity ratio and E 5 the modulus of elasticity. If for a steel beam, E 5 200,000 MPa, ecyk2 5 0.2, and sm 5 250 MPa, compute PyA for Lyk 5 100. Recall that sec x 5 1ycos x. 8.15 In environmental engineering (a specialty area in civil engineering), the following equation can be used to compute the oxygen level c (mg/L) in a river downstream from a sewage discharge: c 5 10 2 20 ( e20.2x 2 e20.75x )

218

CASE STUDIES: ROOTS OF EQUATIONS

where x is the distance downstream in kilometers. (a) Determine the distance downstream where the oxygen level first falls to a reading of 5 mg/L. (Hint: It is within 2 km of the discharge.) Determine your answer to a 1% error. Note that levels of oxygen below 5 mg/L are generally harmful to gamefish such as trout and salmon. (b) Determine the distance downstream at which the oxygen is at a minimum. What is the concentration at that location? 8.16 The concentration of pollutant bacteria c in a lake decreases according to

where the hyperbolic cosine can be computed by cosh x 5

Use a numerical method to calculate a value for the parameter TA given values for the parameters w 5 10 and y0 5 5, such that the cable has a height of y 5 15 at x 5 50. 8.18 Figure P8.18a shows a uniform beam subject to a linearly increasing distributed load. The equation for the resulting elastic curve is (see Fig. P8.18b)

c 5 70e21.5t 1 25e20.075t

dx

2

5

(P8.18.1)

Use bisection to determine the point of maximum deflection (that is, the value of x where dy/dx 5 0). Then substitute this value into Eq. (P8.18.1) to determine the value of the maximum deflection. Use the following parameter values in your computation: L 5 450 cm, E 5 50,000 kN/cm2, I 5 30,000 cm4, and w0 5 1.75 kN/cm. 8.19 The displacement of a structure is defined by the following equation for a damped oscillation: y 5 8e2kt cos vt where k 5 0.5 and 5 3. (a) Use the graphical method to make an initial estimate of the time required for the displacement to decrease to 4. (b) Use the Newton-Raphson method to determine the root to s 5 0.01%. (c) Use the secant method to determine the root to s 5 0.01%. 8.20 The Manning equation can be written for a rectangular open channel as

dy 2 w 11a b TA B dx

Calculus can be employed to solve this equation for the height y of the cable as a function of distance x, y5

w0 ( 2x5 1 2L2x3 2 L4x ) 120EIL

y5

Determine the time required for the bacteria concentration to be reduced to 9 using (a) the graphical method and (b) using the Newton-Raphson method with an initial guess of t 5 10 and a stopping criterion of 0.5%. Check your result. 8.17 A catenary cable is one that is hung between two points not in the same vertical line. As depicted in Fig. P8.17a, it is subject to no loads other than its own weight. Thus, its weight (N/m) acts as a uniform load per unit length along the cable. A free-body diagram of a section AB is depicted in Fig. P8.17b, where TA and TB are the tension forces at the end. Based on horizontal and vertical force balances, the following differential equation model of the cable can be derived: d 2y

1 x ( e 1 e2x ) 2

TA TA w cosh a xb 1 y0 2 w TA w

Q5

1S(BH) 5y3 n(B 1 2H) 2y3

y

TB

B

FIGURE P8.17 (a) Forces acting on a section AB of a ﬂexible hanging cable. The load is uniform along the cable (but not uniform per the horizontal distance x). (b) A freebody diagram of section AB.

A w

W = ws

y0

TA x

(a)

(b)

PROBLEMS

219 formula relating present worth P, annual payments A, number of years n, and interest rate i is

w0

A5P

8.23 Many fields of engineering require accurate population estimates. For example, transportation engineers might find it necessary to determine separately the population growth trends of a city and adjacent suburb. The population of the urban area is declining with time according to

L

(a) (x = L, y = 0)

(x = 0, y = 0)

Pu (t) 5 Pu, maxe2kut 1 Pu, min x

while the suburban population is growing, as in Ps (t) 5

(b) FIGURE P8.18

where Q 5 flow [m3/s], S 5 slope [m/m], H 5 depth [m], and n 5 the Manning roughness coefficient. Develop a fixed-point iteration scheme to solve this equation for H given Q 5 5, S 5 0.0002, B 5 20, and n 5 0.03. Prove that your scheme converges for all initial guesses greater than or equal to zero. 8.21 In ocean engineering, the equation for a reflected standing wave in a harbor is given by 5 16, t 5 12, 5 48: h 5 h0 c sin a

i(1 1 i) n (1 1 i) n 2 1

Ps, max 1 1 [Ps, max yP0 2 1]e2kst

where Pu, max, ku, Ps, max, P0, and ks 5 empirically derived parameters. Determine the time and corresponding values of Pu(t) and Ps(t) when the suburbs are 20% larger than the city. The parameter values are Pu, max 5 75,000, ku 5 0.045/yr, Pu, min 5 100,000 people, Ps, max 5 300,000 people, P0 5 10,000 people, ks 5 0.08/yr. To obtain your solutions, use (a) graphical, (b) false-position, and (c) modified secant methods. 8.24 A simply supported beam is loaded as shown in Fig. P8.24. Using singularity functions, the shear along the beam can be expressed by the equation: V(x) 5 20[kx 2 0l1 2 kx 2 5l1 ] 2 15 kx 2 8l0 2 57 By definition, the singularity function can be expressed as follows:

2px 2pty b cos a b 1 e2x d l l

kx 2 aln 5 e

Solve for the lowest positive value of x if h 5 0.4h0. 8.22 You buy a $20,000 piece of equipment for nothing down and $4000 per year for 6 years. What interest rate are you paying? The

(x 2 a) n 0

Use a numerical method to find the point(s) where the shear equals zero.

FIGURE P8.24 20 kips/ft

150 kip-ft

5’

when x . a f when x # a

2’

15 kips

1’

2’

220

CASE STUDIES: ROOTS OF EQUATIONS

8.25 Using the simply supported beam from Prob. 8.24, the moment along the beam, M(x), is given by: M(x) 5 210[kx 2 0l2 2 kx 2 5l2 ] 1 15 kx 2 8l1 1 150 kx 2 7l0 1 57x Use a numerical method to find the point(s) where the moment equals zero. 8.26 Using the simply supported beam from Prob. 8.24, the slope along the beam is given by: duy dx

(x) 5

210 15 [kx 2 0l3 2 kx 2 5l3 ] 1 kx 2 8l2 3 2 1 150 kx 2 7l1 1

57 2 x 2 238.25 2

Use a numerical method to find the point(s) where the slope equals zero. 8.27 Using the simply supported beam from Prob. 8.24, the displacement along the beam is given by: uy (x) 5

25 15 [kx 2 0l4 2 kx 2 5l4 ] 1 kx 2 8l3 6 6

r5

1 qnm

where n5

57 3 1 75 kx 2 7l2 1 x 2 238.25x 6 (a) Find the point(s) where the displacement equals zero. (b) How would you use a root location technique to determine the location of the minimum displacement? 8.28 Although we did not mention it in Sec. 8.2, Eq. (8.10) is actually an expression of electroneutrality; that is, that positive and negative charges must balance. This can be seen more clearly by expressing it as 2 [H 1 ] 5 [HCO32 ] 1 2[CO22 3 ] 1 [OH ]

1 T 22.42 ( N 1 2N2 1 4n2i ) and m 5 m0 a b 2 T0

Determine N, given T0 5 300 K, T 5 1000 K, 0 5 1300 cm2 (V s)21, q 5 1.6 3 10219 C, ni 5 6.21 3 109 cm23, and a desired 5 6 3 106 V s cm/C. Use (a) bisection and (b) the modified secant method. 8.32 A total charge Q is uniformly distributed around a ring-shaped conductor with radius a. A charge q is located at a distance x from the center of the ring (Fig. P8.32). The force exerted on the charge by the ring is given by F5

In other words, the positive charges must equal the negative charges. Thus, when you compute the pH of a natural water body such as a lake, you must also account for other ions that may be present. For the case where these ions originate from nonreactive salts, the net negative minus positive charges due to these ions are lumped together in a quantity called alkalinity, and the equation is reformulated as 2 Alk 1 [H 1 ] 5 [HCO23 ] 1 2[CO22 3 ] 1 [OH ]

Electrical Engineering 8.29 Perform the same computation as in Sec. 8.3, but determine the value of L required for the circuit to dissipate to 1% of its original value in t 5 0.05 s, given R 5 280 V, and C 5 1024 F. Use (a) a graphical approach, (b) bisection, and (c) root location software such as the Excel Solver, the MATLAB function fzero, or the Mathcad function root. 8.30 An oscillating current in an electric circuit is described by i 5 9e2t sin(2 t), where t is in seconds. Determine the lowest value of t such that i 5 3.5. 8.31 The resistivity of doped silicon is based on the charge q on an electron, the electron density n, and the electron mobility . The electron density is given in terms of the doping density N and the intrinsic carrier density ni. The electron mobility is described by the temperature T, the reference temperature T0, and the reference mobility 0. The equations required to compute the resistivity are

qQx 1 2 4pe0 ( x 1 a2 ) 3y2

where e0 5 8.85 3 10212 C2/(N m2). Find the distance x where the force is 1N if q and Q are 2 3 1025 C for a ring with a radius of 0.9 m.

FIGURE P8.32

(P8.28)

where Alk 5 alkalinity (eq/L). For example, the alkalinity of Lake Superior is approximately 0.4 3 1023 eq/L. Perform the same calculations as in Sec. 8.2 to compute the pH of Lake Superior in 2008. Assume that just like the raindrops, the lake is in equilibrium with atmospheric CO2, but account for the alkalinity as in Eq. (P8.28).

a x

Q

q

PROBLEMS

221

8.33 Figure P8.33 shows a circuit with a resistor, an inductor, and a capacitor in parallel. Kirchhoff’s rules can be used to express the impedance of the system as

h

1 1 1 2 5 b 1 avC 2 2 Z BR vL where Z 5 impedance (V) and v 5 the angular frequency. Find the that results in an impedance of 75 V using both bisection and false position with initial guesses of 1 and 1000 for the following parameters: R 5 225 V, C 5 0.6 3 1026 F, and L 5 0.5 H. Determine how many iterations of each technique are necessary to determine the answer to s 5 0.1%. Use the graphical approach to explain any difficulties that arise.

ⵑ

R

h+d d

L

C

(a)

(b)

FIGURE P8.35

8.36 Mechanical engineers, as well as most other engineers, use thermodynamics extensively in their work. The following polynomial can be used to relate the zero-pressure specific heat of dry air, cp kJ/(kg K), to temperature (K): cp 5 0.99403 1 1.671 3 1024T 1 9.7215 3 1028T 2 29.5838 3 10211T 3 1 1.9520 3 10214T 4

FIGURE P8.33

Mechanical and Aerospace Engineering 8.34 Beyond the Colebrook equation, other relationships, such as the Fanning friction factor f, are available to estimate friction in pipes. The Fanning friction factor is dependent on a number of parameters related to the size of the pipe and the fluid, which can all be represented by another dimensionless quantity, the Reynolds number Re. A formula that predicts f given Re is the von Karman equation, 1 5 4 log10 (Re 1f ) 2 0.4 1f Typical values for the Reynolds number for turbulent flow are 10,000 to 500,000 and for the Fanning friction factor are 0.001 to 0.01. Develop a function that uses bisection to solve for f given a user-supplied value of Re between 2500 and 1,000,000. Design the function so that it ensures that the absolute error in the result is Ea,d , 0.000005. 8.35 Real mechanical systems may involve the deflection of nonlinear springs. In Fig. P8.35, a mass m is released a distance h above a nonlinear spring. The resistance force F of the spring is given by

Determine the temperature that corresponds to a specific heat of 1.2 kJ/(kg K). 8.37 Aerospace engineers sometimes compute the trajectories of projectiles like rockets. A related problem deals with the trajectory of a thrown ball. The trajectory of a ball is defined by the (x, y) coordinates, as displayed in Fig. P8.37. The trajectory can be modeled as y 5 (tan u0 )x 2

g 2y20

cos2 u0

x2 1 y0

Find the appropriate initial angle u0, if the initial velocity 0 5 20 m/s and the distance to the catcher x is 40 m. Note that the ball leaves the thrower’s hand at an elevation of y0 5 1.8 m and the catcher receives it at 1 m. Express the final result in degrees. Use a value of 9.81 m/s2 for g and employ the graphical method to develop your initial guesses.

FIGURE P8.37 y

F 5 2 ( k1d 1 k2d 3y2 ) Conservation of energy can be used to show that 05

2k2d 5y2 1 1 k1d2 2 mgd 2 mgh 5 2

Solve for d, given the following parameter values: k1 5 40,000 g/s2, k2 5 40 g/(s2 m0.5), m 5 95 g, g 5 9.81 m/s2, and h 5 0.43 m.

v0 0 x

222

CASE STUDIES: ROOTS OF EQUATIONS

8.38 The general form for a three-dimensional stress field is given by sxx £ sxy sxz

sxy syy syz

sxz syz § szz

where the diagonal terms represent tensile or compressive stresses and the off-diagonal terms represent shear stresses. A stress field (in MPa) is given by 10 £ 14 25

14 7 15

10 2 s 14 25

14 72s 15

25 15 § 16 2 s

s3 2 Is2 1 IIs 2 III 5 0 where I 5 sxx 1 syy 1 szz II 5 sxxsyy 1 sxxszz 1 syyszz 2 s2xy 2 s2xz 2 s2yz III 5 sxxsyyszz 2 sxxs2yz 2 syys2xz 2 szzs2xy 1 2sxysxzsyz I, II, and III are known as the stress invariants. Find 1, 2, and 3 using a root-finding technique. 8.39 The upward velocity of a rocket can be computed by the following formula: m0 y 5 u ln 2 gt m0 2 qt where 5 upward velocity, u 5 the velocity at which fuel is expelled relative to the rocket, m0 5 the initial mass of the rocket at time t 5 0, q 5 the fuel consumption rate, and g 5 the downward acceleration of gravity (assumed constant 5 9.81 m/s2). If u 5 2200 m/s, m0 5 160,000 kg, and q 5 2680 kg/s, compute the time at which 5 1000 m/s. (Hint: t is somewhere between 10 and 50 s.) Determine your result so that it is within 1% of the true value. Check your answer. 8.40 The phase angle between the forced vibration caused by the rough road and the motion of the car is given by 2(cycc ) (vyp) 1 2 (vyp)

and the heat capacity of fluid B is given by:

where cp is in units of cal/mol K, and T is in units of K. Note that

1, 2, and 3 can be solved from the equation

tan f 5

cp 5 3.381 1 1.804 3 1022T 2 4.300 3 1026T 2

cp 5 8.592 1 1.290 3 1021T 2 4.078 3 1025T 2

25 15 § 16

To solve for the principal stresses, it is necessary to construct the following matrix (again in MPa): £

As a mechanical engineer, you would like to know if there are cases where 5 y2 2 1. Use the other parameters from the section to set up the equation as a roots problem and solve for . 8.41 Two fluids at different temperatures enter a mixer and come out at the same temperature. The heat capacity of fluid A is given by:

2

¢H 5

#

T2

cpdT

T1

A enters the mixer at 4008C. B enters the mixer at 6008C. There is twice as much A as there is B entering into the mixer. At what temperature do the two fluids exit the mixer? 8.42 A compressor is operating at compression ratio Rc of 3.0 (the pressure of gas at the outlet is three times greater than the pressure of the gas at the inlet). The power requirements of the compressor Hp can be determined from the equation below. Assuming that the power requirements of the compressor are exactly equal to zRT1yMW, find the polytropic efficiency n of the compressor. The parameter z is compressibility of the gas under operating conditions of the compressor, R is the gas constant, T1 is the temperature of the gas at the compressor inlet, and MW is the molecular weight of the gas. z RT1 n ( Rc(n21)yn 2 1 ) MW n 2 1

HP 5

8.43 In the thermos shown in Fig. P8.43, the innermost compartment is separated from the middle container by a vacuum. There is a final shell around the thermos. This final shell is separated from the middle layer by a thin layer of air. The outside of the final shell comes in contact with room air. Heat transfer from the inner compartment to the next layer q1 is by radiation only (since the space is evacuated). Heat transfer between the middle layer and outside shell q2 is by convection in a small space. Heat transfer from the outside shell to the air q3 is by natural convection. The heat flux from each region of the thermos must be equal— that is, q1 5 q2 5 q3. Find the temperatures T1 and T2 at steady state. T0 is 5008C and T3 5 258C. q1 5 1029 [ (T0 1 273) 4 2 (T1 1 273) 4 ] q2 5 4(T1 2 T2 ) q3 5 1.3(T2 2 T3 ) 4y3

PROBLEMS

223

Q1

Q3

Q2

Q10

T3

T0 T1

T2

Q5

Q4

Q9

Q6

Q7

Q8

FIGURE P8.45

FIGURE P8.43

8.45 A fluid is pumped into the network of pipes shown in Fig. P8.45. At steady state, the following flow balances must hold,

8.44 Figure P8.44 shows three reservoirs connected by circular pipes. The pipes, which are made of asphalt-dipped cast iron (ε 5 0.0012 m), have the following characteristics: Pipe Length, m Diameter, m Flow, m3/s

1 1800 0.4 ?

2 500 0.25 0.1

3 1400 0.2 ?

Q1 5 Q2 1 Q3 Q3 5 Q4 1 Q5 Q5 5 Q6 1 Q7 where Qi 5 flow in pipe i(m3/s). In addition, the pressure drops around the three right-hand loops must equal zero. The pressure drop in each circular pipe length can be computed with ¢P 5

16 f Lr 2 Q p2 2 D5

If the water surface elevations in Reservoirs A and C are 200 and 172.5 m, respectively, determine the elevation in Reservoir B and the flows in pipes 1 and 3. Note that the kinematic viscosity of water is 1 3 1026 m2/s and use the Colebrook equation to determine the friction factor (recall Prob. 8.13). FIGURE P8.44 h1 A

h2 1 Q1

1 5 4 log10 (Re 1f ) 2 0.4 1f

B

2

h3

Q2

where DP 5 the pressure drop (Pa), f 5 the friction factor (dimensionless), L 5 the pipe length (m), 5 the fluid density (kg/m3), and D 5 pipe diameter (m). Write a program (or develop an algorithm in a mathematics software package) that will allow you to compute the flow in every pipe length given that Q1 5 1 m3/s and 5 1.23 kg/m3. All the pipes have D 5 500 mm and f 5 0.005. The pipe lengths are: L3 5 L5 5 L8 5 L9 5 2 m; L2 5 L4 5 L6 5 4 m; and L7 5 8 m. 8.46 Repeat Prob. 8.45, but incorporate the fact that the friction factor can be computed with the von Karman equation,

3

where Re 5 the Reynolds number Re 5

Q3 C

rVD m

where V 5 the velocity of the fluid in the pipe (m/s) and 5 dynamic viscosity (N ? s/m2). Note that for a circular pipe

224

CASE STUDIES: ROOTS OF EQUATIONS

V 5 4Q/ D2. Also, assume that the fluid has a viscosity of 1.79 3 1025 N ? s/m2. 8.47 The space shuttle, at lift-off from the launch pad, has four forces acting on it, which are shown on the free-body diagram (Fig. P8.47). The combined weight of the two solid rocket boosters and external fuel tank is WB 5 1.663 3 106 lb. The weight of

28’ 4’

External tank

Solid rocket booster Orbiter

G

38’ WB

WS

the orbiter with a full payload is WS 5 0.23 3 106 lb. The combined thrust of the two solid rocket boosters is TB 5 5.30 3 106 lb. The combined thrust of the three liquid fuel orbiter engines is TS 5 1.125 3 106 lb. At liftoff, the orbiter engine thrust is directed at angle to make the resultant moment acting on the entire craft assembly (external tank, solid rocket boosters, and orbiter) equal to zero. With the resultant moment equal to zero, the craft will not rotate about its mass center G at liftoff. With these forces, the craft will have a resultant force with components in both the vertical and horizontal direction. The vertical resultant force component is what allows the craft to lift off from the launch pad and fly vertically. The horizontal resultant force component causes the craft to fly horizontally. The resultant moment acting on the craft will be zero when is adjusted to the proper value. If this angle is not adjusted properly, and there is some resultant moment acting on the craft, the craft will tend to rotate about it mass center. (a) Resolve the orbiter thrust TS into horizontal and vertical components, and then sum moments about point G, the craft mass center. Set the resulting moment equation equal to zero. This equation can now be solved for the value of required for liftoff. (b) Derive an equation for the resultant moment acting on the craft in terms of the angle . Plot the resultant moment as a function of the angle over a range of 25 radians to 15 radians. (c) Write a computer program to solve for the angle using Newton’s method to find the root of the resultant moment equation. Make an initial first guess at the root of interest using the plot. Terminate your iterations when the value of has better than five significant figures. (d) Repeat the program for the minimum payload weight of the orbiter of WS 5 195,000 lb. 8.48 Determining the velocity of particles settling through fluids is of great importance of many areas of engineering and science. Such calculations depend on the flow regime as represented by the dimensionless Reynolds number, Re 5

TS TB

(P8.48.1)

where 5 the fluid’s density (kg/m3), d 5 the particle diameter (m), y 5 the particle’s settling velocity (m/s), and 5 the fluid’s dynamic viscosity (N s/m2). Under laminar conditions (Re , 0.1), the settling velocity of a spherical particle can be computed with the following formula based on Stokes law, y5

FIGURE P8.47

rdy m

g rs 2 r 2 a bd m 18

(P8.48.2)

where g 5 the gravitational constant (5 9.81 m/s2), and s 5 the particle’s density (kg/m3). For turbulent conditions (i.e., higher

PROBLEMS

225

Reynolds numbers), an alternative approach can be used based on the following formula: y5

B

4g(rs 2 r)d 3CDr

(P8.48.3)

where CD 5 the drag coefficient, which depends on the Reynolds number as in CD 5

24 3 1 0.34 1 Re 1Re

(P8.48.4)

(a) Combine Eqs. (P8.48.2), (P8.48.3), and (P8.48.4) to express the determination of y as a roots of equations problem. That is, express the combined formula in the format f(y) 5 0.

(b) Use the modified secant method with d 5 1023 and εS 5 0.05% to determine y for a spherical iron particle settling in water, where d 5 200 m, 5 1 g/cm3, s 5 7.874 g/cm3, and 5 0.014 g/(cm?s). Employ Eq. (P8.48.2) to generate your initial guess. (c) Based on the result of (b), compute the Reynolds number and the drag coefficient, and use the latter to confirm that the flow regime is not laminar. (d) Develop a fixed-point iteration solution for the conditions outlined in (b). (e) Use a graphical approach to illustrate that the formulation developed in (d) will converge for any positive guess.

EPILOGUE: PART TWO PT2.4

TRADE-OFFS Table PT2.3 provides a summary of the trade-offs involved in solving for roots of algebraic and transcendental equations. Although graphical methods are time-consuming, they provide insight into the behavior of the function and are useful in identifying initial guesses and potential problems such as multiple roots. Therefore, if time permits, a quick sketch (or better yet, a computerized graph) can yield valuable information regarding the behavior of the function. The numerical methods themselves are divided into two general categories: bracketing and open methods. The former requires two initial guesses that are on either side of a root. This “bracketing” is maintained as the solution proceeds, and thus, these techniques are always convergent. However, a price is paid for this property in that the rate of convergence is relatively slow.

TABLE PT2.3 Comparison of the characteristics of alternative methods for ﬁnding roots of algebraic and transcendental equations. The comparisons are based on general experience and do not account for the behavior of speciﬁc functions. Method

Type

Guesses

Convergence

Stability

Programming

Comments

Direct Graphical Bisection False-position Modiﬁed FP Fixed-point iteration Newton-Raphson

Analytical Visual Bracketing Bracketing Bracketing Open

— — 2 2 2 1

— — Slow Slow/medium Medium Slow

— — Always Always Always Possibly divergent

— Easy Easy Easy Easy

Imprecise

Open

1

Fast

Possibly divergent

Easy

Modiﬁed NewtonRaphson

Open

1

Fast (multiple), medium (single)

Possibly divergent

Easy

Secant

Open

2

Medium/fast

Possibly divergent

Easy

Requires evaluation of f’(x) Requires evaluation of f’(x) and f”(x) Initial guesses do not have to bracket the root

Modiﬁed secant Brent

Open Hybrid

1 1 or 2

Medium/fast Medium

Easy Moderate

Müller Bairstow

Polynomials Polynomials

2 2

Medium/fast Fast

Possibly divergent Always (for 2 guesses) Possibly divergent Possibly divergent

226

Moderate Moderate

Robust

PT2.6

ADVANCED METHODS AND ADDITIONAL REFERENCES

227

Open techniques differ from bracketing methods in that they use information at a single point (or two values that need not bracket the root to extrapolate to a new root estimate). This property is a double-edged sword. Although it leads to quicker convergence, it also allows the possibility that the solution may diverge. In general, the convergence of open techniques is partially dependent on the quality of the initial guess and the nature of the function. The closer the guess is to the true root, the more likely the methods will converge. Of the open techniques, the standard Newton-Raphson method is often used because of its property of quadratic convergence. However, its major shortcoming is that it requires the derivative of the function be obtained analytically. For some functions this is impractical. In these cases, the secant method, which employs a finite-difference representation of the derivative, provides a viable alternative. Because of the finite-difference approximation, the rate of convergence of the secant method is initially slower than for the Newton-Raphson method. However, as the root estimate is refined, the difference approximation becomes a better representation of the true derivative, and convergence accelerates rapidly. The modified Newton-Raphson technique can be used to attain rapid convergence for multiple roots. However, this technique requires an analytical expression for both the first and second derivatives. Of particular interest are hybrid methods that combine the reliability of bracketing with the speed of open methods. Brent’s method does this by combining bisection with several open methods. All the methods are easy-to-moderate to program on computers and require minimal time to determine a single root. On this basis, you might conclude that simple methods such as bisection would be good enough for practical purposes. This would be true if you were exclusively interested in determining the root of an equation once. However, there are many cases in engineering where numerous root locations are required and where speed becomes important. For these cases, slow methods are very time-consuming and, hence, costly. On the other hand, the fast open methods may diverge, and the accompanying delays can also be costly. Some computer algorithms attempt to capitalize on the strong points of both classes of techniques by initially employing a bracketing method to approach the root, then switching to an open method to rapidly refine the estimate. Whether a single approach or a combination is used, the trade-offs between convergence and speed are at the heart of the choice of a root-location technique.

PT2.5

IMPORTANT RELATIONSHIPS AND FORMULAS Table PT2.4 summarizes important information that was presented in Part Two. This table can be consulted to quickly access important relationships and formulas.

PT2.6

ADVANCED METHODS AND ADDITIONAL REFERENCES The methods in this text have focused on determining a single real root of an algebraic or transcendental equation based on foreknowledge of its approximate location. In addition, we have also described methods expressly designed to determine both the real

228

EPILOGUE: PART TWO

TABLE PT2.4 Summary of important information presented in Part Two. Method

Formulation

Bisection

xl 1 xu xr 5 2

Graphical Interpretation

Errors and Stopping Criteria

Bracketing methods: f (x)

Stopping criterion:

Root xl

If f (xl)f (xr) , 0, xu 5 xr f (xl)f (xr) . 0, xl 5 xr

xu

L

x

`

x rnew 2 xrold ` 100% # es xrnew

L/2 L/4

False position

xr 5 xu 2

f (xu ) (xl 2 xu )

f (x)

Stopping criterion:

f (xl ) 2 f (xu )

`

d

If f (xl)f (xr) , 0, xu 5 xr f (xl)f (xr) . 0, xl 5 xr

r ho

xr

C

xl

xu

x rnew 2 x old r ` 100% # es xrnew

x

f (x) Tangent

Newton-Raphson

Stopping criterion: xi11 2 xi ` ` 100% # es xi11

f (xi ) xi11 5 xi 2 f ¿(xi )

Error: Ei11 5 0(E 2i ) xi + 1

xi

x

f (x)

Secant xi11 5 xi 2

Stopping criterion: xi11 2 xi ` ` 100% # es xi11

f (xi ) (xi21 2 xi ) f (xi21 ) 2 f (xi )

xi + 1

xi xi – 1

x

and complex roots of polynomials. Additional references on the subject are Ralston and Rabinowitz (1978) and Carnahan, Luther, and Wilkes (1969). In addition to Müller’s and Bairstow’s methods, several techniques are available to determine all the roots of polynomials. In particular, the quotient difference (QD) algorithm (Henrici, 1964, and Gerald and Wheatley, 2004) determines all roots without initial guesses. Ralston and Rabinowitz (1978) and Carnahan, Luther, and Wilkes (1969)

PT2.6

ADVANCED METHODS AND ADDITIONAL REFERENCES

229

contain discussions of this method as well as of other techniques for locating roots of polynomials. As discussed in the text, the Jenkins-Traub and Laguerre’s methods are widely employed. In summary, the foregoing is intended to provide you with avenues for deeper exploration of the subject. Additionally, all the above references provide descriptions of the basic techniques covered in Part Two. We urge you to consult these alternative sources to broaden your understanding of numerical methods for root location.1

1

Books are referenced only by author here, a complete bibliography will be found at the back of this text.

PART THREE

LINEAR ALGEBRAIC EQUATIONS PT3.1

MOTIVATION In Part Two, we determined the value x that satisfied a single equation, f(x) 5 0. Now, we deal with the case of determining the values x1, x2, . . . , xn that simultaneously satisfy a set of equations f1 (x1, x2, p , xn ) 5 0 f2 (x1, x2, p , xn ) 5 0 . . . . . . fn (x1, x2, p , xn ) 5 0 Such systems can be either linear or nonlinear. In Part Three, we deal with linear algebraic equations that are of the general form a11 x1 1 a12 x2 1 p 1 a1n xn 5 b1 a21x1 1 a22 x2 1 p 1 a2n xn 5 b2 . . . . . . p an1x1 1 an2 x2 1 1 ann xn 5 bn

(PT3.1)

where the a’s are constant coefficients, the b’s are constants, and n is the number of equations. All other equations are nonlinear. Nonlinear systems were discussed in Chap. 6 and will be covered briefly again in Chap. 9. PT3.1.1 Noncomputer Methods for Solving Systems of Equations For small numbers of equations (n # 3), linear (and sometimes nonlinear) equations can be solved readily by simple techniques. Some of these methods will be reviewed at the beginning of Chap. 9. However, for four or more equations, solutions become arduous and computers must be utilized. Historically, the inability to solve all but the smallest sets of equations by hand has limited the scope of problems addressed in many engineering applications. Before computers, techniques to solve linear algebraic equations were time-consuming and awkward. These approaches placed a constraint on creativity because the methods were often difficult to implement and understand. Consequently, the techniques were sometimes overemphasized at the expense of other aspects of the problem-solving process such as formulation and interpretation (recall Fig. PT1.1 and accompanying discussion). 231

232

LINEAR ALGEBRAIC EQUATIONS

The advent of easily accessible computers makes it possible and practical for you to solve large sets of simultaneous linear algebraic equations. Thus, you can approach more complex and realistic examples and problems. Furthermore, you will have more time to test your creative skills because you will be able to place more emphasis on problem formulation and solution interpretation. PT3.1.2 Linear Algebraic Equations and Engineering Practice Many of the fundamental equations of engineering are based on conservation laws (recall Table 1.1). Some familiar quantities that conform to such laws are mass, energy, and momentum. In mathematical terms, these principles lead to balance or continuity equations that relate system behavior as represented by the levels or response of the quantity being modeled to the properties or characteristics of the system and the external stimuli or forcing functions acting on the system. As an example, the principle of mass conservation can be used to formulate a model for a series of chemical reactors (Fig. PT3.1a). For this case, the quantity being modeled is the mass of the chemical in each reactor. The system properties are the reaction characteristics of the chemical and the reactors’ sizes and flow rates. The forcing functions are the feed rates of the chemical into the system. In Part Two, you saw how single-component systems result in a single equation that can be solved using root-location techniques. Multicomponent systems result in a coupled set of mathematical equations that must be solved simultaneously. The equations are

FIGURE PT3.1 Two types of systems that can be modeled using linear algebraic equations: (a) lumped variable system that involves coupled ﬁnite components and (b) distributed variable system that involves a continuum.

x3

x1

Feed

x5

x2

x4

(a)

Feed

x1

…

xi⫺1

x1

(b)

xi⫹1

…

xn

PT3.2

MATHEMATICAL BACKGROUND

233

coupled because the individual parts of the system are influenced by other parts. For example, in Fig. PT3.1a, reactor 4 receives chemical inputs from reactors 2 and 3. Consequently, its response is dependent on the quantity of chemical in these other reactors. When these dependencies are expressed mathematically, the resulting equations are often of the linear algebraic form of Eq. (PT3.1). The x’s are usually measures of the magnitudes of the responses of the individual components. Using Fig. PT3.1a as an example, x1 might quantify the amount of mass in the first reactor, x2 might quantify the amount in the second, and so forth. The a’s typically represent the properties and characteristics that bear on the interactions between components. For instance, the a’s for Fig. PT3.1a might be reflective of the flow rates of mass between the reactors. Finally, the b’s usually represent the forcing functions acting on the system, such as the feed rate in Fig. PT3.1a. The applications in Chap. 12 provide other examples of such equations derived from engineering practice. Multicomponent problems of the above types arise from both lumped (macro-) or distributed (micro-) variable mathematical models (Fig. PT3.1). Lumped variable problems involve coupled finite components. Examples include trusses (Sec. 12.2), reactors (Fig. PT3.1a and Sec. 12.1), and electric circuits (Sec. 12.3). These types of problems use models that provide little or no spatial detail. Conversely, distributed variable problems attempt to describe spatial detail of systems on a continuous or semicontinuous basis. The distribution of chemicals along the length of an elongated, rectangular reactor (Fig. PT3.1b) is an example of a continuous variable model. Differential equations derived from the conservation laws specify the distribution of the dependent variable for such systems. These differential equations can be solved numerically by converting them to an equivalent system of simultaneous algebraic equations. The solution of such sets of equations represents a major engineering application area for the methods in the following chapters. These equations are coupled because the variables at one location are dependent on the variables in adjoining regions. For example, the concentration at the middle of the reactor is a function of the concentration in adjoining regions. Similar examples could be developed for the spatial distribution of temperature or momentum. We will address such problems when we discuss differential equations later in the book. Aside from physical systems, simultaneous linear algebraic equations also arise in a variety of mathematical problem contexts. These result when mathematical functions are required to satisfy several conditions simultaneously. Each condition results in an equation that contains known coefficients and unknown variables. The techniques discussed in this part can be used to solve for the unknowns when the equations are linear and algebraic. Some widely used numerical techniques that employ simultaneous equations are regression analysis (Chap. 17) and spline interpolation (Chap. 18).

PT3.2

MATHEMATICAL BACKGROUND All parts of this book require some mathematical background. For Part Three, matrix notation and algebra are useful because they provide a concise way to represent and manipulate linear algebraic equations. If you are already familiar with matrices, feel free to skip to Sec. PT3.3. For those who are unfamiliar or require a review, the following material provides a brief introduction to the subject.

234

LINEAR ALGEBRAIC EQUATIONS

Column 3 a11 a21 . [A] 5 F . . an1

a12 a22 . . . an2

a13 a23

p p

an3

p

a1m a2m . V . . anm

Row 2

FIGURE PT3.2 A matrix.

PT3.2.1 Matrix Notation A matrix consists of a rectangular array of elements represented by a single symbol. As depicted in Fig. PT3.2, [A] is the shorthand notation for the matrix and aij designates an individual element of the matrix. A horizontal set of elements is called a row and a vertical set is called a column. The first subscript i always designates the number of the row in which the element lies. The second subscript j designates the column. For example, element a23 is in row 2 and column 3. The matrix in Fig. PT3.2 has n rows and m columns and is said to have a dimension of n by m (or n 3 m). It is referred to as an n by m matrix. Matrices with row dimension n 5 1, such as [B] 5 [b1

b2

p

bm ]

are called row vectors. Note that for simplicity, the first subscript of each element is dropped. Also, it should be mentioned that there are times when it is desirable to employ a special shorthand notation to distinguish a row matrix from other types of matrices. One way to accomplish this is to employ special open-topped brackets, as in :B; . Matrices with column dimension m 5 1, such as c1 c2 . [C] 5 F V . . cn are referred to as column vectors. For simplicity, the second subscript is dropped. As with the row vector, there are occasions when it is desirable to employ a special shorthand notation to distinguish a column matrix from other types of matrices. One way to accomplish this is to employ special brackets, as in {C}.

PT3.2

MATHEMATICAL BACKGROUND

235

Matrices where n 5 m are called square matrices. For example, a 4 by 4 matrix is a11 a21 [A] 5 ≥ a31 a41

a12 a22 a32 a42

a13 a23 a33 a43

a14 a24 ¥ a34 a44

The diagonal consisting of the elements a11, a22, a33, and a44 is termed the principal or main diagonal of the matrix. Square matrices are particularly important when solving sets of simultaneous linear equations. For such systems, the number of equations (corresponding to rows) and the number of unknowns (corresponding to columns) must be equal for a unique solution to be possible. Consequently, square matrices of coefficients are encountered when dealing with such systems. Some special types of square matrices are described in Box PT3.1.

Box PT3.1

Special Types of Square Matrices

There are a number of special forms of square matrices that are important and should be noted: A symmetric matrix is one where aij 5 aji for all i’s and j’s. For example, 5 [A] 5 £ 1 2

1 3 7

a11

2 7§ 8

[A] 5 ≥

is a 3 by 3 symmetric matrix. A diagonal matrix is a square matrix where all elements off the main diagonal are equal to zero, as in a11 a22

[A] 5 ≥

¥

a33 a44

Note that where large blocks of elements are zero, they are left blank. An identity matrix is a diagonal matrix where all elements on the main diagonal are equal to 1, as in 1

¥

1 1

a12 a22

a13 a23 a33

a14 a24 ¥ a34 a44

A lower triangular matrix is one where all elements above the main diagonal are zero, as in a11 a21 [A] 5 ≥ a31 a41

a22 a32 a42

a33 a43

¥ a44

A banded matrix has all elements equal to zero, with the exception of a band centered on the main diagonal: [A] 5 ≥

1 [I] 5 ≥

The symbol [I] is used to denote the identity matrix. The identity matrix has properties similar to unity. An upper triangular matrix is one where all the elements below the main diagonal are zero, as in

a12 a21

a12 a22 a32

a23 a33 a43

a34 a44

¥

The above matrix has a bandwidth of 3 and is given a special name—the tridiagonal matrix.

236

LINEAR ALGEBRAIC EQUATIONS

PT3.2.2 Matrix Operating Rules Now that we have specified what we mean by a matrix, we can define some operating rules that govern its use. Two n by m matrices are equal if, and only if, every element in the first is equal to every element in the second, that is, [A] 5 [B] if aij 5 bij for all i and j. Addition of two matrices, say, [A] and [B], is accomplished by adding corresponding terms in each matrix. The elements of the resulting matrix [C] are computed, cij 5 aij 1 bij for i 5 1, 2, . . . , n and j 5 1, 2, . . . , m. Similarly, the subtraction of two matrices, say, [E] minus [F], is obtained by subtracting corresponding terms, as in dij 5 eij 2 fij for i 5 1, 2, . . . , n and j 5 1, 2, . . . , m. It follows directly from the above definitions that addition and subtraction can be performed only between matrices having the same dimensions. Both addition and subtraction are commutative: [A] 1 [B] 5 [B] 1 [A] Addition and subtraction are also associative, that is, ([A] 1 [B]) 1 [C] 5 [A] 1 ([B] 1 [C]) The multiplication of a matrix [A] by a scalar g is obtained by multiplying every element of [A] by g, as in ga11 ga21 . [D] 5 g[A] 5 F . . gan1

ga12 ga22 . . . gan2

p p

p

ga1m ga2m . V . . ganm

The product of two matrices is represented as [C] 5 [A][B], where the elements of [C] are defined as (see Box PT3.2 for a simple way to conceptualize matrix multiplication) n

cij 5 a aikbkj

(PT3.2)

k51

where n 5 the column dimension of [A] and the row dimension of [B]. That is, the cij element is obtained by adding the product of individual elements from the ith row of the first matrix, in this case [A], by the jth column of the second matrix [B]. According to this definition, multiplication of two matrices can be performed only if the first matrix has as many columns as the number of rows in the second matrix. Thus, if [A] is an n by m matrix, [B] could be an m by l matrix. For this case, the resulting [C] matrix would have the dimension of n by l. However, if [B] were an l by m matrix, the multiplication could not be performed. Figure PT3.3 provides an easy way to check whether two matrices can be multiplied.

PT3.2

Box PT3.2

MATHEMATICAL BACKGROUND

A Simple Method for Multiplying Two Matrices

Although Eq. (PT3.2) is well suited for implementation on a computer, it is not the simplest means for visualizing the mechanics of multiplying two matrices. What follows gives more tangible expression to the operation. Suppose that we want to multiply [X] by [Y] to yield [Z], 3 [Z] 5 [X] [Y] 5 £ 8 0

1 5 6§ c 7 4

237

c

3 £8 0

1 S 3 3 5 1 1 3 7 5 22 6§ £ § 4

Thus, z11 is equal to 22. Element z21 can be computed in a similar fashion, as in c

A 5 c 7 3 [X] S £ 8 0

9 d d [Y] 2

3 £8 0

1 6 § £ ? § d [Z] 4

Now the answer [Z] can be computed in the space vacated by [Y]. This format has utility because it aligns the appropriate rows and columns that are to be multiplied. For example, according to Eq. (PT3.2), the element z11 is obtained by multiplying the first row of [X] by the first column of [Y]. This amounts to adding the product of x11 and y11 to the product of x12 and y21, as in

9 d 2

T

9 d 2

A simple way to visualize the computation of [Z] is to raise [Y], as in

5 7

5 7

9 d 2

T 1 22 6 § S £ 8 3 5 1 6 3 7 5 82 § 4

The computation can be continued in this way, following the alignment of the rows and columns, to yield the result 22 [Z] 5 £ 82 28

29 84 § 8

Note how this simple method makes it clear why it is impossible to multiply two matrices if the number of columns of the first matrix does not equal the number of rows in the second matrix. Also, note how it demonstrates that the order of multiplication matters (that is, matrix multiplication is not commutative).

FIGURE PT3.3

[A]n ⴛ m

[B]m ⴛ l ⴝ [C]n ⴛ l

Interior dimensions are equal; multiplication is possible Exterior dimensions define the dimensions of the result

238

LINEAR ALGEBRAIC EQUATIONS

If the dimensions of the matrices are suitable, matrix multiplication is associative, ([A][B])[C] 5 [A]([B][C]) and distributive, [A]([B] 1 [C]) 5 [A][B] 1 [A][C] or ([A] 1 [B])[C] 5 [A][C] 1 [B][C] However, multiplication is not generally commutative: [A][B] ? [B][A] That is, the order of multiplication is important. Figure PT3.4 shows pseudocode to multiply an n by m matrix [A], by an m by l matrix [B], and store the result in an n by l matrix [C]. Notice that, instead of the inner product being directly accumulated in [C], it is collected in a temporary variable, sum. This is done for two reasons. First, it is a bit more efficient, because the computer need determine the location of ci, j only n 3 l times rather than n 3 l 3 m times. Second, the precision of the multiplication can be greatly improved by declaring sum as a double precision variable (recall the discussion of inner products in Sec. 3.4.2). Although multiplication is possible, matrix division is not a defined operation. However, if a matrix [A] is square and nonsingular, there is another matrix [A]21, called the inverse of [A], for which [A][A] 21 5 [A] 21 [A] 5 [I]

(PT3.3)

Thus, the multiplication of a matrix by the inverse is analogous to division, in the sense that a number divided by itself is equal to 1. That is, multiplication of a matrix by its inverse leads to the identity matrix (recall Box PT3.1). The inverse of a two-dimensional square matrix can be represented simply by [A] 21 5

FIGURE PT3.4

1 a22 c a11a22 2 a12a21 2a21

SUBROUTINE Mmult (a, b, c, m, n, l) DOFOR i 5 1, n DOFOR j 5 1, l sum 5 0. DOFOR k 5 1, m sum 5 sum 1 a(i,k) ? b(k,j) END DO c(i,j) 5 sum END DO END DO

2a12 d a11

(PT3.4)

PT3.2

MATHEMATICAL BACKGROUND

239

Similar formulas for higher-dimensional matrices are much more involved. Sections in Chaps. 10 and 11 will be devoted to techniques for using numerical methods and the computer to calculate the inverse for such systems. Two other matrix manipulations that will have utility in our discussion are the transpose and the trace of a matrix. The transpose of a matrix involves transforming its rows into columns and its columns into rows. For example, for the 4 3 4 matrix, a11 a21 [A] 5 ≥ a31 a41

a12 a22 a32 a42

a13 a23 a33 a43

a14 a24 ¥ a34 a44

the transpose, designated [A]T, is defined as a11 a12 [A] T 5 ≥ a13 a14

a21 a22 a23 a24

a31 a32 a33 a34

a41 a42 ¥ a43 a44

In other words, the element aij of the transpose is equal to the aji element of the original matrix. The transpose has a variety of functions in matrix algebra. One simple advantage is that it allows a column vector to be written as a row. For example, if c1 c2 {C} 5 μ ∂ c3 c4 then

{C} T 5 :c1

c2

c3

c4 ;

where the superscript T designates the transpose. For example, this can save space when writing a column vector in a manuscript. In addition, the transpose has numerous mathematical applications. The trace of a matrix is the sum of the elements on its principal diagonal. It is designated as tr [A] and is computed as n

tr [A] 5 a aii i51

The trace will be used in our discussion of eigenvalues in Chap. 27. The final matrix manipulation that will have utility in our discussion is augmentation. A matrix is augmented by the addition of a column (or columns) to the original matrix. For example, suppose we have a matrix of coefficients: a11 [A] 5 £ a21 a31

a12 a22 a32

a13 a23 § a33

240

LINEAR ALGEBRAIC EQUATIONS

We might wish to augment this matrix [A] with an identity matrix (recall Box PT3.1) to yield a 3-by-6-dimensional matrix: a11 [A] 5 £ a21 a31

a12 a22 a32

a13 1 a23 0 a33 0

0 1 0

0 0§ 1

Such an expression has utility when we must perform a set of identical operations on two matrices. Thus, we can perform the operations on the single augmented matrix rather than on the two individual matrices. PT3.2.3 Representing Linear Algebraic Equations in Matrix Form It should be clear that matrices provide a concise notation for representing simultaneous linear equations. For example, Eq. (PT3.1) can be expressed as [A]{X} 5 {B}

(PT3.5)

where [A] is the n by n square matrix of coefficients, a11 a21 . [A] 5 F . . an1

a12 a22 . . . an2

p p

p

a1n a2n . V . . ann

{B} is the n by 1 column vector of constants, {B} T 5 :b1

b2

p

bn ;

and {X} is the n by 1 column vector of unknowns: {X} T 5 :x1

x2

p

xn ;

Recall the definition of matrix multiplication [Eq. (PT3.2) or Box PT3.2] to convince yourself that Eqs. (PT3.1) and (PT3.5) are equivalent. Also, realize that Eq. (PT3.5) is a valid matrix multiplication because the number of columns, n, of the first matrix [A] is equal to the number of rows, n, of the second matrix {X}. This part of the book is devoted to solving Eq. (PT3.5) for {X}. A formal way to obtain a solution using matrix algebra is to multiply each side of the equation by the inverse of [A] to yield [A] 21 [A]{X} 5 [A] 21 {B} Because [A]21[A] equals the identity matrix, the equation becomes {X} 5 [A] 21 {B}

(PT3.6)

Therefore, the equation has been solved for {X}. This is another example of how the inverse plays a role in matrix algebra that is similar to division. It should be noted that this is not a very efficient way to solve a system of equations. Thus, other approaches

PT3.3

ORIENTATION

241

are employed in numerical algorithms. However, as discussed in Chap. 10, the matrix inverse itself has great value in the engineering analyses of such systems. Finally, we will sometimes find it useful to augment [A] with {B}. For example, if n 5 3, this results in a 3-by-4-dimensional matrix: a11 [A] 5 £ a21 a31

a12 a22 a32

a13 a23 a33

b1 b2 § b3

(PT3.7)

Expressing the equations in this form is useful because several of the techniques for solving linear systems perform identical operations on a row of coefficients and the corresponding right-hand-side constant. As expressed in Eq. (PT3.7), we can perform the manipulation once on an individual row of the augmented matrix rather than separately on the coefficient matrix and the right-hand-side vector.

PT3.3

ORIENTATION Before proceeding to the numerical methods, some further orientation might be helpful. The following is intended as an overview of the material discussed in Part Three. In addition, we have formulated some objectives to help focus your efforts when studying the material. PT3.3.1 Scope and Preview Figure PT3.5 provides an overview for Part Three. Chapter 9 is devoted to the most fundamental technique for solving linear algebraic systems: Gauss elimination. Before launching into a detailed discussion of this technique, a preliminary section deals with simple methods for solving small systems. These approaches are presented to provide you with visual insight and because one of the methods—the elimination of unknowns— represents the basis for Gauss elimination. After the preliminary material, “naive’’ Gauss elimination is discussed. We start with this “stripped-down” version because it allows the fundamental technique to be elaborated on without complicating details. Then, in subsequent sections, we discuss potential problems of the naive approach and present a number of modifications to minimize and circumvent these problems. The focus of this discussion will be the process of switching rows, or partial pivoting. Chapter 10 begins by illustrating how Gauss elimination can be formulated as an LU decomposition solution. Such solution techniques are valuable for cases where many right-hand-side vectors need to be evaluated. It is shown how this attribute allows efficient calculation of the matrix inverse, which has tremendous utility in engineering practice. Finally, the chapter ends with a discussion of matrix condition. The condition number is introduced as a measure of the loss of significant digits of accuracy that can result when solving ill-conditioned matrices. The beginning of Chap. 11 focuses on special types of systems of equations that have broad engineering application. In particular, efficient techniques for solving tridiagonal systems are presented. Then, the remainder of the chapter focuses on an alternative to elimination methods called the Gauss-Seidel method. This technique is similar in spirit to

242

LINEAR ALGEBRAIC EQUATIONS

PT 3.1 Motivation

PT 3.6 Advanced methods

PT 3.2 Mathematical background

PT 3.3 Orientation

9.1 Small systems

PART 3

9.2 Naive Gauss elimination 9.3 Pitfalls

Linear Algebraic Equations

PT 3.5 Important formulas

9.4 Remedies

CHAPTER 9 EPILOGUE

9.5 Complex systems

Gauss Elimination

PT 3.4 Trade-offs 9.7 Gauss-Jordan

9.6 Nonlinear systems

10.1 LU decomposition

12.4 Mechanical engineering

CHAPTER 10 CHAPTER 12 12.3 Electrical engineering

LU Decomposition and Matrix Inversion

Engineering Case Studies

10.2 Matrix inverse

CHAPTER 11 12.2 Civil engineering

10.3 System condition

Special Matrices and Gauss-Seidel 12.1 Chemical engineering

11.3 Software

11.1 Special matrices 11.2 GaussSeidel

FIGURE PT3.5 Schematic of the organization of the material in Part Three: Linear Algebraic Equations.

PT3.3

ORIENTATION

243

the approximate methods for roots of equations that were discussed in Chap. 6. That is, the technique involves guessing a solution and then iterating to obtain a refined estimate. The chapter ends with information related to solving linear algebraic equations with software packages. Chapter 12 demonstrates how the methods can actually be applied for problem solving. As with other parts of the book, applications are drawn from all fields of engineering. Finally, an epilogue is included at the end of Part Three. This review includes discussion of trade-offs that are relevant to implementation of the methods in engineering practice. This section also summarizes the important formulas and advanced methods related to linear algebraic equations. As such, it can be used before exams or as a refresher after you have graduated and must return to linear algebraic equations as a professional. PT3.3.2 Goals and Objectives Study Objectives. After completing Part Three, you should be able to solve problems involving linear algebraic equations and appreciate the application of these equations in many fields of engineering. You should strive to master several techniques and assess their reliability. You should understand the trade-offs involved in selecting the “best” method (or methods) for any particular problem. In addition to these general objectives, the specific concepts listed in Table PT3.1 should be assimilated and mastered. Computer Objectives. Your most fundamental computer objectives are to be able to solve a system of linear algebraic equations and to evaluate the matrix inverse. You will

TABLE PT3.1 Speciﬁc study objectives for Part Three. 1. Understand the graphical interpretation of ill-conditioned systems and how it relates to the determinant. 2. Be familiar with terminology: forward elimination, back substitution, pivot equation, and pivot coefﬁcient. 3. Understand the problems of division by zero, round-off error, and ill-conditioning. 4. Know how to compute the determinant using Gauss elimination. 5. Understand the advantages of pivoting; realize the difference between partial and complete pivoting. 6. Know the fundamental difference between Gauss elimination and the Gauss-Jordan method and which is more efﬁcient. 7. Recognize how Gauss elimination can be formulated as an LU decomposition. 8. Know how to incorporate pivoting and matrix inversion into an LU decomposition algorithm. 9. Know how to interpret the elements of the matrix inverse in evaluating stimulus response computations in engineering. 10. Realize how to use the inverse and matrix norms to evaluate system condition. 11. Understand how banded and symmetric systems can be decomposed and solved efﬁciently. 12. Understand why the Gauss-Seidel method is particularly well suited for large, sparse systems of equations. 13. Know how to assess diagonal dominance of a system of equations and how it relates to whether the system can be solved with the Gauss-Seidel method. 14. Understand the rationale behind relaxation; know where underrelaxation and overrelaxation are appropriate.

244

LINEAR ALGEBRAIC EQUATIONS

want to have subprograms developed for LU decomposition of both full and tridiagonal matrices. You may also want to have your own software to implement the Gauss-Seidel method. You should know how to use packages to solve linear algebraic equations and find the matrix inverse. You should become familiar with how the same evaluations can be implemented on popular software packages such as Excel, MATLAB software, and Mathcad.

9

CHAPTER

Gauss Elimination

This chapter deals with simultaneous linear algebraic equations that can be represented generally as a11x1 1 a12x2 1 p 1 a1nxn 5 b1 a21x1 1 a22 x2 1 p 1 a2n xn 5 b2 . . . . . . an1x1 1 an2 x2 1 p 1 ann xn 5 bn

(9.1)

where the a’s are constant coefficients and the b’s are constants. The technique described in this chapter is called Gauss elimination because it involves combining equations to eliminate unknowns. Although it is one of the earliest methods for solving simultaneous equations, it remains among the most important algorithms in use today and is the basis for linear equation solving on many popular software packages.

9.1

SOLVING SMALL NUMBERS OF EQUATIONS Before proceeding to the computer methods, we will describe several methods that are appropriate for solving small (n # 3) sets of simultaneous equations and that do not require a computer. These are the graphical method, Cramer’s rule, and the elimination of unknowns. 9.1.1 The Graphical Method A graphical solution is obtainable for two equations by plotting them on Cartesian coordinates with one axis corresponding to x1 and the other to x2. Because we are dealing with linear systems, each equation is a straight line. This can be easily illustrated for the general equations a11 x1 1 a12 x2 5 b1 a21 x1 1 a22 x2 5 b2 245

246

GAUSS ELIMINATION

Both equations can be solved for x2: x2 5 2a

a11 b1 bx 1 a12 1 a12

x2 5 2a

a21 b2 b x1 1 a22 a22

Thus, the equations are now in the form of straight lines; that is, x2 5 (slope) x1 1 intercept. These lines can be graphed on Cartesian coordinates with x2 as the ordinate and x1 as the abscissa. The values of x1 and x2 at the intersection of the lines represent the solution. EXAMPLE 9.1

The Graphical Method for Two Equations Problem Statement. Use the graphical method to solve 3x1 1 2x2 5 18 2x1 1 2x2 5 2

(E9.1.1) (E9.1.2)

Solution. Let x1 be the abscissa. Solve Eq. (E9.1.1) for x2: 3 x2 5 2 x1 1 9 2 which, when plotted on Fig. 9.1, is a straight line with an intercept of 9 and a slope of 23y2.

FIGURE 9.1 Graphical solution of a set of two simultaneous linear algebraic equations. The intersection of the lines represents the solution.

x2

8 3x 1 2x 2 18

6

Solution: x1 4; x2 3 4

x2

0

2

2 x 1

2

0

2

4

6

x1

9.1

SOLVING SMALL NUMBERS OF EQUATIONS

247

Equation (E9.1.2) can also be solved for x2: x2 5

1 x1 1 1 2

which is also plotted on Fig. 9.1. The solution is the intersection of the two lines at x1 5 4 and x2 5 3. This result can be checked by substituting these values into the original equations to yield 3(4) 1 2(3) 5 18 2(4) 1 2(3) 5 2 Thus, the results are equivalent to the right-hand sides of the original equations.

For three simultaneous equations, each equation would be represented by a plane in a three-dimensional coordinate system. The point where the three planes intersect would represent the solution. Beyond three equations, graphical methods break down and, consequently, have little practical value for solving simultaneous equations. However, they sometimes prove useful in visualizing properties of the solutions. For example, Fig. 9.2 depicts three cases that can pose problems when solving sets of linear equations. Figure 9.2a shows the case where the two equations represent parallel lines. For such situations, there is no solution because the lines never cross. Figure 9.2b depicts the case where the two lines are coincident. For such situations there is an infinite number of solutions. Both types of systems are said to be singular. In addition, systems that are very close to being singular (Fig. 9.2c) can also cause problems. These systems are said to be ill-conditioned. Graphically, this corresponds to the fact that it is difficult to identify the exact point at which the lines intersect. Ill-conditioned systems will also pose problems when they are encountered during the numerical solution of linear equations. This is because they will be extremely sensitive to round-off error (recall Sec. 4.2.3).

FIGURE 9.2 Graphical depiction of singular and ill-conditioned systems: (a) no solution, (b) inﬁnite solutions, and (c) ill-conditioned system where the slopes are so close that the point of intersection is difﬁcult to detect visually.

x2

x2

x2 1 x1 2

x2

1 1 x1 2

1

x2

x 1

2 x2 1 x1

1

2x 2

2

2.3 x 1 5

x2 1 x1

2

1.1

1

2

x1

(a)

x2

x1

x1

(b)

(c)

248

GAUSS ELIMINATION

9.1.2 Determinants and Cramer’s Rule Cramer’s rule is another solution technique that is best suited to small numbers of equations. Before describing this method, we will briefly introduce the concept of the determinant, which is used to implement Cramer’s rule. In addition, the determinant has relevance to the evaluation of the ill-conditioning of a matrix. Determinants.

The determinant can be illustrated for a set of three equations:

[A]{X} 5 {B} where [A] is the coefficient matrix: a11 [A] 5 £ a21 a31

a12 a22 a32

a13 a23 § a33

The determinant D of this system is formed from the coefficients of the equation, as in a11 D 5 † a21 a31

a12 a22 a32

a13 a23 † a33

(9.2)

Although the determinant D and the coefficient matrix [A] are composed of the same elements, they are completely different mathematical concepts. That is why they are distinguished visually by using brackets to enclose the matrix and straight lines to enclose the determinant. In contrast to a matrix, the determinant is a single number. For example, the value of the second-order determinant D5 `

a11 a21

a12 ` a22

is calculated by D 5 a11a22 2 a12a21

(9.3)

For the third-order case [Eq. (9.2)], a single numerical value for the determinant can be computed as D 5 a11 `

a22 a32

a23 a21 ` 2a12 ` a33 a31

a23 a21 ` 1a13 ` a33 a31

a22 ` a32

(9.4)

where the 2 by 2 determinants are called minors. EXAMPLE 9.2

Determinants Problem Statement. in Figs. 9.1 and 9.2.

Compute values for the determinants of the systems represented

Solution. For Fig. 9.1: D5 `

3 21

2 ` 5 3(2) 2 2(21) 5 8 2

9.1

SOLVING SMALL NUMBERS OF EQUATIONS

249

For Fig. 9.2a: D5`

21y2 21y2

1 21 21 `5 (1) 2 1 a b50 1 2 2

For Fig. 9.2b: D5`

21y2 21

1 21 ` 5 (2) 2 1(21) 5 0 2 2

For Fig. 9.2c: D5`

21y2 1 21 22.3 `5 (1) 2 1 a b 5 20.04 22.3y5 1 2 5

In the foregoing example, the singular systems had zero determinants. Additionally, the results suggest that the system that is almost singular (Fig. 9.2c) has a determinant that is close to zero. These ideas will be pursued further in our subsequent discussion of ill-conditioning (Sec. 9.3.3). Cramer’s Rule. This rule states that each unknown in a system of linear algebraic equations may be expressed as a fraction of two determinants with denominator D and with the numerator obtained from D by replacing the column of coefficients of the unknown in question by the constants b1, b2, . . . , bn. For example, x1 would be computed as

x1 5 EXAMPLE 9.3

b1 † b2 b3

a12 a22 a32 D

a13 a23 † a33

Cramer’s Rule Problem Statement. Use Cramer’s rule to solve 0.3x1 1 0.52x2 1 x3 5 20.01 0.5x1 1 x2 1 1.9x3 5 0.67 0.1x1 1 0.3x2 1 0.5x3 5 20.44 Solution. The determinant D can be written as [Eq. (9.2)] 0.3 D 5 † 0.5 0.1

0.52 1 0.3

1 1.9 † 0.5

The minors are [Eq. (9.3)] A1 5 `

1 0.3

1.9 ` 5 1(0.5) 2 1.9(0.3) 5 20.07 0.5

A2 5 `

0.5 0.1

1.9 ` 5 0.5(0.5) 2 1.9(0.1) 5 0.06 0.5

(9.5)

250

GAUSS ELIMINATION

A3 5 `

0.5 0.1

1 ` 5 0.5(0.3) 2 1(0.1) 5 0.05 0.3

These can be used to evaluate the determinant, as in [Eq. (9.4)] D 5 0.3(20.07) 2 0.52(0.06) 1 1(0.05) 5 20.0022 Applying Eq. (9.5), the solution is 20.01 0.52 1 † 0.67 1 1.9 † 0.03278 20.44 0.3 0.5 x1 5 5 5 214.9 20.0022 20.0022

x2 5

x3 5

0.3 † 0.5 0.1

20.01 1 0.67 1.9 † 20.44 0.5 0.0649 5 5 229.5 20.0022 20.0022

0.3 † 0.5 0.1

0.52 20.01 1 0.67 † 0.3 20.44 20.04356 5 5 19.8 20.0022 20.0022

For more than three equations, Cramer’s rule becomes impractical because, as the number of equations increases, the determinants are time consuming to evaluate by hand (or by computer). Consequently, more efficient alternatives are used. Some of these alternatives are based on the last noncomputer solution technique covered in the next section—the elimination of unknowns. 9.1.3 The Elimination of Unknowns The elimination of unknowns by combining equations is an algebraic approach that can be illustrated for a set of two equations: a11 x1 1 a12 x2 5 b1

(9.6)

a21 x1 1 a22 x2 5 b2

(9.7)

The basic strategy is to multiply the equations by constants so that one of the unknowns will be eliminated when the two equations are combined. The result is a single equation that can be solved for the remaining unknown. This value can then be substituted into either of the original equations to compute the other variable. For example, Eq. (9.6) might be multiplied by a21 and Eq. (9.7) by a11 to give a11 a21 x1 1 a12 a21 x2 5 b1a21 a21a11x1 1 a22a11x2 5 b2a11

(9.8) (9.9)

9.1

SOLVING SMALL NUMBERS OF EQUATIONS

251

Subtracting Eq. (9.8) from Eq. (9.9) will, therefore, eliminate the x1 term from the equations to yield a22a11x2 2 a12a21x2 5 b2a11 2 b1a21 which can be solved for x2 5

a11b2 2 a21b1 a11a22 2 a12a21

(9.10)

Equation (9.10) can then be substituted into Eq. (9.6), which can be solved for x1 5

a22b1 2 a12b2 a11a22 2 a12a21

(9.11)

Notice that Eqs. (9.10) and (9.11) follow directly from Cramer’s rule, which states b1 b2 x1 5 a11 ` a21

a12 ` b1a22 2 a12b2 a22 5 a11a22 2 a12 a21 a12 ` a22

a11 a21 x2 5 a11 ` a21

b1 ` a11b22 2 b1a21 b2 5 a11a22 2 a12 a21 a12 ` a22

`

`

EXAMPLE 9.4

Elimination of Unknowns Problem Statement. Use the elimination of unknowns to solve (recall Example 9.1) 3x1 1 2x2 5 18 2x1 1 2x2 5 2 Solution. Using Eqs. (9.11) and (9.10), x1 5

2(18) 2 2(2) 54 3(2) 2 2(21)

x2 5

3(2) 2 (21)18 53 3(2) 2 2(21)

which is consistent with our graphical solution (Fig. 9.1).

The elimination of unknowns can be extended to systems with more than two or three equations. However, the numerous calculations that are required for larger systems make the method extremely tedious to implement by hand. However, as described in the next section, the technique can be formalized and readily programmed for the computer.

252

GAUSS ELIMINATION

9.2

NAIVE GAUSS ELIMINATION In the previous section, the elimination of unknowns was used to solve a pair of simultaneous equations. The procedure consisted of two steps: 1. The equations were manipulated to eliminate one of the unknowns from the equations. The result of this elimination step was that we had one equation with one unknown. 2. Consequently, this equation could be solved directly and the result back-substituted into one of the original equations to solve for the remaining unknown. This basic approach can be extended to large sets of equations by developing a systematic scheme or algorithm to eliminate unknowns and to back-substitute. Gauss elimination is the most basic of these schemes. This section includes the systematic techniques for forward elimination and back substitution that comprise Gauss elimination. Although these techniques are ideally suited for implementation on computers, some modifications will be required to obtain a reliable algorithm. In particular, the computer program must avoid division by zero. The following method is called “naive” Gauss elimination because it does not avoid this problem. Subsequent sections will deal with the additional features required for an effective computer program. The approach is designed to solve a general set of n equations: a11x1 1 a12 x2 1 a13 x3 1 p 1 a1n xn 5 b1 a21x1 1 a22 x2 1 a23 x3 1 p 1 a2n xn 5 b2 . . . . . . an1x1 1 an2 x2 1 an3 x3 1 p 1 ann xn 5 bn

(9.12a) (9.12b)

(9.12c)

As was the case with the solution of two equations, the technique for n equations consists of two phases: elimination of unknowns and solution through back substitution. Forward Elimination of Unknowns. The first phase is designed to reduce the set of equations to an upper triangular system (Fig. 9.3). The initial step will be to eliminate the first unknown, x1, from the second through the nth equations. To do this, multiply Eq. (9.12a) by a21Ya11 to give a21x1 1

a21 a21 a21 a x 1p1 a x 5 b a11 12 2 a11 1n n a11 1

(9.13)

Now, this equation can be subtracted from Eq. (9.12b) to give aa22 2

a21 a21 a21 a b x 1 p 1 aa2n 2 a b x 5 b2 2 b a11 12 2 a11 1n n a11 1

or a¿22 x2 1 p 1 a¿2n xn 5 b¿2 where the prime indicates that the elements have been changed from their original values. The procedure is then repeated for the remaining equations. For instance, Eq. (9.12a) can be multiplied by a31ya11 and the result subtracted from the third equation. Repeating

9.2

NAIVE GAUSS ELIMINATION

a11 £ a21 a31

FIGURE 9.3 The two phases of Gauss elimination: forward elimination and back substitution. The primes indicate the number of times that the coefﬁcients and constants have been modiﬁed.

a12 a22 a32

a13 a23 a33

b1 b2 § b3

2 b1 a11 a12 a13 a'22 a'23 b'2 § £ a''33 b''3 2 x3 5 b''3ya''33 x2 5 (b'2 2 a'2333)ya'22 x1 5 (b1 2 a1232 2 a1333)ya11

253

Forward elimination

Back substitution

the procedure for the remaining equations results in the following modified system: a11x1 1 a12 x2 1 a13 x3 1 a¿22 x2 1 a¿23 x3 1 a¿32 x2 1 a¿33 x3 1 . . . . . . a¿n2 x2 1 a¿n3 x3 1

p 1 a1n xn 5 b1 p 1 a¿2n xn 5 b¿2 p 1 a¿3n xn 5 b¿3

(9.14a)

p 1 a¿nn xn 5 b¿n

(9.14d)

(9.14b) (9.14c)

For the foregoing steps, Eq. (9.12a) is called the pivot equation and a11 is called the pivot coefficient or element. Note that the process of multiplying the first row by a21ya11 is equivalent to dividing it by a11 and multiplying it by a21. Sometimes the division operation is referred to as normalization. We make this distinction because a zero pivot element can interfere with normalization by causing a division by zero. We will return to this important issue after we complete our description of naive Gauss elimination. Now repeat the above to eliminate the second unknown from Eq. (9.14c) through (9.14d). To do this multiply Eq. (9.14b) by a932ya922 and subtract the result from Eq. (9.14c). Perform a similar elimination for the remaining equations to yield a11 x1 1 a12 x2 1 a13 x3 1 p 1 a1n xn a¿22 x2 1 a¿23 x3 1 p 1 a¿2n xn a–33 x3 1 p 1 a–3n xn . . . . . . a–n3 x3 1 p 1 a–nn xn

5 b1 5 b¿2 5 b–2

5 b–n

where the double prime indicates that the elements have been modified twice.

254

GAUSS ELIMINATION

The procedure can be continued using the remaining pivot equations. The final manipulation in the sequence is to use the (n 2 1)th equation to eliminate the xn21 term from the nth equation. At this point, the system will have been transformed to an upper triangular system (recall Box PT3.1): a11x1 1 a12x2 1 a13 x3 1 p 1 a1n xn 5 b1 (9.15a) p a¿22 x2 1 a¿23 x3 1 1 a¿2n xn 5 b¿2 (9.15b) a–33 x3 1 p 1 a–3n xn 5 b–3 (9.15c) .

. . .

. .

(n21) ann xn 5 bn(n21)

(9.15d)

Pseudocode to implement forward elimination is presented in Fig. 9.4a. Notice that three nested loops provide a concise representation of the process. The outer loop moves down the matrix from one pivot row to the next. The middle loop moves below the pivot row to each of the subsequent rows where elimination is to take place. Finally, the innermost loop progresses across the columns to eliminate or transform the elements of a particular row. Back Substitution. Equation (9.15d) can now be solved for xn: xn 5

bn(n21)

(9.16)

(n21) ann

This result can be back-substituted into the (n 2 l)th equation to solve for xn21. The procedure, which is repeated to evaluate the remaining x’s, can be represented by the following formula: n

bi(i21) 2 a aij(i21)xj xi 5

FIGURE 9.4 Pseudocode to perform (a) forward elimination and (b) back substitution.

(a)

(b)

j5i11

aii(i21)

for i 5 n 2 1, n 2 2, p , 1

DOFOR k 5 1, n 2 1 DOFOR i 5 k 1 1, n factor 5 ai,k y ak,k DOFOR j 5 k 1 1 to n ai,j 5 ai,j 2 factor ? ak,j END DO bi 5 bi 2 factor ? bk END DO END DO xn 5 bn y an,n DOFOR i 5 n 2 1, 1, 21 sum 5 bi DOFOR j 5 i 1 1, n sum 5 sum 2 ai,j ? xj END DO xi 5 sum y ai,i END DO

(9.17)

9.2

NAIVE GAUSS ELIMINATION

255

Pseudocode to implement Eqs. (9.16) and (9.17) is presented in Fig. 9.4b. Notice the similarity between this pseudocode and that in Fig. PT3.4 for matrix multiplication. As with Fig. PT3.4, a temporary variable, sum, is used to accumulate the summation from Eq. (9.17). This results in a somewhat faster execution time than if the summation were accumulated in bi. More importantly, it allows efficient improvement in precision if the variable, sum, is declared in double precision. EXAMPLE 9.5

Naive Gauss Elimination Problem Statement.

Use Gauss elimination to solve

3x1 2 0.1x2 2 0.2x3 5 7.85

(E9.5.1)

0.1x1 1 7x2 2 0.3x3 5 219.3 0.3x1 2 0.2x2 1 10x3 5 71.4

(E9.5.2) (E9.5.3)

Carry six significant figures during the computation. Solution. The first part of the procedure is forward elimination. Multiply Eq. (E9.5.1) by (0.1)y3 and subtract the result from Eq. (E9.5.2) to give 7.00333x2 2 0.293333x3 5 219.5617 Then multiply Eq. (E9.5.1) by (0.3)y3 and subtract it from Eq. (E9.5.3) to eliminate x1. After these operations, the set of equations is 3x1 20.1x2 20.2x3 5 7.85 7.00333x2 2 0.293333x3 5 219.5617 20.190000x2 1 10.0200x3 5 70.6150

(E9.5.4) (E9.5.5) (E9.5.6)

To complete the forward elimination, x2 must be removed from Eq. (E9.5.6). To accomplish this, multiply Eq. (E9.5.5) by 20.190000y7.00333 and subtract the result from Eq. (E9.5.6). This eliminates x2 from the third equation and reduces the system to an upper triangular form, as in 3x1 20.1x2 20.2x3 5 7.85 7.00333x2 2 0.293333x3 5 219.5617 10.0120x3 5 70.0843

(E9.5.7) (E9.5.8) (E9.5.9)

We can now solve these equations by back substitution. First, Eq. (E9.5.9) can be solved for x3 5

70.0843 5 7.0000 10.0120

(E9.5.10)

This result can be back-substituted into Eq. (E9.5.8): 7.00333x2 2 0.293333(7.0000) 5 219.5617 which can be solved for x2 5

2 19.5617 1 0.293333(7.0000) 5 22.50000 7.00333

(E9.5.11)

256

GAUSS ELIMINATION

Finally, Eqs. (E9.5.10) and (E9.5.11) can be substituted into Eq. (E9.5.4): 3x1 2 0.1(22.50000) 2 0.2(7.0000) 5 7.85 which can be solved for x1 5

7.85 1 0.1(22.50000) 1 0.2(7.0000) 5 3.00000 3

The results are identical to the exact solution of x1 5 3, x2 5 22.5, and x3 5 7. This can be verified by substituting the results into the original equation set 3(3) 2 0.1(22.5) 2 0.2(7) 5 7.85 0.1(3) 1 7(22.5) 2 0.3(7) 5 219.3 0.3(3) 2 0.2(22.5) 1 10(7) 5 71.4

9.2.1 Operation Counting The execution time of Gauss elimination depends on the amount of floating-point operations (or flops) involved in the algorithm. On modern computers using math coprocessors, the time consumed to perform addition/subtraction and multiplication/division is about the same. Therefore, totaling up these operations provides insight into which parts of the algorithm are most time consuming and how computation time increases as the system gets larger. Before analyzing naive Gauss elimination, we will first define some quantities that facilitate operation counting: m

m

m

m

m

a cf (i) 5 c a f(i) a f(i) 1 g(i) 5 a f(i) 1 a g(i)

i51 m

i51

i51

i51

m

p 1 1 5 m a15111111 a15m2k11

i51 m

ai5112131

i51 m

(9.18a,b)

i51

(9.18c,d)

i5k

2 p 1 m 5 m(m 1 1) 5 m 1 O(m) 2 2

3 2 2 2 2 p 1 m2 5 m(m 1 1)(2m 1 1) 5 m 1 O(m2 ) i 5 1 1 2 1 3 1 a 6 3 i51

(9.18e) (9.18f )

where O(mn) means “terms of order mn and lower.” Now let us examine the naive Gauss elimination algorithm (Fig. 9.4a) in detail. We will first count the flops in the elimination stage. On the first pass through the outer loop, k 5 1. Therefore, the limits on the middle loop are from i 5 2 to n. According to Eq. (9.18d), this means that the number of iterations of the middle loop will be n

a15n22115n21

(9.19)

i52

For every one of these iterations, there is one division to define the factor. The interior loop then performs a single multiplication and subtraction for each iteration from j 5 2 to n. Finally, there is one additional multiplication and subtraction for the right-hand-side value.

9.2

NAIVE GAUSS ELIMINATION

257

Thus, for every iteration of the middle loop, the number of multiplications is 1 1 [n 2 2 1 1] 1 1 5 1 1 n

(9.20)

The total multiplications for the first pass through the outer loop is therefore obtained by multiplying Eq. (9.19) by (9.20) to give [n 2 1](1 1 n). In like fashion, the number of subtractions is computed as [n 2 1](n). Similar reasoning can be used to estimate the flops for the subsequent iterations of the outer loop. These can be summarized as Outer Loop k 1 2 . . . k . . . n 2 1

Middle Loop i

Addition/Subtraction ﬂops

Multiplication/Division ﬂops

2, n 3, n . . . k 1 1, n . . . n, n

(n 2 1)(n) (n 2 2)(n – 1)

(n 2 1)(n 1 1) (n 2 2)(n)

(n 2 k)(n 1 1 2 k)

(n 2 k)(n 1 2 2 k)

(1)(2)

(1) (3)

Therefore, the total addition/subtraction flops for elimination can be computed as n21

n21

2 a (n 2 k)(n 1 1 2 k) 5 a [n(n 1 1) 2 k(2n 1 1) 1 k ]

k51

k51

or n21

n21

n21

n(n 1 1) a 1 2 (2n 1 1) a k 1 a k2 k51

k51

k51

Applying some of the relationships from Eq. (9.18) yields 1 n3 [n3 1 O(n)] 2 [n3 1 O ( n2 ) ] 1 c n3 1 O ( n2 ) d 5 1 O(n) 3 3

(9.21)

A similar analysis for the multiplication/division flops yields 1 n3 [n3 1 O ( n2 ) ] 2 [n3 1 O(n)] 1 c n3 1 O ( n2 ) d 5 1 O ( n2 ) 3 3

(9.22)

Summing these results gives 2n3 1 O ( n2 ) 3 Thus, the total number of flops is equal to 2n3y3 plus an additional component proportional to terms of order n2 and lower. The result is written in this way because as n gets large, the O(n2) and lower terms become negligible. We are therefore justified in concluding that for large n, the effort involved in forward elimination converges on 2n3/3. Because only a single loop is used, back substitution is much simpler to evaluate. The number of addition/subtraction flops is equal to n(n 2 1)y2. Because of the extra

258

GAUSS ELIMINATION TABLE 9.1 Number of Flops for Gauss Elimination. n

Elimination

10 100 1000

705 671550 6.67 3 108

Back Substitution 100 10000 1 3 106

Total Flops

2n3/3

Percent Due to Elimination

805 681550 6.68 3 108

667 666667 6.67 3 108

87.58% 98.53% 99.85%

division prior to the loop, the number of multiplication/division flops is n(n 1 1)y2. These can be added to arrive at a total of n2 1 O(n) Thus, the total effort in naive Gauss elimination can be represented as 3 2n3 as n increases 2n 1 O ( n2 ) 1 n2 1 O(n) ————S 1 O ( n2 ) 3 3

Forward elimination

(9.23)

Backward substitution

Two useful general conclusions can be drawn from this analysis: 1. As the system gets larger, the computation time increases greatly. As in Table 9.1, the amount of flops increases nearly three orders of magnitude for every order of magnitude increase in the dimension. 2. Most of the effort is incurred in the elimination step. Thus, efforts to make the method more efficient should probably focus on this step.

9.3

PITFALLS OF ELIMINATION METHODS Whereas there are many systems of equations that can be solved with naive Gauss elimination, there are some pitfalls that must be explored before writing a general computer program to implement the method. Although the following material relates directly to naive Gauss elimination, the information is relevant for other elimination techniques as well. 9.3.1 Division by Zero The primary reason that the foregoing technique is called “naive” is that during both the elimination and the back-substitution phases, it is possible that a division by zero can occur. For example, if we use naive Gauss elimination to solve 2x2 1 3x3 5 8 4x1 1 6x2 1 7x3 5 23 2x1 1 x2 1 6x3 5 5 the normalization of the first row would involve division by a11 5 0. Problems also can arise when a coefficient is very close to zero. The technique of pivoting has been developed to partially avoid these problems. It will be described in Sec. 9.4.2.

9.3

PITFALLS OF ELIMINATION METHODS

259

9.3.2 Round-Off Errors Even though the solution in Example 9.5 was close to the true answer, there was a slight discrepancy in the result for x3 [Eq. (E9.5.10)]. This discrepancy, which amounted to a relative error of 20.00043 percent, was due to our use of six significant figures during the computation. If we had used more significant figures, the error in the results would be reduced further. If we had used fractions instead of decimals (and consequently avoided round-off altogether), the answers would have been exact. However, because computers carry only a limited number of significant figures (recall Sec. 3.4.1), round-off errors can occur and must be considered when evaluating the results. The problem of round-off error can become particularly important when large numbers of equations are to be solved. This is due to the fact that every result is dependent on previous results. Consequently, an error in the early steps will tend to propagate—that is, it will cause errors in subsequent steps. Specifying the system size where round-off error becomes significant is complicated by the fact that the type of computer and the properties of the equations are determining factors. A rough rule of thumb is that round-off error may be important when dealing with 100 or more equations. In any event, you should always substitute your answers back into the original equations to check whether a substantial error has occurred. However, as discussed below, the magnitudes of the coefficients themselves can influence whether such an error check ensures a reliable result. 9.3.3 Ill-Conditioned Systems The adequacy of the solution depends on the condition of the system. In Sec. 9.1.1, a graphical depiction of system condition was developed. As discussed in Sec. 4.2.3, well-conditioned systems are those where a small change in one or more of the coefficients results in a similar small change in the solution. Ill-conditioned systems are those where small changes in coefficients result in large changes in the solution. An alternative interpretation of ill-conditioning is that a wide range of answers can approximately satisfy the equations. Because round-off errors can induce small changes in the coefficients, these artificial changes can lead to large solution errors for ill-conditioned systems, as illustrated in the following example. EXAMPLE 9.6

Ill-Conditioned Systems Problem Statement.

Solve the following system:

x1 1 2x2 5 10 1.1x1 1 2x2 5 10.4

(E9.6.1) (E9.6.2)

Then, solve it again, but with the coefficient of x1 in the second equation modified slightly to 1.05. Solution. Using Eqs. (9.10) and (9.11), the solution is 2(10) 2 2(10.4) 54 1(2) 2 2(1.1) 1(10.4) 2 1.1(10) 53 x2 5 1(2) 2 2(1.1) x1 5

260

GAUSS ELIMINATION

However, with the slight change of the coefficient a21 from 1.1 to 1.05, the result is changed dramatically to x1 5

2(10) 2 2(10.4) 58 1(2) 2 2(1.05)

x2 5

1(10.4) 2 1.1(10) 51 1(2) 2 2(1.05)

Notice that the primary reason for the discrepancy between the two results is that the denominator represents the difference of two almost-equal numbers. As illustrated previously in Sec. 3.4.2, such differences are highly sensitive to slight variations in the numbers being manipulated. At this point, you might suggest that substitution of the results into the original equations would alert you to the problem. Unfortunately, for ill-conditioned systems this is often not the case. Substitution of the erroneous values of x1 5 8 and x2 5 1 into Eqs. (E9.6.1) and (E9.6.2) yields 8 1 2(1) 5 10 5 10 1.1(8) 1 2(1) 5 10.8 > 10.4 Therefore, although x1 5 8 and x2 5 1 is not the true solution to the original problem, the error check is close enough to possibly mislead you into believing that your solutions are adequate.

As was done previously in the section on graphical methods, a visual representative of ill-conditioning can be developed by plotting Eqs. (E9.6.1) and (E9.6.2) (recall Fig. 9.2). Because the slopes of the lines are almost equal, it is visually difficult to see exactly where they intersect. This visual difficulty is reflected quantitatively in the nebulous results of Example 9.6. We can mathematically characterize this situation by writing the two equations in general form: a11x1 1 a12x2 5 b1 a21x1 1 a22x2 5 b2

(9.24) (9.25)

Dividing Eq. (9.24) by a12 and Eq. (9.25) by a22 and rearranging yields alternative versions that are in the format of straight lines [x2 5 (slope) x1 1 intercept]: x2 5 2

a11 b1 x 1 a12 1 a12

x2 5 2

a21 b2 x 1 a22 1 a22

Consequently, if the slopes are nearly equal, a11 a21 > a12 a22

9.3

PITFALLS OF ELIMINATION METHODS

261

or, cross-multiplying, a11a22 > a12a21 which can be also expressed as a11a22 2 a12a21 > 0

(9.26)

Now, recalling that a11a22 2 a12a2l is the determinant of a two-dimensional system [Eq. (9.3)], we arrive at the general conclusion that an ill-conditioned system is one with a determinant close to zero. In fact, if the determinant is exactly zero, the two slopes are identical, which connotes either no solution or an infinite number of solutions, as is the case for the singular systems depicted in Fig. 9.2a and b. It is difficult to specify how close to zero the determinant must be to indicate illconditioning. This is complicated by the fact that the determinant can be changed by multiplying one or more of the equations by a scale factor without changing the solution. Consequently, the determinant is a relative value that is influenced by the magnitude of the coefficients. EXAMPLE 9.7

Effect of Scale on the Determinant Problem Statement. Evaluate the determinant of the following systems: (a) From Example 9.1: 3x1 1 2x2 5 18 2x1 1 2x2 5 2

(E9.7.1) (E9.7.2)

(b) From Example 9.6: x1 1 2x2 5 10 1.1x1 1 2x2 5 10.4

(E9.7.3) (E9.7.4)

(c) Repeat (b) but with the equations multiplied by 10. Solution. (a) The determinant of Eqs. (E9.7.1) and (E9.7.2), which are well-conditioned, is D 5 3(2) 2 2(21) 5 8 (b) The determinant of Eqs. (E9.7.3) and (E9.7.4), which are ill-conditioned, is D 5 1(2) 2 2(1.1) 5 20.2 (c) The results of (a) and (b) seem to bear out the contention that ill-conditioned systems have near-zero determinants. However, suppose that the ill-conditioned system in (b) is multiplied by 10 to give 10x1 1 20x2 5 100 11x1 1 20x2 5 104 The multiplication of an equation by a constant has no effect on its solution. In addition, it is still ill-conditioned. This can be verified by the fact that multiplying by

262

GAUSS ELIMINATION

a constant has no effect on the graphical solution. However, the determinant is dramatically affected: D 5 10(20) 2 20(11) 5 220 Not only has it been raised two orders of magnitude, but it is now over twice as large as the determinant of the well-conditioned system in (a). As illustrated by the previous example, the magnitude of the coefficients interjects a scale effect that complicates the relationship between system condition and determinant size. One way to partially circumvent this difficulty is to scale the equations so that the maximum element in any row is equal to 1. EXAMPLE 9.8

Scaling Problem Statement. Scale the systems of equations in Example 9.7 to a maximum value of 1 and recompute their determinants. Solution. (a) For the well-conditioned system, scaling results in x1 1 0.667x2 5 6 20.5x1 1 x2 5 1 for which the determinant is D 5 1(1) 2 0.667(20.5) 5 1.333 (b) For the ill-conditioned system, scaling gives 0.5x1 1 x2 5 5 0.55x1 1 x2 5 5.2 for which the determinant is D 5 0.5(1) 2 1(0.55) 5 20.05 (c) For the last case, scaling changes the system to the same form as in (b) and the determinant is also 20.05. Thus, the scale effect is removed. In a previous section (Sec. 9.1.2), we suggested that the determinant is difficult to compute for more than three simultaneous equations. Therefore, it might seem that it does not provide a practical means for evaluating system condition. However, as described in Box 9.1, there is a simple algorithm that results from Gauss elimination that can be used to evaluate the determinant. Aside from the approach used in the previous example, there are a variety of other ways to evaluate system condition. For example, there are alternative methods for normalizing the elements (see Stark, 1970). In addition, as described in the next chapter (Sec. 10.3), the matrix inverse and matrix norms can be employed to evaluate system condition. Finally, a simple (but time-consuming) test is to modify the coefficients

9.3

Box 9.1

PITFALLS OF ELIMINATION METHODS

Determinant Evaluation Using Gauss Elimination

In Sec. 9.1.2, we stated that determinant evaluation by expansion of minors was impractical for large sets of equations. Thus, we concluded that Cramer’s rule would be applicable only to small systems. However, as mentioned in Sec. 9.3.3, the determinant has value in assessing system condition. It would, therefore, be useful to have a practical method for computing this quantity. Fortunately, Gauss elimination provides a simple way to do this. The method is based on the fact that the determinant of a triangular matrix can be simply computed as the product of its diagonal elements: D 5 a11a22a33 p ann

(B9.1.1)

The validity of this formulation can be illustrated for a 3 by 3 system: a11 D5 † 0 0

a12 a22 0

a13 a23 † a33

a22 0

0 a23 ` 2a12 ` a33 0

Recall that the forward-elimination step of Gauss elimination results in an upper triangular system. Because the value of the determinant is not changed by the forward-elimination process, the determinant can be simply evaluated at the end of this step via (n21) D 5 a11a¿22 a–33 p ann

a23 0 ` 1a13 ` a33 0

a22 ` 0

or, by evaluating the minors (that is, the 2 by 2 determinants),

(B9.1.2)

where the superscripts signify the number of times that the elements have been modified by the elimination process. Thus, we can capitalize on the effort that has already been expended in reducing the system to triangular form and, in the bargain, come up with a simple estimate of the determinant. There is a slight modification to the above approach when the program employs partial pivoting (Sec. 9.4.2). For this case, the determinant changes sign every time a row is pivoted. One way to represent this is to modify Eq. (B9.1.2): (n21) D 5 a11a¿22 a–33 p ann (21) p

where the determinant can be evaluated as [recall Eq. (9.4)] D 5 a11 `

263

(B9.1.3)

where p represents the number of times that rows are pivoted. This modification can be incorporated simply into a program; merely keep track of the number of pivots that take place during the course of the computation and then use Eq. (B9.1.3) to evaluate the determinant.

D 5 a11a22 a33 2 a12 (0) 1 a13 (0) 5 a11a12 a33

slightly and repeat the solution. If such modifications lead to drastically different results, the system is likely to be ill-conditioned. As you might gather from the foregoing discussion, ill-conditioned systems are problematic. Fortunately, most linear algebraic equations derived from engineering-problem settings are naturally well-conditioned. In addition, some of the techniques outlined in Sec. 9.4 help to alleviate the problem. 9.3.4 Singular Systems In the previous section, we learned that one way in which a system of equations can be ill-conditioned is when two or more of the equations are nearly identical. Obviously, it is even worse when the two are identical. In such cases, we would lose one degree of freedom, and would be dealing with the impossible case of n 2 1 equations with n unknowns. Such cases might not be obvious to you, particularly when dealing with large equation sets. Consequently, it would be nice to have some way of automatically detecting singularity. The answer to this problem is neatly offered by the fact that the determinant of a singular system is zero. This idea can, in turn, be connected to Gauss elimination by recognizing that after the elimination step, the determinant can be evaluated as the product of the diagonal elements (recall Box 9.1). Thus, a computer algorithm can test to discern whether a zero diagonal element is created during the elimination stage. If one is discovered, the calculation can be immediately terminated and a message displayed

264

GAUSS ELIMINATION

alerting the user. We will show the details of how this is done when we present a full algorithm for Gauss elimination later in this chapter.

9.4

TECHNIQUES FOR IMPROVING SOLUTIONS The following techniques can be incorporated into the naive Gauss elimination algorithm to circumvent some of the pitfalls discussed in the previous section. 9.4.1 Use of More Signiﬁcant Figures The simplest remedy for ill-conditioning is to use more significant figures in the computation. If your application can be extended to handle larger word size, such a feature will greatly reduce the problem. However, a price must be paid in the form of the computational and memory overhead connected with using extended precision (recall Sec. 3.4.1). 9.4.2 Pivoting As mentioned at the beginning of Sec. 9.3, obvious problems occur when a pivot element is zero because the normalization step leads to division by zero. Problems may also arise when the pivot element is close to, rather than exactly equal to, zero because if the magnitude of the pivot element is small compared to the other elements, then round-off errors can be introduced. Therefore, before each row is normalized, it is advantageous to determine the largest available coefficient in the column below the pivot element. The rows can then be switched so that the largest element is the pivot element. This is called partial pivoting. If columns as well as rows are searched for the largest element and then switched, the procedure is called complete pivoting. Complete pivoting is rarely used because switching columns changes the order of the x’s and, consequently, adds significant and usually unjustified complexity to the computer program. The following example illustrates the advantages of partial pivoting. Aside from avoiding division by zero, pivoting also minimizes round-off error. As such, it also serves as a partial remedy for ill-conditioning.

EXAMPLE 9.9

Partial Pivoting Problem Statement. Use Gauss elimination to solve 0.0003x1 1 3.0000x2 5 2.0001 1.0000x1 1 1.0000x2 5 1.0000 Note that in this form the first pivot element, a11 5 0.0003, is very close to zero. Then repeat the computation, but partial pivot by reversing the order of the equations. The exact solution is x1 5 1y3 and x2 5 2y3. Solution. Multiplying the first equation by 1y(0.0003) yields x1 1 10,000x2 5 6667 which can be used to eliminate x1 from the second equation: 29999x2 5 26666

9.4

TECHNIQUES FOR IMPROVING SOLUTIONS

265

which can be solved for x2 5

2 3

This result can be substituted back into the first equation to evaluate x1: x1 5

2.0001 2 3(2y3) 0.0003

(E9.9.1)

However, due to subtractive cancellation, the result is very sensitive to the number of significant figures carried in the computation:

Signiﬁcant Figures

x2

3 4 5 6 7

0.667 0.6667 0.66667 0.666667 0.6666667

x1 23.33 0.0000 0.30000 0.330000 0.3330000

Absolute Value of Percent Relative Error for x1 1099 100 10 1 0.1

Note how the solution for x1 is highly dependent on the number of significant figures. This is because in Eq. (E9.9.1), we are subtracting two almost-equal numbers. On the other hand, if the equations are solved in reverse order, the row with the larger pivot element is normalized. The equations are 1.0000x1 1 1.0000x2 5 1.0000 0.0003x1 1 3.0000x2 5 2.0001 Elimination and substitution yield x2 5 2y3. For different numbers of significant figures, x1 can be computed from the first equation, as in x1 5

1 2 (2y3) 1

(E9.9.2)

This case is much less sensitive to the number of significant figures in the computation:

Signiﬁcant Figures

x2

x1

3 4 5 6 7

0.667 0.6667 0.66667 0.666667 0.6666667

0.333 0.3333 0.33333 0.333333 0.3333333

Thus, a pivot strategy is much more satisfactory.

Absolute Value of Percent Relative Error for x1 0.1 0.01 0.001 0.0001 0.00001

266

GAUSS ELIMINATION

p5 k big 5 |ak,k| DOFOR ii 5 k11, n dummy 5 |aii,k| IF (dummy . big) big 5 dummy p 5 ii END IF END DO IF (p ﬁ k) DOFOR jj 5 k, n dummy 5 ap,jj ap,jj 5 ak,jj ak,jj 5 dummy END DO dummy 5 bp bp 5 bk bk 5 dummy END IF

FIGURE 9.5 Pseudocode to implement partial pivoting.

EXAMPLE 9.10

General-purpose computer programs must include a pivot strategy. Figure 9.5 provides a simple algorithm to implement such a strategy. Notice that the algorithm consists of two major loops. After storing the current pivot element and its row number as the variables, big and p, the first loop compares the pivot element with the elements below it to check whether any of these is larger than the pivot element. If so, the new largest element and its row number are stored in big and p. Then, the second loop switches the original pivot row with the one with the largest element so that the latter becomes the new pivot row. This pseudocode can be integrated into a program based on the other elements of Gauss elimination outlined in Fig. 9.4. The best way to do this is to employ a modular approach and write Fig. 9.5 as a subroutine (or procedure) that would be called directly after the beginning of the first loop in Fig. 9.4a. Note that the second IF/THEN construct in Fig. 9.5 physically interchanges the rows. For large matrices, this can become quite time consuming. Consequently, most codes do not actually exchange rows but rather keep track of the pivot rows by storing the appropriate subscripts in a vector. This vector then provides a basis for specifying the proper row ordering during the forward-elimination and back-substitution operations. Thus, the operations are said to be implemented in place. 9.4.3 Scaling In Sec. 9.3.3, we proposed that scaling had value in standardizing the size of the determinant. Beyond this application, it has utility in minimizing round-off errors for those cases where some of the equations in a system have much larger coefficients than others. Such situations are frequently encountered in engineering practice when widely different units are used in the development of simultaneous equations. For instance, in electriccircuit problems, the unknown voltages can be expressed in units ranging from microvolts to kilovolts. Similar examples can arise in all fields of engineering. As long as each equation is consistent, the system will be technically correct and solvable. However, the use of widely differing units can lead to coefficients of widely differing magnitudes. This, in turn, can have an impact on round-off error as it affects pivoting, as illustrated by the following example. Effect of Scaling on Pivoting and Round-Off Problem Statement. (a) Solve the following set of equations using Gauss elimination and a pivoting strategy: 2x1 1 100,000x2 5 100,000 x1 1 x2 5 2 (b) Repeat the solution after scaling the equations so that the maximum coefficient in each row is 1. (c) Finally, use the scaled coefficients to determine whether pivoting is necessary. However, actually solve the equations with the original coefficient values. For all cases, retain only three significant figures. Note that the correct answers are x1 5 1.00002 and x2 5 0.99998 or, for three significant figures, x1 5 x2 5 1.00.

9.4

TECHNIQUES FOR IMPROVING SOLUTIONS

267

Solution. (a) Without scaling, forward elimination is applied to give 2x1 1 100,000x2 5 100,000 250,000x2 5 250,000 which can be solved by back substitution for x2 5 1.00 x1 5 0.00 Although x2 is correct, x1 is 100 percent in error because of round-off. (b) Scaling transforms the original equations to 0.00002x1 1 x2 5 1 x1 1 x2 5 2 Therefore, the rows should be pivoted to put the greatest value on the diagonal. x1 1 x2 5 2 0.00002x1 1 x2 5 1 Forward elimination yields x1 1 x2 5 2 x2 5 1.00 which can be solved for x1 5 x2 5 1 Thus, scaling leads to the correct answer. (c) The scaled coefficients indicate that pivoting is necessary. We therefore pivot but retain the original coefficients to give x1 1

x2 5 2

2x1 1 100,000x2 5 100,000 Forward elimination yields x1 1

x2 5 2

100,000x2 5 100,000 which can be solved for the correct answer: x1 5 x2 5 1. Thus, scaling was useful in determining whether pivoting was necessary, but the equations themselves did not require scaling to arrive at a correct result.

268

GAUSS ELIMINATION

SUB Gauss (a, b, n, x, tol, er) DIMENSION s(n) er 5 0 DOFOR i 5 1, n si 5 ABS(ai,1) DOFOR j 5 2, n IF ABS(ai,j).si THEN si 5 ABS(ai,j) END DO END DO CALL Eliminate(a, s, n, b, tol, er) IF er ? 21 THEN CALL Substitute(a, n, b, x) END IF END Gauss

SUB Eliminate (a, s, n, b, tol, er) DOFOR k 5 1, n 2 1 CALL Pivot (a, b, s, n, k) IF ABS (ak,k/sk) , tol THEN er 5 21 EXIT DO END IF DOFOR i 5 k 1 1, n factor 5 ai,k/ak,k DOFOR j 5 k 1 1, n ai,j 5 ai,j 2 factor*ak,j END DO bi 5 bi 2 factor * bk END DO END DO IF ABS(an,n/sn) , to1 THEN er 5 21 END Eliminate

FIGURE 9.6 Pseudocode to implement Gauss elimination with partial pivoting.

SUB Pivot (a, b, s, n, k) p5 k big 5 ABS(ak,k/sk) DOFOR ii 5 k 1 1, n dummy 5 ABS(aii,k/sii) IF dummy . big THEN big 5 dummy p 5 ii END IF END DO IF p ? k THEN DOFOR jj 5 k, n dummy 5 ap,jj ap,jj 5 ak,jj ak,jj 5 dummy END DO dummy 5 bp bp 5 bk bk 5 dummy dummy 5 sp sp 5 sk sk 5 dummy END IF END pivot

SUB Substitute (a, n, b, x) xn 5 bn/an,n DOFOR i 5 n 2 1, 1, 21 sum 5 0 DOFOR j 5 i 1 1, n sum 5 sum 1 ai,j * xj END DO xn 5 (bn 2 sum) / an,n END DO END Substitute

9.4

TECHNIQUES FOR IMPROVING SOLUTIONS

269

As in the previous example, scaling has utility in minimizing round-off. However, it should be noted that scaling itself also leads to round-off. For example, given the equation 2x1 1 300,000x2 5 1 and using three significant figures, scaling leads to 0.00000667x1 1 x2 5 0.00000333 Thus, scaling introduces a round-off error to the first coefficient and the right-hand-side constant. For this reason, it is sometimes suggested that scaling should be employed only as in part (c) of the preceding example. That is, it is used to calculate scaled values for the coefficients solely as a criterion for pivoting, but the original coefficient values are retained for the actual elimination and substitution computations. This involves a tradeoff if the determinant is being calculated as part of the program. That is, the resulting determinant will be unscaled. However, because many applications of Gauss elimination do not require determinant evaluation, it is the most common approach and will be used in the algorithm in the next section. 9.4.4 Computer Algorithm for Gauss Elimination The algorithms from Figs. 9.4 and 9.5 can now be combined into a larger algorithm to implement the entire Gauss elimination algorithm. Figure 9.6 shows an algorithm for a general subroutine to implement Gauss elimination. Note that the program includes modules for the three primary operations of the Gauss elimination algorithm: forward elimination, back substitution, and pivoting. In addition, there are several aspects of the code that differ and represent improvements over the pseudocodes from Figs. 9.4 and 9.5. These are: The equations are not scaled, but scaled values of the elements are used to determine whether pivoting is to be implemented. The diagonal term is monitored during the pivoting phase to detect near-zero occurrences in order to flag singular systems. If it passes back a value of er 5 21, a singular matrix has been detected and the computation should be terminated. A parameter tol is set by the user to a small number in order to detect near-zero occurrences. EXAMPLE 9.11

Solution of Linear Algebraic Equations Using the Computer Problem Statement. A computer program to solve linear algebraic equations such as one based on Fig. 9.6 can be used to solve a problem associated with the falling parachutist example discussed in Chap. 1. Suppose that a team of three parachutists is connected by a weightless cord while free-falling at a velocity of 5 m/s (Fig. 9.7).

270

GAUSS ELIMINATION c2v

c3v

3

R

c1v

T

R

m3g

m 2g

R

T 2

3

2

m1g 1

FIGURE 9.8 Free-body diagrams for each of the three falling parachutists.

Calculate the tension in each section of cord and the acceleration of the team, given the following: Parachutist

Mass, kg

1 2 3

70 60 40

T

a 1

Drag Coefﬁcient, kg/s 10 14 17

Solution. Free-body diagrams for each of the parachutists are depicted in Fig. 9.8. Summing the forces in the vertical direction and using Newton’s second law gives a set of three simultaneous linear equations: m1g 2 T 2 c1y 5 m1a m2g 1 T 2 c2y 2 R 5 m2a m3g 2 c3y 1 R 5 m3a These equations have three unknowns: a, T, and R. After substituting the known values, the equations can be expressed in matrix form as (g 5 9.81 m/s2),

FIGURE 9.7 Three parachutists free-falling while connected by weightless cords.

70 £ 60 40

1 21 0

0 a 636.7 1 § • T ¶ 5 • 518.6 ¶ 21 R 307.4

This system can be solved using your own software. The result is a 5 8.6041 m/s2; T 5 34.4118 N; and R 5 36.7647 N.

9.6

9.5

NONLINEAR SYSTEMS OF EQUATIONS

271

COMPLEX SYSTEMS In some problems, it is possible to obtain a complex system of equations [C]{Z} 5 {W}

(9.27)

where [C] 5 [A] 1 i[B] {Z} 5 {X} 1 i{Y} {W} 5 {U} 1 i{V}

(9.28)

where i 5 121. The most straightforward way to solve such a system is to employ one of the algorithms described in this part of the book, but replace all real operations with complex ones. Of course, this is only possible for those languages, such as Fortran, that allow complex variables. For languages that do not permit the declaration of complex variables, it is possible to write a code to convert real to complex operations. However, this is not a trivial task. An alternative is to convert the complex system into an equivalent one dealing with real variables. This can be done by substituting Eq. (9.28) into Eq. (9.27) and equating real and complex parts of the resulting equation to yield [A]{X} 2 [B]{Y} 5 {U}

(9.29)

[B]{X} 1 [A]{Y} 5 {V}

(9.30)

and

Thus, the system of n complex equations is converted to a set of 2n real ones. This means that storage and execution time will be increased significantly. Consequently, a trade-off exists regarding this option. If you evaluate complex systems infrequently, it is preferable to use Eqs. (9.29) and (9.30) because of their convenience. However, if you use them often and desire to employ a language that does not allow complex data types, it may be worth the up-front programming effort to write a customized equation solver that converts real to complex operations.

9.6

NONLINEAR SYSTEMS OF EQUATIONS Recall that at the end of Chap. 6 we presented an approach to solve two nonlinear equations with two unknowns. This approach can be extended to the general case of solving n simultaneous nonlinear equations. f1 (x1, x2, p , xn ) 5 0 f2 (x1, x2, p , xn ) 5 0 . . . . . . fn (x1, x2, p , xn ) 5 0

(9.31)

272

GAUSS ELIMINATION

The solution of this system consists of the set of x values all the equations equaling zero. As described in Sec. 6.5.2, one approach to solving multidimensional version of the Newton-Raphson method. sion is written for each equation. For example, for the kth fk,i11 5 fk,i 1 (x1,i11 2 x1,i )

0fk,i 0x1

1 (x2,i11 2 x2,i )

0fk,i 0x2

that simultaneously result in such systems is based on a Thus, a Taylor series expanequation,

1 p 1 (xn,i11 2 xn,i )

0fk,i 0xn (9.32)

where the first subscript, k, represents the equation or unknown and the second subscript denotes whether the value or function in question is at the present value (i) or at the next value (i 1 1). Equations of the form of (9.32) are written for each of the original nonlinear equations. Then, as was done in deriving Eq. (6.20) from (6.19), all fk,i11 terms are set to zero as would be the case at the root, and Eq. (9.32) can be written as 2fk,i 1 x1,i 5 x1,i11

0fk,i 0x1

0fk,i 0x1

1 x2,i

1 x2,i11

0fk,i 0x2 0fk,i 0x2

1 p 1 xn,i

0fk,i 0 xn

1 p 1 xn,i11

0fk,i 0 xn

(9.33)

Notice that the only unknowns in Eq. (9.33) are the xk,i11 terms on the right-hand side. All other quantities are located at the present value (i) and, thus, are known at any iteration. Consequently, the set of equations generally represented by Eq. (9.33) (that is, with k 5 1, 2, . . . , n) constitutes a set of linear simultaneous equations that can be solved by methods elaborated in this part of the book. Matrix notation can be employed to express Eq. (9.33) concisely. The partial derivatives can be expressed as 0f1,i

0f1,i

0x1 0f2,i

0x2 0f2,i

0x1 I . [Z] 5 . . 0fn,i

0x2 . . . 0fn,i

0x1

0x2

p

0f1,i

p

0xn 0f2,i

p

0xn . Y . . 0fn,i 0xn

The initial and final values can be expressed in vector form as {Xi } T 5 :x1,i x2,i p xn,i ; and {Xi11 } T 5 :x1,i11 x2,i11 p xn,i11 ;

(9.34)

9.7

GAUSS-JORDAN

273

Finally, the function values at i can be expressed as a11 £ a21 a31

a12 a22 a32

a13 a23 a33

b1 b2 § b3

T 1 £0 0

0 1 0

b(n)1 b(n)2 § b(n)3

0 0 1

T x1 x2 x3

5 5 5

b(n)1 b(n)2 b(n)3

FIGURE 9.9 Graphical depiction of the Gauss-Jordan method. Compare with Fig. 9.3 to elucidate the differences between this technique and Gauss elimination. The superscript (n) means that the elements of the right-handside vector have been modiﬁed n times (for this case, n 5 3).

9.7

{Fi } T 5 : f1,i f2,i p fn,i ; Using these relationships, Eq. (9.33) can be represented concisely as [Z]{Xi11 } 5 2{Fi } 1 [Z]{Xi }

(9.35)

Equation (9.35) can be solved using a technique such as Gauss elimination. This process can be repeated iteratively to obtain refined estimates in a fashion similar to the twoequation case in Sec. 6.5.2. It should be noted that there are two major shortcomings to the foregoing approach. First, Eq. (9.34) is often inconvenient to evaluate. Therefore, variations of the NewtonRaphson approach have been developed to circumvent this dilemma. As might be expected, most are based on using finite-difference approximations for the partial derivatives that comprise [Z]. The second shortcoming of the multiequation Newton-Raphson method is that excellent initial guesses are usually required to ensure convergence. Because these are often difficult to obtain, alternative approaches that are slower than Newton-Raphson but which have better convergence behavior have been developed. One common approach is to reformulate the nonlinear system as a single function n

F(x) 5 a [ fi (x1, x2, p , xn )] 2

(9.36)

i51

where fi (xl, x2, . . . , xn) is the ith member of the original system of Eq. (9.31). The values of x that minimize this function also represent the solution of the nonlinear system. As we will see in Chap. 17, this reformulation belongs to a class of problems called nonlinear regression. As such, it can be approached with a number of optimization techniques such as the ones described later in this text (Part Four and specifically Chap. 14).

GAUSS-JORDAN The Gauss-Jordan method is a variation of Gauss elimination. The major difference is that when an unknown is eliminated in the Gauss-Jordan method, it is eliminated from all other equations rather than just the subsequent ones. In addition, all rows are normalized by dividing them by their pivot elements. Thus, the elimination step results in an identity matrix rather than a triangular matrix (Fig. 9.9). Consequently, it is not necessary to employ back substitution to obtain the solution. The method is best illustrated by an example.

EXAMPLE 9.12

Gauss-Jordan Method Problem Statement. Use the Gauss-Jordan technique to solve the same system as in Example 9.5: 3x1 2 0.1x2 2 0.2x3 5 7.85 0.1x1 1 7x2 2 0.3x3 5 219.3 0.3x1 2 0.2x2 1 10x3 5 71.4

274

GAUSS ELIMINATION

Solution. First, express the coefficients and the right-hand side as an augmented matrix: 3 £ 0.1 0.3

20.1 7 20.2

20.2 20.3 10

7.85 219.3 § 71.4

Then normalize the first row by dividing it by the pivot element, 3, to yield 1 £ 0.1 0.3

20.0333333 7 20.2

20.066667 20.3 10

2.61667 219.3 § 71.4

The x1 term can be eliminated from the second row by subtracting 0.1 times the first row from the second row. Similarly, subtracting 0.3 times the first row from the third row will eliminate the x1 term from the third row: 1 £0 0

20.0333333 7.00333 20.190000

20.066667 20.293333 10.0200

2.61667 219.5617 § 70.6150

Next, normalize the second row by dividing it by 7.00333: 1 £0 0

20.0333333 1 20.190000

20.066667 20.0418848 10.0200

2.61667 22.79320 § 70.6150

Reduction of the x2 terms from the first and third equations gives 1 £0 0

0 1 0

20.0680629 20.0418848 10.01200

2.52356 22.79320 § 70.0843

The third row is then normalized by dividing it by 10.0120: 1 £0 0

0 1 0

20.0680629 20.0418848 1

2.52356 22.79320 § 7.0000

Finally, the x3 terms can be reduced from the first and the second equations to give 1 £0 0

0 1 0

0 0 1

3.0000 22.5000 § 7.0000

Thus, as depicted in Fig. 9.9, the coefficient matrix has been transformed to the identity matrix, and the solution is obtained in the right-hand-side vector. Notice that no back substitution was required to obtain the solution.

All the material in this chapter regarding the pitfalls and improvements in Gauss elimination also applies to the Gauss-Jordan method. For example, a similar pivoting strategy can be used to avoid division by zero and to reduce round-off error.

PROBLEMS

275

Although the Gauss-Jordan technique and Gauss elimination might appear almost identical, the former requires more work. Using a similar approach to Sec. 9.2.1, it can be determined that the number of flops involved in naive Gauss-Jordan is as n increases

n3 1 n2 2 n ————S n3 1 O ( n2 )

(9.37)

Thus, Gauss-Jordan involves approximately 50 percent more operations than Gauss elimination [compare with Eq. (9.23)]. Therefore, Gauss elimination is the simple elimination method of preference for obtaining solutions of linear algebraic equations. One of the primary reasons that we have introduced the Gauss-Jordan, however, is that it is still used in engineering as well as in some numerical algorithms.

9.8

SUMMARY In summary, we have devoted most of this chapter to Gauss elimination, the most fundamental method for solving simultaneous linear algebraic equations. Although it is one of the earliest techniques developed for this purpose, it is nevertheless an extremely effective algorithm for obtaining solutions for many engineering problems. Aside from this practical utility, this chapter also provided a context for our discussion of general issues such as round-off, scaling, and conditioning. In addition, we briefly presented material on the Gauss-Jordan method, as well as complex and nonlinear systems. Answers obtained using Gauss elimination may be checked by substituting them into the original equations. However, this does not always represent a reliable check for illconditioned systems. Therefore, some measure of condition, such as the determinant of the scaled system, should be computed if round-off error is suspected. Using partial pivoting and more significant figures in the computation are two options for mitigating round-off error. In the next chapter, we will return to the topic of system condition when we discuss the matrix inverse.

PROBLEMS 9.1 (a) Write the following set of equations in matrix form: 8 5 6x3 1 2x2

1 [E] 5 £ 7 4

5 2 0

8 3§ 6

2 2 x1 5 x3

[F] 5 c

3 1

0 7

1 d :G ; 5 :7 3

5x2 1 8xl 5 13 (b) Multiply the matrix of coefficients by its transpose; i.e., [A][A]T. 9.2 A number of matrices are defined as 4 [A] 5 £ 1 5

7 4 2 § [B] 5 £ 1 6 2

3 9 {C} 5 • 6 ¶ [D] 5 c 2 1

3 2 0 4 21

7 7§ 4 3 7

26 d 5

6

4;

Answer the following questions regarding these matrices: (a) What are the dimensions of the matrices? (b) Identify the square, column, and row matrices. (c) What are the values of the elements: a12, b23, d32, e22, f12, g12? (d) Perform the following operations: (1) (2) (3) (4)

[E] 1 [B] [A] 3 [F] [B] 2 [E] 7 3 [B]

(5) (6) (7) (8)

[E] 3 [B] {C} T [B] 3 [A] [D] T

276

GAUSS ELIMINATION (e) Solve again, but with a11 modified slightly to 0.52. Interpret your results. 9.8 Given the equations

(9) [A] 3 {C} (11) [E] T [E] (10) [I] 3 [B] (12) {C} T {C} 9.3 Three matrices are defined as 1 [A] 5 £ 3 7

6 1 10 § [B] 5 c 0.5 4

10x1 1 2x2 2 x3 5 27 3 2 d [C] 5 c 2 23

22 d 1

(a) Perform all possible multiplications that can be computed between pairs of these matrices. (b) Use the method in Box PT3.2 to justify why the remaining pairs cannot be multiplied. (c) Use the results of (a) to illustrate why the order of multiplication is important. 9.4 Use the graphical method to solve

23x1 2 6x2 1 2x3 5 261.5 x1 1 x2 1 5x3 5 221.5 (a) Solve by naive Gauss elimination. Show all steps of the computation. (b) Substitute your results into the original equations to check your answers. 9.9 Use Gauss elimination to solve: 8x1 1 2x2 2 2x3 5 22 10x1 1 2x2 1 4x3 5 4

4x1 2 8x2 5 224

12x1 1 2x2 1 2x3 5 6

2x1 1 6x2 5 34 Check your results by substituting them back into the equations. 9.5 Given the system of equations

Employ partial pivoting and check your answers by substituting them into the original equations. 9.10 Given the system of equations

21.1x1 1 10x2 5 120

23x2 1 7x3 5 2

22x1 1 17.4x2 5 174 (a) Solve graphically and check your results by substituting them back into the equations. (b) On the basis of the graphical solution, what do you expect regarding the condition of the system? (c) Compute the determinant. (d) Solve by the elimination of unknowns. 9.6 For the set of equations 2x2 1 5x3 5 9 2x1 1 x2 1 x3 5 9 3x1 1 x2 5 10 (a) Compute the determinant. (b) Use Cramer’s rule to solve for the x’s. (c) Substitute your results back into the original equation to check your results. 9.7 Given the equations 0.5x1 2 x2 5 29.5 1.02x1 2 2x2 5 218.8 (a) Solve graphically. (b) Compute the determinant. (c) On the basis of (a) and (b), what would you expect regarding the system’s condition? (d) Solve by the elimination of unknowns.

x1 1 2x2 2 x3 5 3 5x1 2 2x2 5 2 (a) (b) (c) (d)

Compute the determinant. Use Cramer’s rule to solve for the x’s. Use Gauss elimination with partial pivoting to solve for the x’s. Substitute your results back into the original equations to check your solution. 9.11 Given the equations 2x1 2 6x2 2 x3 5 238 23x1 2 x2 1 7x3 5 234 28x1 1 x2 2 2x3 5 220 (a) Solve by Gauss elimination with partial pivoting. Show all steps of the computation. (b) Substitute your results into the original equations to check your answers. 9.12 Use Gauss-Jordan elimination to solve: 2x1 1 x2 2 x3 5 1 5x1 1 2x2 1 2x3 5 24 3x1 1 x2 1 x3 5 5 Do not employ pivoting. Check your answers by substituting them into the original equations.

PROBLEMS 9.13 Solve: x1 1 x2 2 x3 5 23 6x1 1 2x2 1 2x3 5 2 23x1 1 4x2 1 x3 5 1 with (a) naive Gauss elimination, (b) Gauss elimination with partial pivoting, and (c) Gauss-Jordan without partial pivoting. 9.14 Perform the same computation as in Example 9.11, but use five parachutists with the following characteristics:

277 9.17 Develop, debug, and test a program in either a high-level language or macro language of your choice to generate the transpose of a matrix. Test it on the matrices from Prob. 9.3. 9.18 Develop, debug, and test a program in either a high-level language or macro language of your choice to solve a system of equations with Gauss elimination with partial pivoting. Base the program on the pseudocode from Fig. 9.6. Test the program using the following system (which has an answer of x1 5 x2 5 x3 5 1), x1 1 2x2 2 x3 5 2 5x1 1 2x2 1 2x3 5 9

Parachutist 1 2 3 4 5

Mass, kg

Drag Coefﬁcient, kg/s

55 75 60 75 90

10 12 15 16 10

The parachutists have a velocity of 9 m/s. 9.15 Solve c

3 1 2i 2i

4 z1 21i de f 5 e f 1 z2 3

9.16 Develop, debug, and test a program in either a high-level language or macro language of your choice to multiply two matrices— that is, [X] 5 [Y][Z], where [Y] is m by n and [Z] is n by p. Test the program using the matrices from Prob. 9.3.

23x1 1 5x2 2 x3 5 1 9.19 Three masses are suspended vertically by a series of identical springs where mass 1 is at the top and mass 3 is at the bottom. If g 5 9.81 m/s2, m1 5 2 kg, m2 5 3 kg, m3 5 2.5 kg, and the k’s 5 10 kg/s2, solve for the displacements x. 9.20 Develop, debug, and test a program in either a high-level language or macro language of your choice to solve a system of n simultaneous nonlinear equations based on Sec. 9.6. Test the program by solving Prob. 7.12. 9.21 Recall from Sec. 8.2 that determining the chemistry of water exposed to atmospheric CO2 can be determined by solving five nonlinear equations (Eqs. 8.6 through 8.10) for five unknowns: cT, [HCO32], [CO322], [H1], and [OH2]. Employing the parameters from Sec. 8.2 and the program developed in Prob. 9.20, solve this system for conditions in 1958 when the partial pressure of CO2 was 315 ppm. Use your results to compute the pH.

10

CHAPTER

LU Decomposition and Matrix Inversion This chapter deals with a class of elimination methods called LU decomposition techniques. The primary appeal of LU decomposition is that the time-consuming elimination step can be formulated so that it involves only operations on the matrix of coefficients, [A]. Thus, it is well suited for those situations where many right-hand-side vectors {B} must be evaluated for a single value of [A]. Although there are a variety of ways in which this is done, we will focus on showing how the Gauss elimination method can be implemented as an LU decomposition. One motive for introducing LU decomposition is that it provides an efficient means to compute the matrix inverse. The inverse has a number of valuable applications in engineering practice. It also provides a means for evaluating system condition.

10.1

LU DECOMPOSITION As described in Chap. 9, Gauss elimination is designed to solve systems of linear algebraic equations, [A]{X} 5 {B}

(10.1)

Although it certainly represents a sound way to solve such systems, it becomes inefficient when solving equations with the same coefficients [A], but with different right-hand-side constants (the b’s). Recall that Gauss elimination involves two steps: forward elimination and backsubstitution (Fig. 9.3). Of these, the forward-elimination step comprises the bulk of the computational effort (recall Table 9.1). This is particularly true for large systems of equations. LU decomposition methods separate the time-consuming elimination of the matrix [A] from the manipulations of the right-hand side {B}. Thus, once [A] has been “decomposed,” multiple right-hand-side vectors can be evaluated in an efficient manner. Interestingly, Gauss elimination itself can be expressed as an LU decomposition. Before showing how this can be done, let us first provide a mathematical overview of the decomposition strategy. 278

10.1

LU DECOMPOSITION

279

10.1.1 Overview of LU Decomposition Just as was the case with Gauss elimination, LU decomposition requires pivoting to avoid division by zero. However, to simplify the following description, we will defer the issue of pivoting until after the fundamental approach is elaborated. In addition, the following explanation is limited to a set of three simultaneous equations. The results can be directly extended to n-dimensional systems. Equation (10.1) can be rearranged to give [A]{X} 2 {B} 5 0

(10.2)

Suppose that Eq. (10.2) could be expressed as an upper triangular system: u11 £ 0 0

u12 u22 0

u13 x1 d1 u23 § • x2 ¶ 5 • d2 ¶ u33 x3 d3

(10.3)

Recognize that this is similar to the manipulation that occurs in the first step of Gauss elimination. That is, elimination is used to reduce the system to upper triangular form. Equation (10.3) can also be expressed in matrix notation and rearranged to give [U]{X} 2 {D} 5 0

(10.4)

Now, assume that there is a lower diagonal matrix with 1’s on the diagonal, 1 [L] 5 £ l21 l31

0 1 l32

0 0§ 1

(10.5)

that has the property that when Eq. (10.4) is premultiplied by it, Eq. (10.2) is the result. That is, [L]{[U]{X} 2 {D}} 5 [A]{X} 2 {B}

(10.6)

If this equation holds, it follows from the rules for matrix multiplication that [L][U] 5 [A]

(10.7)

[L]{D} 5 {B}

(10.8)

and

A two-step strategy (see Fig. 10.1) for obtaining solutions can be based on Eqs. (10.4), (10.7), and (10.8): 1. 2.

LU decomposition step. [A] is factored or “decomposed” into lower [L] and upper [U] triangular matrices. Substitution step. [L] and [U] are used to determine a solution {X} for a right-handside {B}. This step itself consists of two steps. First, Eq. (10.8) is used to generate an intermediate vector {D} by forward substitution. Then, the result is substituted into Eq. (10.4), which can be solved by back substitution for {X}.

Now, let us show how Gauss elimination can be implemented in this way.

280

LU DECOMPOSITION AND MATRIX INVERSION

X

⫽

B

D

⫽

B

A (a) Decomposition U

L

L

(b) Forward D Substitution U

X ⫽ D (c) Backward X

FIGURE 10.1 The steps in LU decomposition.

10.1.2 LU Decomposition Version of Gauss Elimination Although it might appear at face value to be unrelated to LU decomposition, Gauss elimination can be used to decompose [A] into [L] and [U]. This can be easily seen for [U], which is a direct product of the forward elimination. Recall that the forwardelimination step is intended to reduce the original coefficient matrix [A] to the form a11 [U] 5 £ 0 0

a12 a¿22 0

a13 a¿23 § a–33

(10.9)

which is in the desired upper triangular format. Though it might not be as apparent, the matrix [L] is also produced during the step. This can be readily illustrated for a three-equation system, a11 £ a21 a31

a12 a22 a32

x1 b1 a13 a23 § • x2 ¶ 5 • b2 ¶ a33 x3 b3

The first step in Gauss elimination is to multiply row 1 by the factor [recall Eq. (9.13)] f 21 5

a21 a11

and subtract the result from the second row to eliminate a21. Similarly, row 1 is multiplied by f 31 5

a31 a11

10.1

LU DECOMPOSITION

281

and the result subtracted from the third row to eliminate a31. The final step is to multiply the modified second row by f 32 5

a¿32 a¿22

and subtract the result from the third row to eliminate a¿32. Now suppose that we merely perform all these manipulations on the matrix [A]. Clearly, if we do not want to change the equation, we also have to do the same to the right-hand side {B}. But there is absolutely no reason that we have to perform the manipulations simultaneously. Thus, we could save the f’s and manipulate {B} later. Where do we store the factors f21, f31, and f32? Recall that the whole idea behind the elimination was to create zeros in a21, a31, and a32. Thus, we can store f21 in a21, f31 in a31, and f32 in a32. After elimination, the [A] matrix can therefore be written as a11 £ f21 f31

a12 a¿22 f32

a13 a¿23 § a–33

(10.10)

This matrix, in fact, represents an efficient storage of the LU decomposition of [A], [A] S [L][U]

(10.11)

where a11 [U] 5 £ 0 0

a12 a¿22 0

a13 a¿23 § a–33

1 [L] 5 £ f 21 f 31

0 1 f 32

0 0§ 1

and

The following example confirms that [A] 5 [L][U]. EXAMPLE 10.1

LU Decomposition with Gauss Elimination Problem Statement. Derive an LU decomposition based on the Gauss elimination performed in Example 9.5. Solution. In Example 9.5, we solved the matrix 3 [A] 5 £ 0.1 0.3

20.1 7 20.2

20.2 20.3 § 10

After forward elimination, the following upper triangular matrix was obtained: 3 [U] 5 £ 0 0

20.1 7.00333 0

20.2 20.293333 § 10.0120

282

LU DECOMPOSITION AND MATRIX INVERSION

The factors employed to obtain the upper triangular matrix can be assembled into a lower triangular matrix. The elements a21 and a31 were eliminated by using the factors f 21 5

0.1 0.3 5 0.03333333 f 31 5 5 0.1000000 3 3

and the element a¿32 was eliminated by using the factor f 32 5

20.19 5 20.0271300 7.00333

Thus, the lower triangular matrix is 1 [L] 5 £ 0.0333333 0.100000

0 1 20.0271300

0 0§ 1

Consequently, the LU decomposition is 1 [A] 5 [L][U] 5 £ 0.0333333 0.100000

0 1 20.0271300

0 3 0§ £0 1 0

20.1 7.00333 0

20.2 2 0.293333 § 10.0120

This result can be verified by performing the multiplication of [L][U] to give 3 [L][U] 5 £ 0.0999999 0.3

20.1 7 20.2

20.2 20.3 § 9.99996

where the minor discrepancies are due to round-off.

The following is pseudocode for a subroutine to implement the decomposition phase: SUB Decompose (a, n) DOFOR k 5 1, n 2 1 DOFOR i 5 k 1 1, n factor 5 ai,k/ak,k ai,k 5 factor DOFOR j 5 k 1 1, n ai,j 5 ai,j 2 factor * ak,j END DO END DO END DO END Decompose

Notice that this algorithm is “naive” in the sense that pivoting is not included. This feature will be added later when we develop the full algorithm for LU decomposition. After the matrix is decomposed, a solution can be generated for a particular righthand-side vector {B}. This is done in two steps. First, a forward-substitution step is executed by solving Eq. (10.8) for {D}. It is important to recognize that this merely

10.1

LU DECOMPOSITION

283

amounts to performing the elimination manipulations on {B}. Thus, at the end of this step, the right-hand side will be in the same state that it would have been had we performed forward manipulation on [A] and {B} simultaneously. The forward-substitution step can be represented concisely as i21

di 5 bi 2 a ai j dj for i 5 2, 3, p , n

(10.12)

j51

The second step then merely amounts to implementing back substitution, as in Eq. (10.4). Again, it is important to recognize that this is identical to the back-substitution phase of conventional Gauss elimination. Thus, in a fashion similar to Eqs. (9.16) and (9.17), the back-substitution step can be represented concisely as xn 5 dn yann

(10.13) n

di 2 a ai j xj xi 5

EXAMPLE 10.2

j5i11

aii

for i 5 n 2 1, n 2 2, p , 1

(10.14)

The Substitution Steps Problem Statement. Complete the problem initiated in Example 10.1 by generating the final solution with forward and back substitution. Solution. As stated above, the intent of forward substitution is to impose the elimination manipulations, that we had formerly applied to [A], on the right-hand-side vector {B}. Recall that the system being solved in Example 9.5 was 3 £ 0.1 0.3

20.1 7 20.2

20.2 x1 7.85 20.3 § • x2 ¶ 5 • 219.3 ¶ 10 x3 71.4

and that the forward-elimination phase of conventional Gauss elimination resulted in 3 £0 0

20.1 7.00333 0

20.2 x1 7.85 20.293333 § • x2 ¶ 5 • 219.5617 ¶ x3 10.0120 70.0843

(E10.2.1)

The forward-substitution phase is implemented by applying Eq. (10.7) to our problem, 1 £ 0.0333333 0.100000

0 1 20.0271300

0 d1 7.85 0 § • d2 ¶ 5 • 219.3 ¶ 1 d3 71.4

or multiplying out the left-hand side, d1 5 7.85 0.0333333d1 1 d2 5 219.3 0.1d1 2 0.02713d2 1 d3 5 71.4

284

LU DECOMPOSITION AND MATRIX INVERSION

We can solve the first equation for d1, d1 5 7.85 which can be substituted into the second equation to solve for d2 5 219.3 2 0.0333333(7.85) 5 219.5617 Both d1 and d2 can be substituted into the third equation to give d3 5 71.4 2 0.1(7.85) 1 0.02713(219.5617) 5 70.0843 Thus, 7.85 {D} 5 • 219.5617 ¶ 70.0843 which is identical to the right-hand side of Eq. (E10.2.1). This result can then be substituted into Eq. (10.4), [U]{X} 5 {D}, to give 3 £0 0

20.1 7.00333 0

20.2 x1 7.85 20.293333 § • x2 ¶ 5 • 219.5617 ¶ 10.0120 x3 70.0843

which can be solved by back substitution (see Example 9.5 for details) for the final solution, 3 {X} 5 • 22.5 ¶ 7.00003 The following is pseudocode for a subroutine to implement both substitution phases: SUB Substitute (a, n, b, x) 'forward substitution DOFOR i 5 2, n sum 5 bi DOFOR j 5 1, i 2 1 sum 5 sum 2 ai,j * bj END DO bi 5 sum END DO 'back substitution xn 5 bn/an,n DOFOR i 5 n 2 1, 1, 21 sum 5 0 DOFOR j 5 i 1 1, n sum 5 sum 1 ai,j * xj END DO xi 5 (bi 2 sum)/ai,i END DO END Substitute

10.1

LU DECOMPOSITION

285

The LU decomposition algorithm requires the same total multiply/divide flops as for Gauss elimination. The only difference is that a little less effort is expended in the decomposition phase since the operations are not applied to the right-hand side. Thus, the number of multiply/divide flops involved in the decomposition phase can be calculated as n3 n as n increases n3 2 ————S 1 O(n) 3 3 3

(10.15)

Conversely, the substitution phase takes a little more effort. Thus, the number of flops for forward and back substitution is n2. The total effort is therefore identical to Gauss elimination 3 n3 n as n increases n 2 1 n2 ————S 1 O ( n2 ) 3 3 3

(10.16)

10.1.3 LU Decomposition Algorithm An algorithm to implement an LU decomposition expression of Gauss elimination is listed in Fig. 10.2. Four features of this algorithm bear mention: The factors generated during the elimination phase are stored in the lower part of the matrix. This can be done because these are converted to zeros anyway and are unnecessary for the final solution. This storage saves space. This algorithm keeps track of pivoting by using an order vector o. This greatly speeds up the algorithm because only the order vector (as opposed to the whole row) is pivoted. The equations are not scaled, but scaled values of the elements are used to determine whether pivoting is to be implemented. The diagonal term is monitored during the pivoting phase to detect near-zero occurrences in order to flag singular systems. If it passes back a value of er 5 21, a singular matrix has been detected and the computation should be terminated. A parameter tol is set by the user to a small number in order to detect near-zero occurrences. 10.1.4 Crout Decomposition Notice that for the LU decomposition implementation of Gauss elimination, the [L] matrix has 1’s on the diagonal. This is formally referred to as a Doolittle decomposition, or factorization. An alternative approach involves a [U] matrix with 1’s on the diagonal. This is called Crout decomposition. Although there are some differences between the approaches (Atkinson, 1978; Ralston and Rabinowitz, 1978), their performance is comparable. The Crout decomposition approach generates [U] and [L] by sweeping through the matrix by columns and rows, as depicted in Fig. 10.3. It can be implemented by the following concise series of formulas: li, 1 5 ai, 1 for i 5 1, 2, p , n a1j u1j 5 for j 5 2, 3, p , n l11

(10.17) (10.18)

286

LU DECOMPOSITION AND MATRIX INVERSION SUB Ludecomp (a, b, n, tol, x, er) DIM on, sn er 5 0 CALL Decompose(a, n, tol, o, s, er) IF er ,. 21 THEN CALL Substitute(a, o, n, b, x) END IF END Ludecomp

(a)

(b)

(c)

SUB Decompose (a, n, tol, o, s, er) DOFOR i 5 1, n oi 5 i si 5 ABS(ai,1) DOFOR j 5 2, n IF ABS(ai,j).si THEN si 5 ABS(ai,j) END DO END DO DOFOR k 5 1, n 2 1 CALL Pivot(a, o, s, n, k) IF ABS(ao(k),kyso(k)) , tol THEN er 5 21 PRINT ao(k),kyso(k) EXIT DO END IF DOFOR i 5 k 1 1, n factor 5 ao(i),kyao(k),k ao(i),k 5 factor DOFOR j 5 k 1 1, n ao(i),j 5 ao(i),j 2 factor * ao(k),j END DO END DO END DO IF ABS(ao(k),kyso(k)) , tol THEN er 5 21 PRINT ao(k),kyso(k)

END IF END Decompose SUB Pivot (a, o, s, n, k) p5 k big 5 ABS(ao(k),kyso(k)) DOFOR ii 5 k 1 1, n dummy 5 ABS(ao(ii),kyso(ii)) IF dummy . big THEN big 5 dummy p 5 ii END IF END DO dummy 5 op op 5 ok ok 5 dummy END Pivot SUB Substitute (a, o, n, b, x) DOFOR i 5 2, n sum 5 bo(i) DOFOR j 5 1, i 2 1 sum 5 sum 2 ao(i),j * bo(j) END DO bo(i) 5 sum END DO xn 5 bo(n)yao(n),n DOFOR i 5 n 2 1, 1, 21 sum 5 0 DOFOR j 5 i 1 1, n sum 5 sum 1 ao(i),j * xj END DO xi 5 (bo(i) 2 sum)yao(i),i END DO END Substitute

FIGURE 10.2 Pseudocode for an LU decomposition algorithm.

(d)

For j 5 2, 3, . . . , n 2 1 j21

FIGURE 10.3 A schematic depicting the evaluations involved in Crout LU decomposition.

li j 5 ai j 2 a likukj for i 5 j, j 1 1, p , n

(10.19)

k51

j21

ajk 2 a ljiuik

ujk 5

i51

ljj

for k 5 j 1 1, j 1 2, p , n

(10.20)

10.2

FIGURE 10.4 Pseudocode for Crout’s LU decomposition algorithm.

THE MATRIX INVERSE

287

DOFOR j 5 2, n a1,j 5 a1,jya1,1 END DO DOFOR j 5 2, n 2 1 DOFOR i 5 j, n sum 5 0 DOFOR k 5 1, j 2 1 sum 5 sum 1 ai,k ? ak,j END DO ai,j 5 ai,j 2 sum END DO DOFOR k 5 j 1 1, n sum 5 0 DOFOR i 5 1, j 2 1 sum 5 sum 1 aj,i ? ai,k END DO aj,k 5 (aj,k 2 sum)yaj,j END DO END DO sum 5 0 DOFOR k 5 1, n 2 1 sum 5 sum 1 an,k ? ak,n END DO an,n 5 an,n 2 sum

and n21

lnn 5 ann 2 a lnkukn

(10.21)

k51

Aside from the fact that it consists of a few concise loops, the foregoing approach also has the benefit that storage space can be economized. There is no need to store the 1’s on the diagonal of [U] or the 0’s for [L] or [U] because they are givens in the method. Consequently, the values of [U] can be stored in the zero space of [L]. Further, close examination of the foregoing derivation makes it clear that after each element of [A] is employed once, it is never used again. Therefore, as each element of [L] and [U] is computed, it can be substituted for the corresponding element (as designated by its subscripts) of [A]. Pseudocode to accomplish this is presented in Fig. 10.4. Notice that Eq. (10.17) is not included in the pseudocode because the first column of [L] is already stored in [A]. Otherwise, the algorithm directly follows from Eqs. (10.18) through (10.21).

10.2

THE MATRIX INVERSE In our discussion of matrix operations (Sec. PT3.2.2), we introduced the notion that if a matrix [A] is square, there is another matrix, [A]21, called the inverse of [A], for which [Eq. (PT3.3)] [A][A21 ] 5 [A] 21 [A] 5 [I]

288

LU DECOMPOSITION AND MATRIX INVERSION

Now we will focus on how the inverse can be computed numerically. Then we will explore how it can be used for engineering analysis. 10.2.1 Calculating the Inverse The inverse can be computed in a column-by-column fashion by generating solutions with unit vectors as the right-hand-side constants. For example, if the right-hand-side constant has a 1 in the first position and zeros elsewhere, 1 {b} 5 • 0 ¶ 0 the resulting solution will be the first column of the matrix inverse. Similarly, if a unit vector with a 1 at the second row is used 0 {b} 5 • 1 ¶ 0 the result will be the second column of the matrix inverse. The best way to implement such a calculation is with the LU decomposition algorithm described at the beginning of this chapter. Recall that one of the great strengths of LU decomposition is that it provides a very efficient means to evaluate multiple righthand-side vectors. Thus, it is ideal for evaluating the multiple unit vectors needed to compute the inverse. EXAMPLE 10.3

Matrix Inversion Problem Statement. Employ LU decomposition to determine the matrix inverse for the system from Example 10.2. 3 [A] 5 £ 0.1 0.3

20.1 7 20.2

20.2 20.3 § 10

Recall that the decomposition resulted in the following lower and upper triangular matrices: 3 [U] 5 £ 0 0

20.1 7.00333 0

20.2 1 20.293333 § [L] 5 £ 0.0333333 10.0120 0.100000

0 1 20.0271300

0 0§ 1

Solution. The first column of the matrix inverse can be determined by performing the forward-substitution solution procedure with a unit vector (with 1 in the first row) as the right-hand-side vector. Thus, Eq. (10.8), the lower-triangular system, can be set up as 1 £ 0.0333333 0.100000

0 1 20.0271300

0 d1 1 0 § • d2 ¶ 5 • 0 ¶ 1 d3 0

10.2

THE MATRIX INVERSE

289

and solved with forward substitution for {D} T 5 :1 20.03333 can then be used as the right-hand side of Eq. (10.3), 3 £0 0

20.1 7.00333 0

20.1009;. This vector

20.2 x1 1 20.293333 § • x2 ¶ 5 • 20.03333 ¶ 10.0120 x3 20.1009

which can be solved by back substitution for {X} T 5 :0.33249 which is the first column of the matrix, 0.33249 [A] 21 5 £ 20.00518 20.01008

0 0 0

20.00518

20.01008;,

0 0§ 0

To determine the second column, Eq. (10.8) is formulated as 1 £ 0.0333333 0.100000

0 1 20.0271300

0 d1 0 0 § • d2 ¶ 5 • 1 ¶ 1 d3 0

This can be solved for {D}, and the results are used with Eq. (10.3) to determine {X} T 5 :0.004944 0.142903 0.00271;, which is the second column of the matrix, 0.33249 [A] 21 5 £ 20.00518 20.01008

0.004944 0.142903 0.00271

0 0§ 0

Finally, the forward- and back-substitution procedures can be implemented with {B} T 5 :0 0 1; to solve for {X} T 5 :0.006798 0.004183 0.09988;, which is the final column of the matrix, 0.33249 [A] 21 5 £ 20.00518 20.01008

0.004944 0.142903 0.00271

0.006798 0.004183 § 0.09988

The validity of this result can be checked by verifying that [A][A]21 5 [I].

Pseudocode to generate the matrix inverse is shown in Fig. 10.5. Notice how the decomposition subroutine from Fig. 10.2 is called to perform the decomposition and then generates the inverse by repeatedly calling the substitution algorithm with unit vectors. The effort required for this algorithm is simply computed as n3 n 4n3 n 2 1 n(n2 ) 5 2 3 3 3 4 decomposition 1 n 3 substitutions

(10.22)

where from Sec. 10.1.2, the decomposition is defined by Eq. (10.15) and the effort involved with every right-hand-side evaluation involves n2 multiply/divide flops.

290

LU DECOMPOSITION AND MATRIX INVERSION CALL Decompose (a, n, tol, o, s, er) IF er 5 0 THEN DOFOR i 5 1, n DOFOR j 5 1, n IF i 5 j THEN b(j) 5 1 ELSE b(j) 5 0 END IF END DO CALL Substitute (a, o, n, b, x) DOFOR j 5 1, n ai(j, i) 5 x(j) END DO END DO Output ai, if desired ELSE PRINT "ill-conditioned system" END IF

FIGURE 10.5 Driver program that uses some of the subprograms from Fig. 10.2 to generate a matrix inverse.

10.2.2 Stimulus-Response Computations As discussed in Sec. PT3.1.2, many of the linear systems of equations confronted in engineering practice are derived from conservation laws. The mathematical expression of these laws is some form of balance equation to ensure that a particular property—mass, force, heat, momentum, or other—is conserved. For a force balance on a structure, the properties might be horizontal or vertical components of the forces acting on each node of the structure (see Sec. 12.2). For a mass balance, the properties might be the mass in each reactor of a chemical process (see Sec. 12.1). Other fields of engineering would yield similar examples. A single balance equation can be written for each part of the system, resulting in a set of equations defining the behavior of the property for the entire system. These equations are interrelated, or coupled, in that each equation may include one or more of the variables from the other equations. For many cases, these systems are linear and, therefore, of the exact form dealt with in this chapter: [A]{X} 5 {B}

(10.23)

Now, for balance equations, the terms of Eq. (10.23) have a definite physical interpretation. For example, the elements of {X} are the levels of the property being balanced for each part of the system. In a force balance of a structure, they represent the horizontal and vertical forces in each member. For the mass balance, they are the mass of chemical in each reactor. In either case, they represent the system’s state or response, which we are trying to determine. The right-hand-side vector {B} contains those elements of the balance that are independent of behavior of the system—that is, they are constants. As such, they often represent the external forces or stimuli that drive the system.

10.3

ERROR ANALYSIS AND SYSTEM CONDITION

291

Finally, the matrix of coefficients [A] usually contains the parameters that express how the parts of the system interact or are coupled. Consequently, Eq. (10.23) might be reexpressed as [Interactions]{response} 5 {stimuli} Thus, Eq. (10.23) can be seen as an expression of the fundamental mathematical model that we formulated previously as a single equation in Chap. 1 [recall Eq. (1.1)]. We can now see that Eq. (10.23) represents a version that is designed for coupled systems involving several dependent variables {X}. As we know from this chapter and Chap. 9, there are a variety of ways to solve Eq. (10.23). However, using the matrix inverse yields a particularly interesting result. The formal solution can be expressed as {X} 5 [A] 21 {B} or (recalling our definition of matrix multiplication from Box PT3.2) 21 21 x1 5 a21 11 b1 1 a12 b2 1 a13 b3 21 21 x2 5 a21 21 b1 1 a22 b2 1 a23 b3 21 21 x3 5 a21 31 b1 1 a32 b2 1 a33 b3

Thus, we find that the inverted matrix itself, aside from providing a solution, has extremely useful properties. That is, each of its elements represents the response of a single part of the system to a unit stimulus of any other part of the system. Notice that these formulations are linear and, therefore, superposition and proportionality hold. Superposition means that if a system is subject to several different stimuli (the b’s), the responses can be computed individually and the results summed to obtain a total response. Proportionality means that multiplying the stimuli by a quantity results in the response to those stimuli being multiplied by the same quantity. Thus, the coefficient a21 11 is a proportionality constant that gives the value of x1 due to a unit level of b1. This result is independent of the effects of b2 and b3 on x1, which are reflected in the 21 coefficients a21 12 and a13 , respectively. Therefore, we can draw the general conclusion 21 that the element aij of the inverted matrix represents the value of xi due to a unit quantity of bj. Using the example of the structure, element a21 ij of the matrix inverse would represent the force in member i due to a unit external force at node j. Even for small systems, such behavior of individual stimulus-response interactions would not be intuitively obvious. As such, the matrix inverse provides a powerful technique for understanding the interrelationships of component parts of complicated systems. This power will be demonstrated in Secs. 12.1 and 12.2.

10.3

ERROR ANALYSIS AND SYSTEM CONDITION Aside from its engineering applications, the inverse also provides a means to discern whether systems are ill-conditioned. Three methods are available for this purpose: 1. Scale the matrix of coefficients [A] so that the largest element in each row is 1. Invert the scaled matrix and if there are elements of [A]21 that are several orders of magnitude greater than one, it is likely that the system is ill-conditioned (see Box 10.1).

292

LU DECOMPOSITION AND MATRIX INVERSION

Box 10.1

Interpreting the Elements of the Matrix Inverse as a Measure of Ill-Conditioning

One method for assessing a system’s condition is to scale [A] so that the largest element in each row is 1 and then compute [A]21. If elements of [A]21 are several orders of magnitude greater than the elements of the original scaled matrix, it is likely that the system is ill-conditioned. Insight into this approach can be gained by recalling that a way to check whether an approximate solution {X} is acceptable is to substitute it into the original equations and see whether the original right-hand-side constants result. This is equivalent to {R} 5 {B} 2 [A]{X˜ }

(B10.1.1)

where {R} is the residual between the right-hand-side constants and the values computed with the solution {X˜ }. If {R} is small, we might conclude that the {X˜ } values are adequate. However, suppose that {X} is the exact solution that yields a zero residual, as in {0} 5 {B} 2 [A] {X}

(B10.1.2)

Subtracting Eq. (B10.1.2) from (B10.1.1) yields {R} 5 [A] {X} 2 {X˜ } Multiplying both sides of this equation by [A]21 gives {X} 2 {X˜ } 5 [A] 21 {R} This result indicates why checking a solution by substitution can be misleading. For cases where elements of [A]21 are large, a small discrepancy in the right-hand-side residual {R} could correspond to a large error {X} 2 {X˜ } in the calculated value of the unknowns. In other words, a small residual does not guarantee an accurate solution. However, we can conclude that if the largest element of [A]21 is on the order of magnitude of unity, the system can be considered to be well-conditioned. Conversely, if [A]21 includes elements much larger than unity, we conclude that the system is ill-conditioned.

2. Multiply the inverse by the original coefficient matrix and assess whether the result is close to the identity matrix. If not, it indicates ill-conditioning. 3. Invert the inverted matrix and assess whether the result is sufficiently close to the original coefficient matrix. If not, it again indicates that the system is ill-conditioned. Although these methods can indicate ill-conditioning, it would be preferable to obtain a single number (such as the condition number from Sec. 4.2.3) that could serve as an indicator of the problem. Attempts to formulate such a matrix condition number are based on the mathematical concept of the norm. 10.3.1 Vector and Matrix Norms A norm is a real-valued function that provides a measure of the size or “length” of multicomponent mathematical entities such as vectors and matrices (see Box 10.2). A simple example is a vector in three-dimensional Euclidean space (Fig. 10.6) that can be represented as :F; 5 :a

b

c;

where a, b, and c are the distances along the x, y, and z axes, respectively. The length of this vector—that is, the distance from the coordinate (0, 0, 0) to (a, b, c)—can be simply computed as BFBe 5 2a2 1 b2 1 c2 where the nomenclature BFBe indicates that this length is referred to as the Euclidean norm of [F].

10.3

Box 10.2

ERROR ANALYSIS AND SYSTEM CONDITION

Matrix Norms which defines the norm as the element with the largest absolute value. Using a similar approach, norms can be developed for matrices. For example,

As developed in this section, Euclidean norms can be employed to quantify the size of a vector, n

B X Be 5

293

2

a xi B i51

n

B A B1 5 max a Z aij Z 1#j#n

or matrix,

i51

n

B A Be 5

n 2

That is, a summation of the absolute values of the coefficients is performed for each column, and the largest of these summations is taken as the norm. This is called the column-sum norm. A similar determination can be made for the rows, resulting in a uniform-matrix or row-sum norm,

a a ai, j B i51 j51

For vectors, there are alternatives called p norms that can be represented generally by n

1yp

B X B p 5 a a Z xi Z p b

n

B A Bq 5 max a Z aij Z 1#i#n

i51

j51

We can also see that the Euclidean norm and the 2 norm, B XB2 , are identical for vectors. Other important examples are

It should be noted that, in contrast to vectors, the 2 norm and the Euclidean norm for a matrix are not the same. Whereas the Euclidean norm B ABe can be easily determined by Eq. (10.24), the matrix 2 norm B AB2 is calculated as

n

B X B 1 5 a Z xi Z i51

B AB2 5 (mmax ) 1y2

which represents the norm as the sum of the absolute values of the elements. Another is the maximum-magnitude or uniform-vector norm.

where mmax is the largest eigenvalue of [A]T [A]. In Chap. 27, we will learn more about eigenvalues. For the time being, the important point is that the B AB2 , or spectral, norm is the minimum norm and, therefore, provides the tightest measure of size (Ortega 1972).

B X Bq 5 max Z xi Z 1#i#n

FIGURE 10.6 Graphical depiction of a vector :F ; 5 :a b c; in Euclidean space.

y b 2

2

2

a 储F

⫹

b

⫹

c

= 储e

a x c z

294

LU DECOMPOSITION AND MATRIX INVERSION

Similarly, for an n-dimensional vector :X; 5 :x1 x2 p xn ; , a Euclidean norm would be computed as n

B X Be 5

2

a xi B i51

The concept can be extended further to a matrix [A], as in n

B A Be 5

n 2

a a ai, j B i51 j51

(10.24)

which is given a special name—the Frobenius norm. However, as with the other vector norms, it provides a single value to quantify the “size” of [A]. It should be noted that there are alternatives to the Euclidean and Frobenius norms (see Box 10.2). For example, a uniform vector norm is defined as B X Bq 5 max Zxi Z 1#i#n

That is, the element with the largest absolute value is taken as the measure of the vector’s size. Similarly, a uniform matrix norm or row-sum norm is defined as n

B ABq 5 max a Zaij Z 1#i#n

(10.25)

j51

In this case, the sum of the absolute value of the elements is computed for each row, and the largest of these is taken as the norm. Although there are theoretical benefits for using certain of the norms, the choice is sometimes influenced by practical considerations. For example, the uniform-row norm is widely used because of the ease with which it can be calculated and the fact that it usually provides an adequate measure of matrix size. 10.3.2 Matrix Condition Number Now that we have introduced the concept of the norm, we can use it to define Cond [A] 5 B AB # B A21B

(10.26)

where Cond [A] is called the matrix condition number. Note that for a matrix [A], this number will be greater than or equal to 1. It can be shown (Ralston and Rabinowitz, 1978; Gerald and Wheatley, 2004) that B¢X B BXB

# Cond [A]

B¢AB BAB

That is, the relative error of the norm of the computed solution can be as large as the relative error of the norm of the coefficients of [A] multiplied by the condition number. For example, if the coefficients of [A] are known to t-digit precision (that is, rounding errors are on the order of 102t) and Cond [A] 5 10c, the solution [X] may be valid to only t 2 c digits (rounding errors ,10c2t).

10.3

EXAMPLE 10.4

ERROR ANALYSIS AND SYSTEM CONDITION

295

Matrix Condition Evaluation Problem Statement. The Hilbert matrix, which is notoriously ill-conditioned, can be represented generally as 1 1y2 . F . . 1yn

1y2 1y3 . . . 1y(n 1 1)

1y3 1y4 . . . 1y(n 1 2)

p p

p

1yn 1y(n 1 1) . V . . 1y(2n 2 1)

Use the row-sum norm to estimate the matrix condition number for the 3 3 3 Hilbert matrix, 1 [A] 5 £ 1y2 1y3

1y2 1y3 1y4

1y3 1y4 § 1y5

Solution. First, the matrix can be normalized so that the maximum element in each row is 1, 1 [A] 5 £ 1 1

1y2 2y3 3y4

1y3 1y2 § 3y5

Summing each of the rows gives 1.833, 2.1667, and 2.35. Thus, the third row has the largest sum and the row-sum norm is B ABq 5 1 1

3 3 1 5 2.35 4 5

The inverse of the scaled matrix can be computed as [A]

21

9 5 £ 236 30

218 96 290

10 260 § 60

Note that the elements of this matrix are larger than the original matrix. This is also reflected in its row-sum norm, which is computed as BA21Bq 5 Z236Z 1 Z96Z 1 Z260Z 5 192 Thus, the condition number can be calculated as Cond [A] 5 2.35(192) 5 451.2 The fact that the condition number is considerably greater than unity suggests that the system is ill-conditioned. The extent of the ill-conditioning can be quantified by calculating c 5 log 451.2 5 2.65. Computers using IEEE floating-point representation

296

LU DECOMPOSITION AND MATRIX INVERSION

have approximately t 5 log 2224 5 7.2 significant base-10 digits (recall Sec. 3.4.1). Therefore, the solution could exhibit rounding errors of up to 10(2.65-7.2) 5 3 3 1025. Note that such estimates almost always overpredict the actual error. However, they are useful in alerting you to the possibility that round-off errors may be significant.

Practically speaking, the problem with implementing Eq. (10.26) is the computational price required to obtain B A21B. Rice (1983) outlines some possible strategies to mitigate this problem. Further, he suggests an alternative way to assess system condition: run the same solution on two different compilers. Because the resulting codes will likely implement the arithmetic differently, the effect of ill-conditioning should be evident from such an experiment. Finally, it should be mentioned that software packages such as MATLAB software and Mathcad have the capability to conveniently compute matrix condition. We will review these capabilities when we review such packages at the end of Chap. 11. 10.3.3 Iterative Reﬁnement In some cases, round-off errors can be reduced by the following procedure. Suppose that we are solving the following set of equations: a11x1 1 a12x2 1 a13x3 5 b1 a21x1 1 a22x2 1 a23x3 5 b2

(10.27)

a31x1 1 a32x2 1 a33x3 5 b3 For conciseness, we will limit the following discussion to this small (3 3 3) system. However, the approach is generally applicable to larger sets of linear equations. Suppose an approximate solution vector is given by {X˜ } T 5 :x˜ 1 x˜ 2 x˜ 3 ; . This solution can be substituted into Eq. (10.27) to give a11x˜ 1 1 a12x˜ 2 1 a13x˜ 3 5 b˜ 1 a21x˜ 1 1 a22x˜ 2 1 a23x˜ 3 5 b˜ 2

(10.28)

a31x˜ 1 1 a32x˜ 2 1 a33x˜ 3 5 b˜ 3 Now, suppose that the exact solution {X} is expressed as a function of the approximate solution and a vector of correction factors {DX}, where x1 5 x˜ 1 1 ¢x1 x2 5 x˜ 2 1 ¢x2

(10.29)

x3 5 x˜ 3 1 ¢x3 If these results are substituted into Eq. (10.27), the following system results: a11 (x˜ 1 1 ¢x1 ) 1 a12 (x˜ 2 1 ¢x2 ) 1 a13 (x˜ 3 1 ¢x3 ) 5 b1 a21 (x˜ 1 1 ¢x1 ) 1 a22 (x˜ 2 1 ¢x2 ) 1 a23 (x˜ 3 1 ¢x3 ) 5 b2 a31 (x˜ 1 1 ¢x1 ) 1 a32 (x˜ 2 1 ¢x2 ) 1 a33 (x˜ 3 1 ¢x3 ) 5 b3

(10.30)

PROBLEMS

297

Now, Eq. (10.28) can be subtracted from Eq. (10.30) to yield a11 ¢x1 1 a12 ¢x2 1 a13 ¢x3 5 b1 2 b˜ 1 5 E1 a21 ¢x1 1 a22 ¢x2 1 a23 ¢x3 5 b2 2 b˜ 2 5 E2 a31 ¢x1 1 a32 ¢x2 1 a33 ¢x3 5 b3 2 b˜ 3 5 E3

(10.31)

This system itself is a set of simultaneous linear equations that can be solved to obtain the correction factors. The factors can then be applied to improve the solution, as specified by Eq. (10.29). It is relatively straightforward to integrate an iterative refinement procedure into computer programs for elimination methods. It is especially effective for the LU decomposition approaches described earlier, which are designed to evaluate different right-hand-side vectors efficiently. Note that to be effective for correcting ill-conditioned systems, the E’s in Eq. (10.31) must be expressed in double precision.

PROBLEMS 10.1 Use the rules of matrix multiplication to prove that Eqs. (10.7) and (10.8) follow from Eq. (10.6). 10.2 (a) Use naive Gauss elimination to decompose the following system according to the description in Sec. 10.1.2. 10x1 1 2x2 2 x3 5 27 23x1 2 6x2 1 2x3 5 261.5 x1 1 x2 1 5x3 5 221.5 Then, multiply the resulting [L] and [U] matrices to determine that [A] is produced. (b) Use LU decomposition to solve the system. Show all the steps in the computation. (c) Also solve the system for an alternative right-hand-side vector: {B} T 5 :12 18 26 ; . 10.3 (a) Solve the following system of equations by LU decomposition without pivoting 8x1 1 4x2 2 x3 5 11 22x1 1 5x2 1 x3 5 4 2x1 2 x2 1 6x3 5 7 (b) Determine the matrix inverse. Check your results by verifying that [A][A]21 5 [I]. 10.4 Solve the following system of equations using LU decomposition with partial pivoting: 2x1 2 6x2 2 x3 5 238 23x1 2 x2 1 7x3 5 234 28x1 1 x2 2 2x3 5 220 10.5 Determine the total flops as a function of the number of equations n for the (a) decomposition, (b) forward-substitution, and (c) back-substitution phases of the LU decomposition version of Gauss elimination.

10.6 Use LU decomposition to determine the matrix inverse for the following system. Do not use a pivoting strategy, and check your results by verifying that [A][A]21 5 [I]. 10x1 1 2x2 2 x3 5 27 23x1 2 6x2 1 2x3 5 261.5 x1 1 x2 1 5x3 5 221.5 10.7 Perform Crout decomposition on 2x1 2 5x2 1 x3 5 12 2x1 1 3x2 2 x3 5 28 3x1 2 4x2 1 2x3 5 16 Then, multiply the resulting [L] and [U] matrices to determine that [A] is produced. 10.8 The following system of equations is designed to determine concentrations (the c’s in gym3) in a series of coupled reactors as a function of the amount of mass input to each reactor (the right-hand sides in gyday), 15c1 2 3c2 2 c3 5 3800 23c1 1 18c2 2 6c3 5 1200 24c1 2 c2 1 12c3 5 2350 (a) Determine the matrix inverse. (b) Use the inverse to determine the solution. (c) Determine how much the rate of mass input to reactor 3 must be increased to induce a 10 g/m3 rise in the concentration of reactor 1. (d) How much will the concentration in reactor 3 be reduced if the rate of mass input to reactors 1 and 2 is reduced by 500 and 250 g/day, respectively?

298

LU DECOMPOSITION AND MATRIX INVERSION

10.9 Solve the following set of equations with LU decomposition: 3x1 2 2x2 1 x3 5 210 2x1 1 6x2 2 4x3 5 44 2x1 2 2x2 1 5x3 5 226 10.10 (a) Determine the LU decomposition without pivoting by hand for the following matrix and check your results by validating that [L][U] 5 [A]. 8 £3 2

2 7 3

x21 £ x22 x23

1 2§ 9

(b) Employ the result of (a) to compute the determinant. (c) Repeat (a) and (b) using MATLAB. 10.11 Use the following LU decomposition to (a) compute the determinant and (b) solve [A]{x} 5 {b} with {b}T 5 :210 44 226=. 3 22 1 1 [A] 5 [L] [U] 5 £ 0.6667 1 § £ 7.3333 24.6667 § 3.6364 20.3333 20.3636 1 10.12 Determine B A Be, B A B1 , and B ABq for 8 [A] 5 £ 29 15

2 1 21

210 3§ 6

0.125 0.015625 [A] 5 ≥ 0.00463 0.001953

0.25 0.625 0.02777 0.015625

9 16 25 36 49

16 25 36 49 64

25 36 49U 64 81

1 1§ 1

(a) Determine the condition number based on the row-sum norm for the case where x1 5 4, x2 5 2, and x3 5 7. (b) Use MATLAB or Mathcad software to compute the spectral and Frobenius condition numbers. 10.18 Develop a user-friendly program for LU decomposition based on the pseudocode from Fig. 10.2. 10.19 Develop a user-friendly program for LU decomposition, including the capability to evaluate the matrix inverse. Base the program on Figs. 10.2 and 10.5. 10.20 Use iterative refinement techniques to improve x1 5 2, x2 5 23, and x3 5 8, which are approximate solutions of

5x1 1 2x2 1 x3 5 12

0.5 0.25 0.16667 0.125

1 1 ¥ 1 1

Using the column-sum norm, compute the condition number and how many suspect digits would be generated by this matrix. 10.15 (a) Determine the condition number for the following system using the row-sum norm. Do not normalize the system. 4 9 16 25 36

x1 x2 x3

2x1 1 5x2 1 x3 5 25

Scale the matrix by making the maximum element in each row equal to one. 10.13 Determine the Frobenius and the row-sum norms for the systems in Probs. 10.3 and 10.4. Scale the matrices by making the maximum element in each row equal to one. 10.14 A matrix [A] is defined as

1 4 E9 16 25

How many digits of precision will be lost due to ill-conditioning? (b) Repeat (a), but scale the matrix by making the maximum element in each row equal to one. 10.16 Determine the condition number based on the row-sum norm for the normalized 5 3 5 Hilbert matrix. How many significant digits of precision will be lost due to ill-conditioning? 10.17 Besides the Hilbert matrix, there are other matrices that are inherently ill-conditioned. One such case is the Vandermonde matrix, which has the following form:

x1 1 2x2 1 x3 5 3 10.21 Consider vectors: S

S

S

S

S

S

S

S

A 5 2i 2 3 j 1 a k S

S

B 5 bi 1 j 2 4 k S

S

C 5 3i 1 c j 1 2 k S

S

S

Vector S A S is perpendicular to B as well as to C . It is also known that B # C 5 2. Use any method studied in this chapter to solve for the three unknowns, a, b, and c. 10.22 Consider the following vectors: S

S

S

S

A 5 ai 1 b j 1 ck

S

S

S

S

B 5 22 i 1 j 2 4 k

S

S

S

S

C 5 i 1 3 j 1 2k S

where A is an unknown vector. If S

S

S

S

S

S

S

( A 3 B ) 1 (A 3 C ) 5 (5a 1 6) i 1 (3b 2 2) j 1 (24c 1 1) k

use any method learned in this chapter to solve for the three unknowns, a, b, and c.

PROBLEMS 10.23 Let the function be defined on the interval [0, 2] as follows: f (x) 5 e

ax 1 b, cx 1 d,

0#x#1 f 1#x#2

Determine the constants a, b, c, and d so that the function f satisfies the following: f(0) 5 f(2) 5 1. f is continuous on the entire interval. a 1 b 5 4. Derive and solve a system of linear algebraic equations with a matrix form identical to Eq. (10.1). 10.24 (a) Create a 3 3 3 Hilbert matrix. This will be your matrix [A]. Multiply the matrix by the column vector {x} 5 [1, 1, 1]T. The solution of [A]{x} will be another column vector {b}. Using any numerical package and Gauss elimination, find the solution to [A]{x} 5 {b}using the Hilbert matrix and the vector {b} that you calculated. Compare the result to your known {x} vector.

299 Use sufficient precision in displaying results to allow you to detect imprecision. (b) Repeat part (a) using a 7 3 7 Hilbert matrix. (c) Repeat part (a) using a 10 3 10 Hilbert matrix. 10.25 Polynomial interpolation consists of determining the unique (n 2 1)th-order polynomial that fits n data points. Such polynomials have the general form, f (x) 5 p1x n21 1 p2 x n22 1 p 1 pn21 x 1 pn

(P10.25)

where the p’s are constant coefficients. A straightforward way for computing the coefficients is to generate n linear algebraic equations that we can solve simultaneously for the coefficients. Suppose that we want to determine the coefficients of the fourth-order polynomial f (x) 5 p1 x4 1 p2 x3 1 p3 x2 1 p4 x 1 p5 that passes through the following five points: (200, 0.746), (250, 0.675), (300, 0.616), (400, 0.525), and (500, 0.457). Each of these pairs can be substituted into Eq. (P10.25) to yield a system of five equations with five unknowns (the p’s). Use this approach to solve for the coefficients. In addition, determine and interpret the condition number.

11

CHAPTER

Special Matrices and Gauss-Seidel Certain matrices have a particular structure that can be exploited to develop efficient solution schemes. The first part of this chapter is devoted to two such systems: banded and symmetric matrices. Efficient elimination methods are described for both. The second part of the chapter turns to an alternative to elimination methods, that is, approximate, iterative methods. The focus is on the Gauss-Seidel method, which employs initial guesses and then iterates to obtain refined estimates of the solution. The Gauss-Seidel method is particularly well suited for large numbers of equations. In these cases, elimination methods can be subject to round-off errors. Because the error of the Gauss-Seidel method is controlled by the number of iterations, round-off error is not an issue of concern with this method. However, there are certain instances where the GaussSeidel technique will not converge on the correct answer. These and other trade-offs between elimination and iterative methods will be discussed in subsequent pages.

11.1

SPECIAL MATRICES As mentioned in Box PT3.1, a banded matrix is a square matrix that has all elements equal to zero, with the exception of a band centered on the main diagonal. Banded systems are frequently encountered in engineering and scientific practice. For example, they typically occur in the solution of differential equations. In addition, other numerical methods such as cubic splines (Sec. 18.5) involve the solution of banded systems. The dimensions of a banded system can be quantified by two parameters: the bandwidth BW and the half-bandwidth HBW (Fig. 11.1). These two values are related by BW 5 2HBW 1 1. In general, then, a banded system is one for which aij 5 0 if Zi 2 jZ . HBW. Although Gauss elimination or conventional LU decomposition can be employed to solve banded equations, they are inefficient, because if pivoting is unnecessary none of the elements outside the band would change from their original values of zero. Thus, unnecessary space and time would be expended on the storage and manipulation of these useless zeros. If it is known beforehand that pivoting is unnecessary, very efficient algorithms can be developed that do not involve the zero elements outside the band. Because many problems involving banded systems do not require pivoting, these alternative algorithms, as described next, are the methods of choice.

300

11.1

SPECIAL MATRICES

301 HBW + 1

Di

ag

on

al

HBW

BW

(a) Decomposition DOFOR k 5 2, n ek 5 ek yfk21 fk 5 fk 2 ek ? gk21 END DO

FIGURE 11.1 Parameters used to quantify the dimensions of a banded system. BW and HBW designate the bandwidth and the half-bandwidth, respectively.

(b) Forward substitution

11.1.1 Tridiagonal Systems

DOFOR k 5 2, n rk 5 rk 2 ek ? rk21 END DO

A tridiagonal system—that is, one with a bandwidth of 3—can be expressed generally as

(c) Back substitution

f1 e2

g1 f2 e3

G

g2 f3 .

xn 5 rn yfn DOFOR k 5 n 21, 1, 21 xk 5 (rk 2 gk ? xk11)yfk END DO

FIGURE 11.2 Pseudocode to implement the Thomas algorithm, an LU decomposition method for tridiagonal systems.

EXAMPLE 11.1

g3 . .

. . .

. . en21

. fn21 en

x1 r1 x2 r2 x3 r3 . . Wg w5g w . . . . gn21 xn21 rn21 fn xn rn

(11.1)

Notice that we have changed our notation for the coefficients from a’s and b’s to e’s, f’s, g’s, and r’s. This was done to avoid storing large numbers of useless zeros in the square matrix of a’s. This space-saving modification is advantageous because the resulting algorithm requires less computer memory. Figure 11.2 shows pseudocode for an efficient method, called the Thomas algorithm, to solve Eq. (11.1). As with conventional LU decomposition, the algorithm consists of three steps: decomposition and forward and back substitution. Thus, all the advantages of LU decomposition, such as convenient evaluation of multiple right-hand-side vectors and the matrix inverse, can be accomplished by proper application of this algorithm. Tridiagonal Solution with the Thomas Algorithm Problem Statement. Solve the following tridiagonal system with the Thomas algorithm. ≥

2.04 21

21 2.04 21

21 2.04 21

T1 40.8 0.8 T2 ∂ ¥ μ ∂ 5 μ 0.8 21 T3 200.8 2.04 T4

302

SPECIAL MATRICES AND GAUSS-SEIDEL

Solution. First, the decomposition is implemented as e2 f2 e3 f3 e4 f4

5 21y2.04 5 20.49 5 2.04 2 (20.49)(21) 5 1.550 5 21y1.550 5 20.645 5 2.04 2 (20.645)(21) 5 1.395 5 21y1.395 5 20.717 5 2.04 2 (20.717)(21) 5 1.323

Thus, the matrix has been transformed to ≥

2.04 20.49

21 1.550 20.645

21 1.395 20.717

21 1.323

¥

and the LU decomposition is [A] 5 [L][U] 5 ≥

1 20.49

2.04 1 20.645

1 20.717 1

¥ ≥

21 1.550

21 1.395

21 1.323

¥

You can verify that this is correct by multiplying [L][U] to yield [A]. The forward substitution is implemented as r2 5 0.8 2 (20.49)40.8 5 20.8 r3 5 0.8 2 (20.645)20.8 5 14.221 r4 5 200.8 2 (20.717)14.221 5 210.996 Thus, the right-hand-side vector has been modified to 40.8 20.8 μ ∂ 14.221 210.996 which then can be used in conjunction with the [U] matrix to perform back substitution and obtain the solution T4 T3 T2 T1

5 210.996y1.323 5 159.480 5 [14.221 2 (21)159.48]y1.395 5 124.538 5 [20.800 2 (21)124.538]y1.550 5 93.778 5 [40.800 2 (21)93.778]y2.040 5 65.970

11.1.2 Cholesky Decomposition Recall from Box PT3.1 that a symmetric matrix is one where aij 5 aji for all i and j. In other words, [A] 5 [A]T. Such systems occur commonly in both mathematical and

11.1

SPECIAL MATRICES

303

engineering problem contexts. They offer computational advantages because only half the storage is needed and, in most cases, only half the computation time is required for their solution. One of the most popular approaches involves Cholesky decomposition. This algorithm is based on the fact that a symmetric matrix can be decomposed, as in [A] 5 [L][L] T

(11.2)

That is, the resulting triangular factors are the transpose of each other. The terms of Eq. (11.2) can be multiplied out and set equal to each other. The result can be expressed simply by recurrence relations. For the kth row, i21

aki 2 a lij lkj j51

lki 5

for i 5 1, 2, p , k 2 1

lii

(11.3)

and k21

lkk 5

akk 2 a lkj2 B

(11.4)

j51

EXAMPLE 11.2

Cholesky Decomposition Problem Statement. 6 [A] 5 £ 15 55

15 55 225

Apply Cholesky decomposition to the symmetric matrix 55 225 § 979

Solution. For the first row (k 5 1), Eq. (11.3) is skipped and Eq. (11.4) is employed to compute l11 5 1a11 5 16 5 2.4495 For the second row (k 5 2), Eq. (11.3) gives l21 5

a21 15 5 5 6.1237 l11 2.4495

and Eq. (11.4) yields 2 l22 5 2a22 2 l21 5 255 2 (6.1237) 2 5 4.1833

For the third row (k 5 3), Eq. (11.3) gives (i 5 1) l31 5

a31 55 5 5 22.454 l11 2.4495

and (i 5 2) l32 5

a32 2 l21l31 225 2 6.1237(22.454) 5 5 20.917 l22 4.1833

304

SPECIAL MATRICES AND GAUSS-SEIDEL

DOFOR k 5 1, n DOFOR i 5 1, k 2 1 sum 5 0. DOFOR j 5 1, i 2 1 sum 5 sum 1 aij ? akj END DO aki 5 (aki 2 sum)yaii END DO sum 5 0. DOFOR j 5 1, k 2 1 sum 5 sum 1 a2kj END DO akk 5 1akk 2 sum END DO

and Eq. (11.4) yields

FIGURE 11.3 Pseudocode for Cholesky’s LU decomposition algorithm.

Figure 11.3 presents pseudocode for implementing the Cholesky decomposition algorithm. It should be noted that the algorithm in Fig. 11.3 could result in an execution error if the evaluation of akk involves taking the square root of a negative number. However, for cases where the matrix is positive definite,1 this will never occur. Because many symmetric matrices dealt with in engineering are, in fact, positive definite, the Cholesky algorithm has wide application. Another benefit of dealing with positive definite symmetric matrices is that pivoting is not required to avoid division by zero. Thus, we can implement the algorithm in Fig. 11.3 without the complication of pivoting.

11.2

l33 5 2a33 2 l 231 2 l 232 5 2979 2 (22.454) 2 2 (20.917) 2 5 6.1101 Thus, the Cholesky decomposition yields 2.4495 [L] 5 £ 6.1237 22.454

4.1833 20.917

6.1101

§

The validity of this decomposition can be verified by substituting it and its transpose into Eq. (11.2) to see if their product yields the original matrix [A]. This is left for an exercise.

GAUSS-SEIDEL Iterative or approximate methods provide an alternative to the elimination methods described to this point. Such approaches are similar to the techniques we developed to obtain the roots of a single equation in Chap. 6. Those approaches consisted of guessing a value and then using a systematic method to obtain a refined estimate of the root. Because the present part of the book deals with a similar problem—obtaining the values that simultaneously satisfy a set of equations—we might suspect that such approximate methods could be useful in this context. The Gauss-Seidel method is the most commonly used iterative method. Assume that we are given a set of n equations: [A]{X} 5 {B} Suppose that for conciseness we limit ourselves to a 3 3 3 set of equations. If the diagonal elements are all nonzero, the first equation can be solved for x1, the second for x2, and the third for x3 to yield x1 5

b1 2 a12x2 2 a13x3 a11

(11.5a)

A positive definite matrix is one for which the product {X}T [A]{X} is greater than zero for all nonzero vectors {X}. 1

11.2

GAUSS-SEIDEL

305

x2 5

b2 2 a21x1 2 a23 x3 a22

(11.5b)

x3 5

b3 2 a31x1 2 a32x2 a33

(11.5c)

Now, we can start the solution process by choosing guesses for the x’s. A simple way to obtain initial guesses is to assume that they are all zero. These zeros can be substituted into Eq. (11.5a), which can be used to calculate a new value for x1 5 b1ya11. Then, we substitute this new value of x1 along with the previous guess of zero for x3 into Eq. (11.5b) to compute a new value for x2. The process is repeated for Eq. (11.5c) to calculate a new estimate for x3. Then we return to the first equation and repeat the entire procedure until our solution converges closely enough to the true values. Convergence can be checked using the criterion [recall Eq. (3.5)] Zea,i Z 5 `

xij 2 x ij21 ` 100% , es x ij

(11.6)

for all i, where j and j 2 1 are the present and previous iterations. EXAMPLE 11.3

Gauss-Seidel Method Problem Statement. Use the Gauss-Seidel method to obtain the solution of the same system used in Example 10.2: 3x1 2 0.1x2 2 0.2x3 5 7.85 0.1x1 1 7x2 2 0.3x3 5 219.3 0.3x1 2 0.2x2 1 10x3 5 71.4 Recall that the true solution is x1 5 3, x2 5 22.5, and x3 5 7. Solution. First, solve each of the equations for its unknown on the diagonal. 7.85 1 0.1x2 1 0.2x3 3 219.3 2 0.1x1 1 0.3x3 x2 5 7 71.4 2 0.3x1 1 0.2x2 x3 5 10 x1 5

(E11.3.1) (E11.3.2) (E11.3.3)

By assuming that x2 and x3 are zero, Eq. (E11.3.1) can be used to compute x1 5

7.85 1 0 1 0 5 2.616667 3

This value, along with the assumed value of x3 5 0, can be substituted into Eq. (E11.3.2) to calculate x2 5

219.3 2 0.1(2.616667) 1 0 5 22.794524 7

306

SPECIAL MATRICES AND GAUSS-SEIDEL

The first iteration is completed by substituting the calculated values for x1 and x2 into Eq. (E11.3.3) to yield x3 5

71.4 2 0.3(2.616667) 1 0.2(22.794524) 5 7.005610 10

For the second iteration, the same process is repeated to compute 7.85 1 0.1(22.794524) 1 0.2(7.005610) 5 2.990557 Zet Z 5 0.31% 3 219.3 2 0.1(2.990557) 1 0.3(7.005610) x2 5 5 22.499625 Zet Z 5 0.015% 7 71.4 2 0.3(2.990557) 1 0.2(22.499625) x3 5 5 7.000291 Zet Z 5 0.0042% 10 x1 5

The method is, therefore, converging on the true solution. Additional iterations could be applied to improve the answers. However, in an actual problem, we would not know the true answer a priori. Consequently, Eq. (11.6) provides a means to estimate the error. For example, for x1, Zea, 1 Z 5 `

2.990557 2 2.616667 ` 100% 5 12.5% 2.990557

For x2 and x3, the error estimates are Zea, 2Z 5 11.8% and Zea, 3Z 5 0.076%. Note that, as was the case when determining roots of a single equation, formulations such as Eq. (11.6) usually provide a conservative appraisal of convergence. Thus, when they are met, they ensure that the result is known to at least the tolerance specified by es.

As each new x value is computed for the Gauss-Seidel method, it is immediately used in the next equation to determine another x value. Thus, if the solution is converging, the best available estimates will be employed. An alternative approach, called Jacobi iteration, utilizes a somewhat different tactic. Rather than using the latest available x’s, this technique uses Eq. (11.5) to compute a set of new x’s on the basis of a set of old x’s. Thus, as new values are generated, they are not immediately used but rather are retained for the next iteration. The difference between the Gauss-Seidel method and Jacobi iteration is depicted in Fig. 11.4. Although there are certain cases where the Jacobi method is useful, GaussSeidel’s utilization of the best available estimates usually makes it the method of preference. 11.2.1 Convergence Criterion for the Gauss-Seidel Method Note that the Gauss-Seidel method is similar in spirit to the technique of simple fixedpoint iteration that was used in Sec. 6.1 to solve for the roots of a single equation. Recall that simple fixed-point iteration had two fundamental problems: (1) it was sometimes nonconvergent and (2) when it converged, it often did so very slowly. The GaussSeidel method can also exhibit these shortcomings.

11.2

GAUSS-SEIDEL

307 First Iteration

x1 5 (b1 2 a12x2 2 a13x3)ya11 T

x2 5 (b2 2 a21x1 2 a23x3)ya22 T

x3 5 (b3 2 a31x1 2 a32x2)ya33

x1 5 (b1 2 a12x2 2 a13x3)ya11 x2 5 (b2 2 a21x1 2 a23x3)ya22 x3 5 (b3 2 a31x1 2 a32x2)ya33

T

x1 5 (b1 2 a12x2 2 a13x3)ya11 T

x2 5 (b2 2 a21x1 2 a23x3)ya22 T

T

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

Second Interation

x1 5 (b1 2 a12x2 2 a13x3)ya11 x2 5 (b2 2 a21x1 2 a23x3)ya22

x3 5 (b3 2 a31x1 2 a32x2)ya33

x3 5 (b3 2 a31x1 2 a32x2)ya33

(a)

(b)

FIGURE 11.4 Graphical depiction of the difference between (a) the Gauss-Seidel and (b) the Jacobi iterative methods for solving simultaneous linear algebraic equations.

Convergence criteria can be developed by recalling from Sec. 6.5.1 that sufficient conditions for convergence of two nonlinear equations, u(x, y) and y(x, y), are `

0u 0u ` 1 ` ` , 1 0x 0y

(11.7a)

`

0y 0y ` 1 ` ` , 1 0x 0y

(11.7b)

and

These criteria also apply to linear equations of the sort we are solving with the Gauss-Seidel method. For example, in the case of two simultaneous equations, the GaussSeidel algorithm [Eq. (11.5)] can be expressed as u(x1, x2 ) 5

b1 a12 2 x a11 a11 2

(11.8a)

and b2 a21 2 x (11.8b) a22 a22 1 The partial derivatives of these equations can be evaluated with respect to each of the unknowns as y(x1, x2 ) 5

a12 0u 0u 5 0 52 a11 0x1 0x2

308

SPECIAL MATRICES AND GAUSS-SEIDEL

and a21 0y 0y 52 50 a22 0x2 0x1 which can be substituted into Eq. (11.7) to give `

a12 ` , 1 a11

(11.9a)

`

a21 ` , 1 a22

(11.9b)

and

In other words, the absolute values of the slopes of Eq. (11.8) must be less than unity to ensure convergence. This is displayed graphically in Fig. 11.5. Equation (11.9) can also be reformulated as Za11 Z . Za12 Z and Za22 Z . Za21 Z That is, the diagonal element must be greater than the off-diagonal element for each row. The extension of the above to n equations is straightforward and can be expressed as n

Zaii Z . a Zaij Z j51

(11.10)

j? i

FIGURE 11.5 Iteration cobwebs illustrating (a) convergence and (b) divergence of the Gauss-Seidel method. Notice that the same functions are plotted in both cases (u: 11x1 1 13x2 5 286; v: 11x1 2 9x2 5 99). Thus, the order in which the equations are implemented (as depicted by the direction of the ﬁrst arrow from the origin) dictates whether the computation converges. x2

x2

v

v

u

x1

x1 u

(a)

(b)

11.2

GAUSS-SEIDEL

309

That is, the diagonal coefficient in each of the equations must be larger than the sum of the absolute values of the other coefficients in the equation. This criterion is sufficient but not necessary for convergence. That is, although the method may sometimes work if Eq. (11.10) is not met, convergence is guaranteed if the condition is satisfied. Systems where Eq. (11.10) holds are called diagonally dominant. Fortunately, many engineering problems of practical importance fulfill this requirement. 11.2.2 Improvement of Convergence Using Relaxation Relaxation represents a slight modification of the Gauss-Seidel method and is designed to enhance convergence. After each new value of x is computed using Eq. (11.5), that value is modified by a weighted average of the results of the previous and the present iterations: xinew 5 lxinew 1 (1 2 l)xiold

(11.11)

where l is a weighting factor that is assigned a value between 0 and 2. If l 5 1, (1 2 l) is equal to 0 and the result is unmodified. However, if l is set at a value between 0 and 1, the result is a weighted average of the present and the previous results. This type of modification is called underrelaxation. It is typically employed to make a nonconvergent system converge or to hasten convergence by dampening out oscillations. For values of l from 1 to 2, extra weight is placed on the present value. In this instance, there is an implicit assumption that the new value is moving in the correct direction toward the true solution but at too slow a rate. Thus, the added weight of l is intended to improve the estimate by pushing it closer to the truth. Hence, this type of modification, which is called overrelaxation, is designed to accelerate the convergence of an already convergent system. The approach is also called successive or simultaneous overrelaxation, or SOR. The choice of a proper value for l is highly problem-specific and is often determined empirically. For a single solution of a set of equations it is often unnecessary. However, if the system under study is to be solved repeatedly, the efficiency introduced by a wise choice of l can be extremely important. Good examples are the very large systems of partial differential equations that often arise when modeling continuous variations of variables (recall the distributed system depicted in Fig. PT3.1b). We will return to this topic in Part Eight. 11.2.3 Algorithm for Gauss-Seidel An algorithm for the Gauss-Seidel method, with relaxation, is depicted in Fig. 11.6. Note that this algorithm is not guaranteed to converge if the equations are not input in a diagonally dominant form. The pseudocode has two features that bear mentioning. First, there is an initial set of nested loops to divide each equation by its diagonal element. This reduces the total number of operations in the algorithm. Second, notice that the error check is designated by a variable called sentinel. If any of the equations has an approximate error greater than the stopping criterion (es), then the iterations are allowed to continue. The use of the sentinel

310

SPECIAL MATRICES AND GAUSS-SEIDEL

FIGURE 11.6 Pseudocode for Gauss-Seidel with relaxation.

SUBROUTINE Gseid (a,b,n,x,imax,es,lambda) DOFOR i 5 1,n dummy 5 ai,i DOFOR j 5 1,n ai,j 5 ai,j/dummy END DO bi 5 bi/dummy END DO DOFOR i 5 1, n sum 5 bi DOFOR j 5 1, n IF i ﬁ j THEN sum 5 sum 2 ai,j*xj END DO xi5sum END DO iter51 DO sentinel 5 1 DOFOR i 5 1,n old 5 xi sum 5 bi DOFOR j 5 1,n IF i ﬁ j THEN sum 5 sum 2 ai,j*xj END DO xi 5 lambda*sum 1(1.2lambda)*old IF sentinel 5 1 AND xi ﬁ 0. THEN ea 5 ABS((xi 2 old)/xi)*100. IF ea . es THEN sentinel 5 0 END IF END DO iter 5 iter 1 1 IF sentinel 5 1 OR (iter $ imax) EXIT END DO END Gseid

allows us to circumvent unnecessary calculations of error estimates once one of the equations exceeds the criterion. 11.2.4 Problem Contexts for the Gauss-Seidel Method Aside from circumventing the round-off dilemma, the Gauss-Seidel technique has a number of other advantages that make it particularly attractive in the context of certain engineering problems. For example, when the matrix in question is very large and very sparse (that is, most of the elements are zero), elimination methods waste large amounts of computer memory by storing zeros.

11.3

LINEAR ALGEBRAIC EQUATIONS WITH SOFTWARE PACKAGES

311

At the beginning of this chapter, we saw how this shortcoming could be circumvented if the coefficient matrix is banded. For nonbanded systems, there is usually no simple way to avoid large memory requirements when using elimination methods. Because all computers have a finite amount of memory, this inefficiency can place a constraint on the size of systems for which elimination methods are practical. Although a general algorithm such as the one in Fig. 11.6 is prone to the same constraint, the structure of the Gauss-Seidel equations [Eq. (11.5)] permits concise programs to be developed for specific systems. Because only nonzero coefficients need be included in Eq. (11.5), large savings of computer memory are possible. Although this entails more up-front investment in software development, the long-term advantages are substantial when dealing with large systems for which many simulations are to be performed. Both lumped- and distributed-variable systems can result in large, sparse matrices for which the Gauss-Seidel method has utility.

LINEAR ALGEBRAIC EQUATIONS WITH SOFTWARE PACKAGES Software packages have great capabilities for solving systems of linear algebraic equations. Before describing these tools, we should mention that the approaches described in Chap. 7 for solving nonlinear systems can be applied to linear systems. However, in this section, we will focus on the approaches that are expressly designed for linear equations. 11.3.1 Excel There are two ways to solve linear algebraic equations with Excel: (1) using the Solver tool or (2) using matrix inversion and multiplication functions. Recall that one way to determine the solution of linear algebraic equations is {X} 5 [A] 21 {B}

(11.12)

Excel has built-in functions for both matrix inversion and multiplication that can be used to implement this formula. EXAMPLE 11.4

Using Excel to Solve Linear Systems Problem Statement. Recall that in Chap. 10 we introduced the Hilbert matrix. The following system is based on the Hilbert matrix. Note that it is scaled, as was done previously in Example 10.3, so that the maximum coefficient in each row is unity. 1 £1 1

1y2 2y3 3y4

1.833333 1y3 x1 1y2 § • x2 ¶ 5 • 2.166667 ¶ 2.35 3y5 x3

The solution to this system is {X} T 5 :1 1 1; . Use Excel to obtain this solution. Solution. The spreadsheet to solve this problem is displayed in Fig. 11.7. First, the matrix [A] and the right-hand-side constants {B} are entered into the spreadsheet cells. Then, a set of cells of the proper dimensions (in our example 3 3 3) is highlighted by either clicking and dragging the mouse or by using the arrow keys while depressing the shift key. As in Fig. 11.7, we highlight the range: B5. .D7.

SOFTWARE

11.3

SPECIAL MATRICES AND GAUSS-SEIDEL

SOFTWARE

312

FIGURE 11.7

Next, a formula invoking the matrix inverse function is entered, =minverse(B1..D3)

Note that the argument is the range holding the elements of [A]. The Ctrl and Shift keys are held down while the Enter key is depressed. The resulting inverse of [A] will be calculated by Excel and displayed in the range B5. .D7 as shown in Fig. 11.7. A similar approach is used to multiply the inverse by the right-hand-side vector. For this case, the range from F5. .F7 is highlighted and the following formula is entered =mmult(B5..D7,F1..F3)

where the first range is the first matrix to be multiplied, [A]21, and the second range is the second matrix to be multiplied, {B}. By again using the Ctrl-Shift-Enter combination, the solution {X} will be calculated by Excel and displayed in the range F5. .F7, as shown in Fig. 11.7. As can be seen, the correct answer results.

Notice that we deliberately reformatted the results in Example 11.4 to show 15 digits. We did this because Excel uses double-precision to store numerical values. Thus, we see that round-off error occurs in the last two digits. This implies a condition number on the order of 100, which agrees with the result of 451.2 originally calculated in Example 10.3. Excel does not have the capability to calculate a condition number. In most cases, particularly because it employs double-precision numbers, this does not represent a problem. However, for cases where you suspect that the system is ill-conditioned, determination of the condition number is useful. MATLAB and Mathcad software are capable of computing this quantity. 11.3.2 MATLAB As the name implies, MATLAB (short for MATrix LABoratory) was designed to facilitate matrix manipulations. Thus, as might be expected, its capabilities in this area are

11.3

LINEAR ALGEBRAIC EQUATIONS WITH SOFTWARE PACKAGES

313

TABLE 11.1 MATLAB functions to implement matrix analysis and numerical linear algebra. Matrix Analysis

Linear Equations

Function

Description

Function

Description

cond norm rcond rank

\ and / chol lu inv

Linear equation solution; use “help slash” Cholesky factorization Factors from Gauss elimination Matrix inverse

det trace

Matrix condition number Matrix or vector norm LINPACK reciprocal condition estimator Number of linearly independent rows or columns Determinant Sum of diagonal elements

qr qrdelete

null orth rref

Null space Orthogonalization Reduced row echelon form

qrinsert nnls pinv lscov

Orthogonal-triangular decomposition Delete a column from the QR factorization Insert a column in the QR factorization Nonnegative least squares Pseudoinverse Least squares in the presence of known covariance

excellent. Some of the key MATLAB functions related to matrix operations are listed in Table 11.1. The following example illustrates a few of these capabilities. EXAMPLE 11.5

Using MATLAB to Manipulate Linear Algebraic Equations Problem Statement. Explore how MATLAB can be employed to solve and analyze linear algebraic equations. Use the same system as in Example 11.4. Solution. First, we can enter the [A] matrix and the {B}vector, >> A 5 [ 1 A =

1/2

1.0000 0.5000 1.0000 0.6667 1.0000 0.7500

1/3 ; 1

2/3

1/2 ; 1

0.3333 0.5000 0.6000

>> B=[1+1/2+1/3;1+2/3+2/4;1+3/4+3/5] B =

1.8333 2.1667 2.3500

Next, we can determine the condition number for [A], as in >> cond(A) ans = 366.3503

3/4

3/5 ]

SOFTWARE

314

SPECIAL MATRICES AND GAUSS-SEIDEL

This result is based on the spectral, or B AB2, norm discussed in Box 10.2. Note that it is of the same order of magnitude as the condition number 5 451.2 based on the rowsum norm in Example 10.3. Both results imply that between two and three digits of precision could be lost. Now we can solve the system of equations in two different ways. The most direct and efficient way is to employ backslash, or “left division”: >> X=A\B X =

1.0000 1.0000 1.0000

For cases such as ours, MATLAB uses Gauss elimination to solve such systems. As an alternative, we can implement Eq. (PT3.6) directly, as in >> X=inv(A)*B X =

1.0000 1.0000 1.0000

This approach actually determines the matrix inverse first and then performs the matrix multiplication. Hence, it is more time consuming than using the backslash approach.

11.3.3 Mathcad Mathcad contains many special functions that manipulate vectors and matrices. These include common operations such as the dot product, matrix transpose, matrix addition, and matrix multiplication. In addition, it allows calculation of the matrix inverse, determinant, trace, various types of norms, and condition numbers based on different norms. It also has several functions that decompose matrices. Systems of linear equations can be solved in two ways by Mathcad. First, it is possible to use matrix inversion and subsequent multiplication by the right-hand-side as discussed in Chap. 10. In addition, Mathcad has a special function called lsolve(A,b) that is specifically designed to solve linear equations. You can use other built-in functions to evaluate the condition of A to determine if A is nearly singular and thus possibly subject to round-off errors. As an example, let’s use lsolve to solve a system of linear equations. As shown in Fig. 11.8, the first step is to enter the coefficients of the A matrix using the definition symbol and the Insert/Matrix pull down menu. This gives a box that allows you to

11.3

LINEAR ALGEBRAIC EQUATIONS WITH SOFTWARE PACKAGES

315

FIGURE 11.8 Mathcad screen to solve a system of linear algebraic equations.

specify the dimensions of the matrix. For our case, we will select a dimension of 434, and Mathcad places a blank 4-by-4-size matrix on screen. Now, simply click the appropriate cell location and enter values. Repeat similar operations to create the righthand-side b vector. Now the vector x is defined as lsolve(A,b) and the value of x is displayed with the equal sign. We can also solve the same system using the matrix inverse. The inverse can be simply computed by merely raising A to the exponent 21. The result is shown on the right side of Fig. 11.8. The solution is then generated as the product of the inverse times b. Next, let’s use Mathcad to find the inverse and the condition number of the Hilbert matrix. As in Fig. 11.9, the scaled matrix can be entered using the definition symbol and the Insert/Matrix pull down menu. The inverse can again be computed by simply raising H to the exponent 21. The result is shown in Fig. 11.9. We can then use some other Mathcad functions to determine condition numbers by using the definition symbol to define variables c1, c2, ce, and ci as the condition number based on the column-sum (cond1), spectral (cond2), the Euclidean (conde), and the row-sum (condi) norms, respectively. The resulting values are shown at the bottom of Fig. 11.9. As expected, the spectral norm provides the smallest measure of magnitude.

SPECIAL MATRICES AND GAUSS-SEIDEL

SOFTWARE

316

FIGURE 11.9 Mathcad screen to determine the matrix inverse and condition numbers of a scaled 333 Hilbert matrix.

PROBLEMS 11.1 Perform the same calculations as in (a) Example 11.1, and (b) Example 11.3, but for the tridiagonal system, 0.8 £ 20.4

20.4 0.8 20.4

x1 41 20.4 § • x2 ¶ 5 • 25 ¶ 0.8 x3 105

11.2 Determine the matrix inverse for Example 11.1 based on the LU decomposition and unit vectors. 11.3 The following tridiagonal system must be solved as part of a larger algorithm (Crank-Nicolson) for solving partial differential equations:

D

2.01475 20.020875

20.020875 2.01475 20.020875

20.020875 2.01475 20.020875

20.020875 2.01475

T

4.175 T1 T2 0 t 3d t5d T3 0 T4 2.0875 Use the Thomas algorithm to obtain a solution. 11.4 Confirm the validity of the Cholesky decomposition of Example 11.2 by substituting the results into Eq. (11.2) to see if the product of [L] and [L]T yields [A].

PROBLEMS 11.5 Perform the same calculations as in Example 11.2, but for the symmetric system, 6 £ 15 55

15 55 225

55 a0 152.6 225 § • a1 ¶ 5 • 585.6 ¶ 979 a2 2488.8

In addition to solving for the Cholesky decomposition, employ it to solve for the a’s. 11.6 Perform a Cholesky decomposition of the following symmetric system by hand, 8 £ 20 15

20 80 50

15 x1 50 50 § • x2 ¶ 5 • 250 ¶ 60 x3 100

11.7 Compute the Cholesky decomposition of 9 [A] 5 £ 0 0

0 25 0

317 11.13 Use the Gauss-Seidel method (a) without relaxation and (b) with relaxation (l 5 1.2) to solve the following system to a tolerance of es 5 5%. If necessary, rearrange the equations to achieve convergence. 2x1 2 6x2 2 x3 5 238 23x1 2 x2 1 7x3 5 234 28x1 1 x2 2 2x3 5 220 11.14 Redraw Fig. 11.5 for the case where the slopes of the equations are 1 and 21. What is the result of applying Gauss-Seidel to such a system? 11.15 Of the following three sets of linear equations, identify the set(s) that you could not solve using an iterative method such as Gauss-Seidel. Show using any number of iterations that is necessary that your solution does not converge. Clearly state your convergence criteria (how you know it is not converging).

0 0§ 4

Do your results make sense in terms of Eqs. (11.3) and (11.4)? 11.8 Use the Gauss-Seidel method to solve the tridiagonal system from Prob. 11.1 (es 5 5%). Use overrelaxation with l 5 1.2. 11.9 Recall from Prob. 10.8, that the following system of equations is designed to determine concentrations (the c’s in g/m3) in a series of coupled reactors as a function of amount of mass input to each reactor (the right-hand sides in g/d), 15c1 2 3c2 2 c3 5 3800

Set One

Set Two

Set Three

8x 1 3y 1 z 5 12 26x 1 7z 5 1 2x 1 4y 2 z 5 5

x 1 y 1 5z 5 7 x 1 4y 2 z 5 4 3x 1 y 2 z 5 4

2x 1 3y 1 5z 5 7 22x 1 4y 2 5z 5 23 2y 2 z 5 1

11.16 Use the software package of your choice to obtain a solution, calculate the inverse, and determine the condition number (without scaling) based on the row-sum norm for (a) 1 £4 9

23c1 1 18c2 2 6c3 5 1200 24c1 2 c2 1 12c3 5 2350 Solve this problem with the Gauss-Seidel method to es 5 5%. 11.10 Repeat Prob. 11.9, but use Jacobi iteration. 11.11 Use the Gauss-Seidel method to solve the following system until the percent relative error falls below es 5 5%, 10x1 1 2x2 2 x3 5 27 23x1 2 6x2 1 2x3 5 261.5 x1 1 x2 1 5x3 5 221.5 11.12 Use the Gauss-Seidel method (a) without relaxation and (b) with relaxation (l 5 0.95) to solve the following system to a tolerance of es 5 5%. If necessary, rearrange the equations to achieve convergence. 23x1 1 x2 1 12x3 5 50 6x1 2 x2 2 x3 5 3 6x1 1 9x2 1 x3 5 40

4 9 16

9 x1 14 16 § • x2 ¶ 5 • 29 ¶ 25 50 x3

(b) 1 4 D 9 16

4 9 16 25

9 16 25 36

16 x1 30 25 x2 54 T d t5d t 36 x3 86 49 x4 126

In both cases, the answers for all the x’s should be 1. 11.17 Given the pair of nonlinear simultaneous equations: f(x, y) 5 4 2 y 2 2x2 g(x, y) 5 8 2 y2 2 4x (a) Use the Excel Solver to determine the two pairs of values of x and y that satisfy these equations. (b) Using a range of initial guesses (x 5 26 to 6 and y 5 26 to 6), determine which initial guesses yield each of the solutions.

318

SPECIAL MATRICES AND GAUSS-SEIDEL

11.18 An electronics company produces transistors, resistors, and computer chips. Each transistor requires four units of copper, one unit of zinc, and two units of glass. Each resistor requires three, three, and one units of the three materials, respectively, and each computer chip requires two, one, and three units of these materials, respectively. Putting this information into table form, we get: Component Transistors Resistors Computer chips

Copper

Zinc

Glass

4 3 2

1 3 1

2 1 3

Supplies of these materials vary from week to week, so the company needs to determine a different production run each week. For example, one week the total amounts of materials available are 960 units of copper, 510 units of zinc, and 610 units of glass. Set up the system of equations modeling the production run, and use Excel, MATLAB, or Mathcad, to solve for the number of transistors, resistors, and computer chips to be manufactured this week. 11.19 Use MATLAB or Mathcad software to determine the spectral condition number for a 10-dimensional Hilbert matrix. How many digits of precision are expected to be lost due to ill-conditioning? Determine the solution for this system for the case where each element of the right-hand-side vector {b} consists of the summation of the coefficients in its row. In other words, solve for the case where all the unknowns should be exactly one. Compare the resulting errors with those expected based on the condition number. 11.20 Repeat Prob. 11.19, but for the case of a six-dimensional Vandermonde matrix (see Prob. 10.17) where x1 5 4, x2 5 2, x3 5 7, x4 5 10, x5 5 3, and x6 5 5. 11.21 Given a square matrix [A], write a single line MATLAB command that will create a new matrix [Aug] that consists of the original matrix [A] augmented by an identity matrix [I]. 11.22 Write the following set of equations in matrix form:

11.24 Develop a user-friendly program in either a high-level or macro language of your choice to obtain a solution for a tridiagonal system with the Thomas algorithm (Fig. 11.2). Test your program by duplicating the results of Example 11.1. 11.25 Develop a user-friendly program in either a high-level or macro language of your choice for Cholesky decomposition based on Fig. 11.3. Test your program by duplicating the results of Example 11.2. 11.26 Develop a user-friendly program in either a high-level or macro language of your choice for the Gauss-Seidel method based on Fig. 11.6. Test your program by duplicating the results of Example 11.3. 11.27 As described in Sec. PT3.1.2, linear algebraic equations can arise in the solution of differential equations. For example, the following differential equation results from a steady-state mass balance for a chemical in a one-dimensional canal, 05D

d2c dc 2 kc 2U 2 dx dx

where c 5 concentration, t 5 time, x 5 distance, D 5 diffusion coefficient, U 5 fluid velocity, and k 5 a first-order decay rate. Convert this differential equation to an equivalent system of simultaneous algebraic equations. Given D 5 2, U 5 1, k 5 0.2, c(0) 5 80 and c(10) 5 20, solve these equations from x 5 0 to 10 with Dx 5 2, and develop a plot of concentration versus distance. 11.28 A pentadiagonal system with a bandwidth of five can be expressed generally as f1 e2 d3

g1 f2 e3

H

h1 g2 f3 .

h2 g3 . .

h3 . . . dn21

50 5 5x3 2 7x2 4x2 1 7x3 1 30 5 0 x1 2 7x3 5 40 2 3x2 1 5x1 Use Excel, MATLAB, or Mathcad to solve for the unknowns. In addition, compute the transpose and the inverse of the coefficient matrix. 11.23 In Sec. 9.2.1, we determined the number of operations required for Gauss elimination without partial pivoting. Make a similar determination for the Thomas algorithm (Fig. 11.2). Develop a plot of operations versus n (from 2 to 20) for both techniques.

. . en21 dn

. fn21 en

x1 r1 x2 r2 x3 r3 . . X h x 5h x . . . . xn21 rn21 gn21 fn xn rn

Develop a program to efficiently solve such systems without pivoting in a similar fashion to the algorithm used for tridiagonal matrices in Sec. 11.1.1. Test it for the following case: 8 22 E21 0 0

22 9 23 24 0

21 24 7 22 27

0 21 21 12 23

0 x1 5 0 x2 2 22U ex3u 5 e0u 25 x4 1 15 x5 5

12

CHAPTER

Case Studies: Linear Algebraic Equations The purpose of this chapter is to use the numerical procedures discussed in Chaps. 9, 10, and 11 to solve systems of linear algebraic equations for some engineering case studies. These systematic numerical techniques have practical significance because engineers frequently encounter problems involving systems of equations that are too large to solve by hand. The numerical algorithms in these applications are particularly convenient to implement on personal computers. Section 12.1 shows how a mass balance can be employed to model a system of reactors. Section 12.2 places special emphasis on the use of the matrix inverse to determine the complex cause-effect interactions between forces in the members of a truss. Section 12.3 is an example of the use of Kirchhoff’s laws to compute the currents and voltages in a resistor circuit. Finally, Sec. 12.4 is an illustration of how linear equations are employed to determine the steady-state configuration of a massspring system.

12.1

STEADY-STATE ANALYSIS OF A SYSTEM OF REACTORS (CHEMICAL/BIO ENGINEERING) Background. One of the most important organizing principles in chemical engineering is the conservation of mass (recall Table 1.1). In quantitative terms, the principle is expressed as a mass balance that accounts for all sources and sinks of a material that pass in and out of a volume (Fig. 12.1). Over a finite period of time, this can be expressed as Accumulation 5 inputs 2 outputs

(12.1)

The mass balance represents a bookkeeping exercise for the particular substance being modeled. For the period of the computation, if the inputs are greater than the outputs, the mass of the substance within the volume increases. If the outputs are greater than the inputs, the mass decreases. If inputs are equal to the outputs, accumulation is zero and mass remains constant. For this stable condition, or steady state, Eq. (12.1) can be expressed as Inputs 5 outputs

(12.2) 319

320

CASE STUDIES: LINEAR ALGEBRAIC EQUATIONS

Volume

Input

Output Accumulation

FIGURE 12.1 A schematic representation of mass balance.

Employ the conservation of mass to determine the steady-state concentrations of a system of coupled reactors. Solution. The mass balance can be used for engineering problem solving by expressing the inputs and outputs in terms of measurable variables and parameters. For example, if we were performing a mass balance for a conservative substance (that is, one that does not increase or decrease due to chemical transformations) in a reactor (Fig. 12.2), we would have to quantify the rate at which mass flows into the reactor through the two inflow pipes and out of the reactor through the outflow pipe. This can be done by taking the product of the flow rate Q (in cubic meters per minute) and the concentration c (in milligrams per cubic meter) for each pipe. For example, for pipe 1 in Fig. 12.2, Q1 5 2 m3/min and c1 5 25 mg/m3; therefore, the rate at which mass flows into the reactor through pipe 1 is Q1c1 5 (2 m3/min)(25 mg/m3) 5 50 mg/min. Thus, 50 mg of chemical flows into the reactor through this pipe each minute. Similarly, for pipe 2 the mass inflow rate can be calculated as Q2c2 5 (1.5 m3/min)(10 mg/m3) 5 15 mg/min. Notice that the concentration out of the reactor through pipe 3 is not specified by Fig. 12.2. This is because we already have sufficient information to calculate it on the basis of the conservation of mass. Because the reactor is at steady state, Eq. (12.2) holds and the inputs should be in balance with the outputs, as in Q1c1 1 Q2c2 5 Q3c3 Substituting the given values into this equation yields 50 1 15 5 3.5c3 which can be solved for c3 5 18.6 mg/m3. Thus, we have determined the concentration in the third pipe. However, the computation yields an additional bonus. Because the reactor is well mixed (as represented by the propeller in Fig. 12.2), the concentration will be uniform, or homogeneous, throughout the tank. Therefore the concentration in pipe 3 should be identical to the concentration throughout the reactor. Consequently, the mass balance has allowed us to compute both the concentration in the reactor and in the

12.1

STEADY-STATE ANALYSIS OF A SYSTEM OF REACTORS

321

Q1 = 2 m3/min c1 = 25 mg/m3

FIGURE 12.2 A steady-state, completely mixed reactor with two inﬂow pipes and one outﬂow pipe. The ﬂows Q are in cubic meters per minute, and the concentrations c are in milligrams per cubic meter.

Q3 = 3.5 m3/min c3 = ? Q2 = 1.5 m3/min c2 = 10 mg/m3

Q15 = 3

Q55 = 2 c5

Q54 = 2

Q25 = 1 Q01 = 5 c01 = 10

c1

Q12 = 3

c2

Q23 = 1

Q03 = 8

FIGURE 12.3 Five reactors linked by pipes.

Q31 = 1

Q24 = 1

c4

Q44 = 11

Q34 = 8

c3

c03 = 20

outflow pipe. Such information is of great utility to chemical and petroleum engineers who must design reactors to yield mixtures of a specified concentration. Because simple algebra was used to determine the concentration for the single reactor in Fig. 12.2, it might not be obvious how computers figure in mass-balance calculations. Figure 12.3 shows a problem setting where computers are not only useful but are a practical necessity. Because there are five interconnected, or coupled, reactors, five simultaneous mass-balance equations are needed to characterize the system. For reactor 1, the rate of mass flow in is 5(10) 1 Q31c3

322

CASE STUDIES: LINEAR ALGEBRAIC EQUATIONS

and the rate of mass flow out is Q12c1 1 Q15c1 Because the system is at steady state, the inflows and outflows must be equal: 5(10) 1 Q31c3 5 Q12c1 1 Q15c1 or, substituting the values for flow from Fig. 12.3, 6c1 2 c3 5 50 Similar equations can be developed for the other reactors: 23c1 1 3c2 5 0 2c2 1 9c3 5 160 2c2 2 8c3 1 11c4 2 2c5 5 0 23c1 2 c2 1 4c5 5 0 A numerical method can be used to solve these five equations for the five unknown concentrations: {C} T 5 :11.51

11.51

19.06

17.00

11.51;

In addition, the matrix inverse can be computed as

[A] 21

0.16981 0.16981 5 E0.01887 0.06003 0.16981

0.00629 0.33962 0.03774 0.07461 0.08962

0.01887 0.01887 0.11321 0.08748 0.01887

0 0 0 0.09091 0

0 0 0 U 0.04545 0.25000

Each of the elements aij signifies the change in concentration of reactor i due to a unit change in loading to reactor j. Thus, the zeros in column 4 indicate that a loading to reactor 4 will have no impact on reactors 1, 2, 3, and 5. This is consistent with the system configuration (Fig. 12.3), which indicates that flow out of reactor 4 does not feed back into any of the other reactors. In contrast, loadings to any of the first three reactors will affect the entire system as indicated by the lack of zeros in the first three columns. Such information is of great utility to engineers who design and manage such systems.

12.2

ANALYSIS OF A STATICALLY DETERMINATE TRUSS (CIVIL/ENVIRONMENTAL ENGINEERING) Background. An important problem in structural engineering is that of finding the forces and reactions associated with a statically determinate truss. Figure 12.4 shows an example of such a truss. The forces (F) represent either tension or compression on the members of the truss. External reactions (H2, V2, and V3) are forces that characterize how the truss interacts with the supporting surface. The hinge at node 2 can transmit both horizontal and vertical forces to the surface, whereas the roller at node 3 transmits only vertical forces. It is observed that the effect of the external loading of 1000 lb is distributed among the various members of the truss.

12.2

ANALYSIS OF A STATICALLY DETERMINATE TRUSS

323

1000 lb 1 F1

H2

60⬚

30⬚

2

F3

90⬚

3

F2

FIGURE 12.4 Forces on a statically determinate truss.

V2

V3

F1,v 1 30⬚

F1,h 60⬚

F1 F2,v H2

F3 60⬚

30⬚

2

FIGURE 12.5 Free-body force diagrams for the nodes of a statically determinate truss.

F3

F1

F2 V2

F2,h

F3,v 3

F3,h

F2 V3

Solution. This type of structure can be described as a system of coupled linear algebraic equations. Free-body force diagrams are shown for each node in Fig. 12.5. The sum of the forces in both horizontal and vertical directions must be zero at each node, because the system is at rest. Therefore, for node 1, g FH 5 0 5 2F1 cos 30° 1 F3 cos 60° 1 F1, h g FV 5 0 5 2F1 sin 30° 2 F3 sin 60° 1 F1, y

(12.3) (12.4)

for node 2, g FH 5 0 5 F2 1 F1 cos 30° 1 F2, h 1 H2 g FV 5 0 5 F1 sin 30° 1 F2, y 1 V2

(12.5) (12.6)

324

CASE STUDIES: LINEAR ALGEBRAIC EQUATIONS

for node 3, g FH 5 0 5 2F2 2 F3 cos 60° 1 F3, h g FV 5 0 5 F3 sin 60° 1 F3, y 1 V3

(12.7) (12.8)

where Fi, h is the external horizontal force applied to node i (where a positive force is from left to right) and F1, y is the external vertical force applied to node i (where a positive force is upward). Thus, in this problem, the 1000-lb downward force on node 1 corresponds to F1, y 5 21000. For this case all other Fi, y’s and Fi, h’s are zero. Note that the directions of the internal forces and reactions are unknown. Proper application of Newton’s laws requires only consistent assumptions regarding direction. Solutions are negative if the directions are assumed incorrectly. Also note that in this problem, the forces in all members are assumed to be in tension and act to pull adjoining nodes together. A negative solution therefore corresponds to compression. This problem can be written as the following system of six equations and six unknowns: 0.866 0.5 20.866 F 20.5 0 0

0 0 21 0 1 0

20.5 0.866 0 0 0.5 20.866

0 0 21 0 0 0

0 0 0 21 0 0

0 F1 0 0 F2 21000 0 F3 0 V f v5f v 0 H2 0 0 V2 0 V3 21 0

(12.9)

Notice that, as formulated in Eq. (12.9), partial pivoting is required to avoid division by zero diagonal elements. Employing a pivot strategy, the system can be solved using any of the elimination techniques discussed in Chap. 9 or 10. However, because this problem is an ideal case study for demonstrating the utility of the matrix inverse, the LU decomposition can be used to compute F1 5 2500 F2 5 433 F3 5 2866 H2 5 0 V2 5 250 V3 5 750 and the matrix inverse is

[A] 21

0.866 0.25 20.5 5F 21 20.433 0.433

0.5 0 20.433 0 0.866 0 0 21 20.25 0 20.75 0

0 0 0 0 21 0

0 1 0 21 0 0

0 0 0 V 0 0 21

Now, realize that the right-hand-side vector represents the externally applied horizontal and vertical forces on each node, as in {F} T 5 :F1, h

F1, y

F2, h

F2, y

F3, h

F3, y ;

(12.10)

Because the external forces have no effect on the LU decomposition, the method need not be implemented over and over again to study the effect of different external forces on the truss. Rather, all that we have to do is perform the forward- and backward-substitution steps for each right-hand-side vector to efficiently obtain alternative solutions. For example,

12.2

ANALYSIS OF A STATICALLY DETERMINATE TRUSS

1000

1000

6

86

0 50

2000

1000

2000

250

433

433

0 50

6

86

325

1250

433

(a)

1000

433

(b)

FIGURE 12.6 Two test cases showing (a) winds from the left and (b) winds from the right.

we might want to study the effect of horizontal forces induced by a wind blowing from left to right. If the wind force can be idealized as two point forces of 1000 lb on nodes 1 and 2 (Fig. 12.6a), the right-hand-side vector is {F} T 5 :21000

0

1000

0

0

0;

which can be used to compute F1 5 866 F2 5 250 F3 5 2500 H2 5 22000 V2 5 2433 V3 5 433 For a wind from the right (Fig. 12.6b), F1, h 5 21000, F3, h 5 21000, and all other external forces are zero, with the result that F1 5 2866 F2 5 21250 F3 5 500 H2 5 2000 V2 5 433 V3 5 2433 The results indicate that the winds have markedly different effects on the structure. Both cases are depicted in Fig. 12.6. The individual elements of the inverted matrix also have direct utility in elucidating stimulus-response interactions for the structure. Each element represents the change of one of the unknown variables to a unit change of one of the external stimuli. For example, element a21 32 indicates that the third unknown (F3) will change 0.866 due to a unit change of the second external stimulus (F1, y). Thus, if the vertical load at the first node were increased by 1, F3 would increase by 0.866. The fact that elements are 0 indicates that certain unknowns are unaffected by some of the external stimuli. For instance a21 32 5 0 means that F1 is unaffected by changes in F2, h. This ability to isolate interactions has a number of engineering applications, including the identification of those components that are most sensitive to external stimuli and, as a consequence, most prone to failure. In addition, it can be used to determine components that may be unnecessary (see Prob. 12.18).

326

CASE STUDIES: LINEAR ALGEBRAIC EQUATIONS

The foregoing approach becomes particularly useful when applied to large complex structures. In engineering practice, it may be necessary to solve trusses with hundreds or even thousands of structural members. Linear equations provide one powerful approach for gaining insight into the behavior of these structures.

12.3

CURRENTS AND VOLTAGES IN RESISTOR CIRCUITS (ELECTRICAL ENGINEERING) Background. A common problem in electrical engineering involves determining the currents and voltages at various locations in resistor circuits. These problems are solved using Kirchhoff’s current and voltage rules. The current (or point) rule states that the algebraic sum of all currents entering a node must be zero (see Fig. 12.7a), or oi 5 0

(12.11)

where all current entering the node is considered positive in sign. The current rule is an application of the principle of conservation of charge (recall Table 1.1). The voltage (or loop) rule specifies that the algebraic sum of the potential differences (that is, voltage changes) in any loop must equal zero. For a resistor circuit, this is expressed as oj 2 oi R 5 0

(12.12)

where j is the emf (electromotive force) of the voltage sources and R is the resistance of any resistors on the loop. Note that the second term derives from Ohm’s law (Fig. 12.7b), which states that the voltage drop across an ideal resistor is equal to the product of the current and the resistance. Kirchhoff’s voltage rule is an expression of the conservation of energy.

FIGURE 12.7 Schematic representations of (a) Kirchhoff’s current rule and (b) Ohm’s law. i1

Solution. Application of these rules results in systems of simultaneous linear algebraic equations because the various loops within a circuit are coupled. For example, consider the circuit shown in Fig. 12.8. The currents associated with this circuit are unknown both in magnitude and direction. This presents no great difficulty because one simply assumes a direction for each current. If the resultant solution from Kirchhoff’s laws is negative, then the assumed direction was incorrect. For example, Fig. 12.9 shows some assumed currents.

i3

FIGURE 12.8 A resistor circuit to be solved using simultaneous linear algebraic equations. i2

3

(a) Vi

Rij iij

(b)

Vj

R = 10 ⍀

2

R=5⍀

R=5⍀

1

V1 = 200 V

R = 10 ⍀

4

R = 15 ⍀

5

R = 20 ⍀

6

V6 = 0 V

12.3

CURRENTS AND VOLTAGES IN RESISTOR CIRCUITS 3

2

1

i32 i43

327

i12 i52

i54

4

i65

5

6

FIGURE 12.9 Assumed currents.

Given these assumptions, Kirchhoff’s current rule is applied at each node to yield i12 1 i52 1 i32 5 0 i65 2 i52 2 i54 5 0 i43 2 i32 5 0 i54 2 i43 5 0 Application of the voltage rule to each of the two loops gives 2i54 R54 2 i43 R43 2 i32 R32 1 i52 R52 5 0 2i65 R65 2 i52 R52 2 i12 R12 2 200 5 0 or, substituting the resistances from Fig. 12.8 and bringing constants to the right-hand side, 215i54 2 5i43 2 10i32 1 10i52 5 0 220i65 2 10i52 1 5i12 5 200 Therefore, the problem amounts to solving the following set of six equations with six unknown currents: 1 0 0 F 0 0 5

1 21 0 0 10 210

1 0 21 0 210 0

0 1 0 0 0 220

0 21 0 1 215 0

0 0 1 V 21 25 0

0 i12 i52 0 0 i32 f v5f v 0 i65 i54 0 200 i43

Although impractical to solve by hand, this system is easily handled using an elimination method. Proceeding in this manner, the solution is i12 5 6.1538 i52 5 24.6154 i32 5 21.5385 i65 5 26.1538 i54 5 21.5385 i43 5 21.5385 Thus, with proper interpretation of the signs of the result, the circuit currents and voltages are as shown in Fig. 12.10. The advantages of using numerical algorithms and computers for problems of this type should be evident.

328

CASE STUDIES: LINEAR ALGEBRAIC EQUATIONS

V = 153.85

V = 169.23

i = 1.5385

V = 146.15

V = 200 i = 6.1538

V = 123.08

V=0

FIGURE 12.10 The solution for currents and voltages obtained using an elimination method.

12.4

SPRING-MASS SYSTEMS (MECHANICAL/AEROSPACE ENGINEERING) Background. Idealized spring-mass systems play an important role in mechanical and other engineering problems. Figure 12.11 shows such a system. After they are released, the masses are pulled downward by the force of gravity. Notice that the resulting displacement of each spring in Fig. 12.11b is measured along local coordinates referenced to its initial position in Fig. 12.11a. As introduced in Chap. 1, Newton’s second law can be employed in conjunction with force balances to develop a mathematical model of the system. For each mass, the second law can be expressed as m

d 2x 5 FD 2 FU d t2

(12.13)

To simplify the analysis, we will assume that all the springs are identical and follow Hooke’s law. A free-body diagram for the first mass is depicted in Fig. 12.12a. The upward force is merely a direct expression of Hooke’s law: FU 5 k x1

(12.14)

The downward component consists of the two spring forces along with the action of gravity on the mass, FD 5 k(x2 2 x1 ) 1 k(x2 2 x1 ) 5 m1 g

(12.15)

Note how the force component of the two springs is proportional to the displacement of the second mass, x2, corrected for the displacement of the first mass, x1. Equations (12.14) and (12.15) can be substituted into Eq. (12.13) to give m1

d 2 x1 dt2

5 2k(x2 2 x1 ) 1 m1 g 2 k x1

(12.16)

Thus, we have derived a second-order ordinary differential equation to describe the displacement of the first mass with respect to time. However, notice that the solution cannot be obtained because the model includes a second dependent variable, x2. Consequently, free-body diagrams must be developed for the second and the third masses (Fig. 12.12b

12.4

SPRING-MASS SYSTEMS

329

k 0

m1 k

m1

k

x1 0

m2 k

m2

m3

x2 0

m3

(a)

x3

(b)

FIGURE 12.11 A system composed of three masses suspended vertically by a series of springs. (a) The system before release, that is, prior to extension or compression of the springs. (b) The system after release. Note that the positions of the masses are referenced to local coordinates with origins at their position before release.

kx1

k(x2 – x1)

m1

k(x2 – x1) m1g k(x2 – x1)

(a)

k(x2 – x1)

m2

m2g

k(x3 – x2)

m3

k(x3 – x2)

(b)

m3g

(c)

FIGURE 12.12 Free-body diagrams for the three masses from Fig. 12.11.

and c) that can be employed to derive m2

d 2 x2 dt 2

5 k(x3 2 x2 ) 1 m2g 2 2k(x2 2 x1 )

(12.17)

5 m3 g 2 k(x3 2 x2 )

(12.18)

and m3

d 2 x3 dt 2

330

CASE STUDIES: LINEAR ALGEBRAIC EQUATIONS

Equations (12.16), (12.17), and (12.18) form a system of three differential equations with three unknowns. With the appropriate initial conditions, they could be used to solve for the displacements of the masses as a function of time (that is, their oscillations). We will discuss numerical methods for obtaining such solutions in Part Seven. For the present, we can obtain the displacements that occur when the system eventually comes to rest, that is, to the steady state. To do this, the derivatives in Eqs. (12.16), (12.17), and (12.18) are set to zero to give 3k x1 22k x1

2 1 2

2k x2 3k x2 k x2

2 1

k x3 k x3

5 5 5

m1 g m2 g m3 g

or, in matrix form, [K]{X} 5 {W} where [K], called the stiffness matrix, is 3k [K ] 5 £ 22k

22k 3k 2k

2k § k

and {X} and {W} are the column vectors of the unknowns X and the weights mg, respectively. Solution. At this point, numerical methods can be employed to obtain a solution. If m1 5 2 kg, m2 5 3 kg, m3 5 2.5 kg, and the k’s 5 10 kg/s2, use LU decomposition to solve for the displacements and generate the inverse of [K]. Substituting the model parameters with g 5 9.81 gives 30 [K ] 5 £ 220

220 30 210

19.62 210 § {W} 5 • 29.43 ¶ 10 24.525

LU decomposition can be employed to solve for x1 5 7.36, x2 5 10.06, and x3 5 12.51. These displacements were used to construct Fig. 12.11b. The inverse of the stiffness matrix is computed as 0.1 [K ] 21 5 £ 0.1 0.1

0.1 0.15 0.15

0.1 0.15 § 0.25

Each element of this matrix k21 ji tells us the displacement of mass i due to a unit force imposed on mass j. Thus, the values of 0.1 in column 1 tell us that a downward unit load to the first mass will displace all of the masses 0.1 m downward. The other elements can be interpreted in a similar fashion. Therefore, the inverse of the stiffness matrix provides a fundamental summary of how the system’s components respond to externally applied forces.

PROBLEMS

331

PROBLEMS Chemical/Bio Engineering 12.1 Perform the same computation as in Sec. 12.1, but change c01 to 20 and c03 to 6. Also change the following flows: Q01 5 6, Q12 5 4, Q24 5 2, and Q44 5 12. 12.2 If the input to reactor 3 in Sec. 12.1 is decreased 25 percent, use the matrix inverse to compute the percent change in the concentration of reactors 2 and 4? 12.3 Because the system shown in Fig. 12.3 is at steady state, what can be said regarding the four flows: Q01, Q03, Q44, and Q55? 12.4 Recompute the concentrations for the five reactors shown in Fig. 12.3, if the flows are changed to Q01 5 5 Q31 5 3 Q25 5 2 Q23 5 2 Q15 5 4 Q55 5 3 Q54 5 3 Q34 5 7 Q12 5 4 Q03 5 8 Q24 5 0 Q44 5 10 12.5 Solve the same system as specified in Prob. 12.4, but set Q12 5 Q54 5 0 and Q15 5 Q34 5 3. Assume that the inflows (Q01, Q03) and outflows (Q44, Q55) are the same. Use conservation of flow to recompute the values for the other flows. 12.6 Figure P12.6 shows three reactors linked by pipes. As indicated, the rate of transfer of chemicals through each pipe is equal to a flow rate (Q, with units of cubic meters per second) multiplied by the concentration of the reactor from which the flow originates (c, with units of milligrams per cubic meter). If the system is at a steady state, the transfer into each reactor will balance the transfer out. Develop massbalance equations for the reactors and solve the three simultaneous linear algebraic equations for their concentrations. 12.7 Employing the same basic approach as in Sec. 12.1, determine the concentration of chloride in each of the Great Lakes using the information shown in Fig. P12.7. 12.8 The Lower Colorado River consists of a series of four reservoirs as shown in Fig. P12.8. Mass balances can be written for each FIGURE P12.6 Three reactors linked by pipes. The rate of mass transfer through each pipe is equal to the product of ﬂow Q and concentration c of the reactor from which the ﬂow originates.

400 mg/s

reservoir and the following set of simultaneous linear algebraic equations results: 13.442 213.442 ≥ 0 0

0 12.252 212.252 0

where the right-hand-side vector consists of the loadings of chloride to each of the four lakes and c1, c2, c3, and c4 5 the resulting chloride concentrations for Lakes Powell, Mead, Mohave, and Havasu, respectively. (a) Use the matrix inverse to solve for the concentrations in each of the four lakes. (b) How much must the loading to Lake Powell be reduced in order for the chloride concentration of Lake Havasu to be 75? (c) Using the column-sum norm, compute the condition number and how many suspect digits would be generated by solving this system. 12.9 A stage extraction process is depicted in Fig. P12.9. In such systems, a stream containing a weight fraction Yin of a chemical enters from the left at a mass flow rate of F1. Simultaneously, a solvent carrying a weight fraction Xin of the same chemical enters from the right at a flow rate of F2. Thus, for stage i, a mass balance can be represented as F1Yi21 1 F2 Xi11 5 F1Yi 1 F2 Xi

(P12.9.1)

At each stage, an equilibrium is assumed to be established between Yi and Xi as in K5

Xi Yi

(P12.9.2)

Q13c1

Q33c3 3

1

Q21c2

0 c1 750.5 0 c2 300 ∂ ¥μ ∂ 5 μ 0 102 c3 11.797 c4 30

0 0 12.377 212.377

Q12c1 2

Q23c2

200 mg/s Q33 Q13 Q12 Q23 Q21

= = = = =

120 40 80 60 20

332

CASE STUDIES: LINEAR ALGEBRAIC EQUATIONS

QSH = 67 QMH = 36 QHE = 161 QEO = 182 QOO = 212

180

740

QSHcS

Superior

3850

QHEcH 710

Huron

4720

QEOcE Superior Erie

FIGURE P12.7 A chloride balance for the Great Lakes. Numbered arrows are direct inputs.

Michigan QMHcM

QOOcO Ontario

12.10 An irreversible, first-order reaction takes place in four wellmixed reactors (Fig. P12.10),

Upper Colorado River

k

ASB Thus, the rate at which A is transformed to B can be represented as

c1 Lake Powell

Rab 5 kVc c2 Lake Mead

The reactors have different volumes, and because they are operated at different temperatures, each has a different reaction rate: c3 Lake Mohave

c4 Lake Havasu

FIGURE P12.8 The Lower Colorado River. where K is called a distribution coefficient. Equation (P12.9.2) can be solved for Xi and substituted into Eq. (P12.9.1) to yield Yi21 2 a1 1

F2 F2 Kb Yi 1 a Kb Yi11 5 0 F1 F1

(P12.9.3)

If F1 5 400 kg/h, Yin 5 0.1, F2 5 800 kg/h, Xin 5 0, and K 5 5, determine the values of Yout and Xout if a five-stage reactor is used. Note that Eq. (P12.9.3) must be modified to account for the inflow weight fractions when applied to the first and last stages.

Reactor

V, L

k, h21

1 2 3 4

25 75 100 25

0.05 0.1 0.5 0.1

Determine the concentration of A and B in each of the reactors at steady state. 12.11 A peristaltic pump delivers a unit flow (Q1) of a highly viscous fluid. The network is depicted in Fig. P12.11. Every pipe section has the same length and diameter. The mass and mechanical energy balance can be simplified to obtain the flows in every pipe. Solve the following system of equations to obtain the flow in every pipe. Q3 1 2Q4 2 2Q2 5 0 Q5 1 2Q6 2 2Q4 5 0 3Q7 2 2Q6 5 0

PROBLEMS

333

Flow = F2

xout

x2

x3

1

2 0

yin

xi

xi + 1 0i

•••

y1

y2

xn – 1

yi

xin

n–1

•••

yi – 1

xn 0 n

yn – 2

yn – 1

yout

Flow = F1

FIGURE P12.9 A stage extraction process.

Q32 = 5

cG0

QG

cG1

cG2

cG3

cG4

cG5

cL4

cL5

QG

D Qin = 10 cA,in = 1

1

2

3

4

Q43 = 3

QL

cL1

cL2

cL3

QL

cL6

FIGURE P12.12

FIGURE P12.10

Q1

Q3 Q2

is passed over a liquid flowing from right to left. The transfer of a chemical from the gas into the liquid occurs at a rate that is proportional to the difference between the gas and liquid concentrations in each reactor. At steady state, a mass balance for the first reactor can be written for the gas as

Q5 Q4

Q6

Q7

QG cG0 2 QG cG1 1 D(cL1 2 cG1 ) 5 0 and for the liquid as

FIGURE P12.11

Q1 5 Q2 1 Q3 Q3 5 Q4 1 Q5 Q5 5 Q6 1 Q7 12.12 Figure P12.12 depicts a chemical exchange process consisting of a series of reactors in which a gas flowing from left to right

QL cL2 2 QL cL1 1 D(cG1 2 cL1 ) 5 0 where QG and QL are the gas and liquid flow rates, respectively, and D 5 the gas-liquid exchange rate. Similar balances can be written for the other reactors. Solve for the concentrations given the following values: QG 5 2, QL 5 1, D 5 0.8, cG0 5 100, cL6 5 20. Civil/Environmental Engineering 12.13 A civil engineer involved in construction requires 4800, 5810, and 5690 m3 of sand, fine gravel, and coarse gravel, respectively, for

334

CASE STUDIES: LINEAR ALGEBRAIC EQUATIONS

a building project. There are three pits from which these materials can be obtained. The composition of these pits is

Pit 1 Pit 2 Pit 3

Sand %

Fine Gravel %

Coarse Gravel %

52 20 25

30 50 20

18 30 55

How many cubic meters must be hauled from each pit in order to meet the engineer’s needs? 12.14 Perform the same computation as in Sec. 12.2, but for the truss depicted in Fig. P12.14. 12.15 Perform the same computation as in Sec. 12.2, but for the truss depicted in Fig. P12.15. 12.16 Calculate the forces and reactions for the truss in Fig. 12.4 if a downward force of 2500 kg and a horizontal force to the right of 2000 kg are applied at node 1.

12.17 In the example for Fig. 12.4, where a 1000-lb downward force is applied at node 1, the external reactions V2 and V3 were calculated. But if the lengths of the truss members had been given, we could have calculated V2 and V3 by utilizing the fact that V2 1 V3 must equal 1000 and by summing moments around node 2. However, because we do know V2 and V3, we can work backward to solve for the lengths of the truss members. Note that because there are three unknown lengths and only two equations, we can solve for only the relationship between lengths. Solve for this relationship. 12.18 Employing the same methods as used to analyze Fig. 12.4, determine the forces and reactions for the truss shown in Fig. P12.18. 12.19 Solve for the forces and reaction for the truss in Fig. P12.19. Determine the matrix inverse for the system. Does the verticalmember force in the middle member seem reasonable? Why?

FIGURE P12.18 800 45⬚

FIGURE P12.14 600

30⬚ 1200

30⬚

250

30⬚

45⬚ 45⬚ 500

FIGURE P12.19 FIGURE P12.15 400

200 60⬚ 60⬚

45⬚

45⬚

60⬚

45⬚

30⬚ 3500

45⬚

PROBLEMS

335 Qd = 100 m3/hr

Qc = 150 m3/hr

cb = 2 mg/m3

FIGURE P12.20 Overhead view of rooms in a restaurant. The one-way arrows represent volumetric airﬂows, whereas the two-way arrows represent diffusive mixing. The smoker and grill loads add carbon monoxide mass to the system but negligible airﬂow.

Qa = 200 m3/hr ca = 2 mg/m3

25 m3/hr

2 (Kids' section)

4

25 m3/hr

1 (Smoking section)

50 m3/hr

Qb = 50 m3/hr

3

Smoker load (1000 mg/hr) Grill load (2000 mg/hr)

12.20 As the name implies, indoor air pollution deals with air contamination in enclosed spaces such as homes, offices, work areas, etc. Suppose that you are designing a ventilation system for a restaurant as shown in Fig. P12.20. The restaurant serving area consists of two square rooms and one elongated room. Room 1 and room 3 have sources of carbon monoxide from smokers and a faulty grill, respectively. Steady-state mass balances can be written for each room. For example, for the smoking section (room 1), the balance can be written as

D

0 5 Wsmoker 1 Qaca 2 Qa c1 1 E13 (c3 2 c1 )

(load) 1 (inflow) 2 (outflow) 1

(mixing)

or substituting the parameters

B

225c1 2 25c3 5 2400 Similar balances can be written for the other rooms. (a) Solve for the steady-state concentration of carbon monoxide in each room. (b) Determine what percent of the carbon monoxide in the kids’ section is due to (i) the smokers, (ii) the grill, and (iii) the air in the intake vents. (c) If the smoker and grill loads are increased to 2000 and 5000 mg/hr, respectively, use the matrix inverse to determine the increase in the concentration in the kids’ section. (d) How does the concentration in the kids’ area change if a screen is constructed so that the mixing between areas 2 and 4 is decreased to 5 m3/hr? 12.21 An upward force of 20 kN is applied at the top of a tripod as depicted in Fig. P12.21. Determine the forces in the legs of the tripod.

A

0.6

2.4 m C

m

x

0.8

m

1m

8

0.

m

y

FIGURE P12.21

12.22 A truss is loaded as shown in Fig. P12.22. Using the following set of equations, solve for the 10 unknowns: AB, BC, AD, BD, CD, DE, CE, Ax, Ay, and Ey.

336

CASE STUDIES: LINEAR ALGEBRAIC EQUATIONS

R

B

54 kN

R = 30 ⍀

3

24 kN C

R=5⍀

=7

R = 35 ⍀

2

⍀

4m

A

R=5⍀ 5

6

3m

3m

D

5⍀

4

E

0⍀

7

10 ⍀ 3

R = 10 ⍀

2

R=2⍀

2

60 ⍀ 8

5

1 V1 = 110

R = 25 ⍀ 5

10 ⍀ 20 ⍀

V1 = 200 volts

15 ⍀

6

5⍀ 1

R=5⍀

R=5⍀ 4

9

5⍀

FIGURE P12.22

R = 20 ⍀

V6 = 200 volts

FIGURE P12.24

20 ⍀

3

V1 = 10 volts

R = 10 ⍀

R = 18 ⍀ 4

1

6

V6 = 0 volts

V2 = 40

FIGURE P12.25

FIGURE P12.23 Metal, Plastic, Rubber, Component g/component g/component g/component

Ax 1 AD 5 0 Ay 1 AB 5 0 74 1 BC 1 (3y5)BD 5 0 2AB 2 (4y5)BD 5 0 2BC 1 (3y5)CE 5 0

224 2 CD 2 (4y5)CE 5 0 2AD 1 DE 2 (3y5)BD 5 0 CD 1 (4y5)BD 5 0 2DE 2 (3y5)CE 5 0 Ey 1 (4y5)CE 5 0

Electrical Engineering 12.23 Perform the same computation as in Sec. 12.3, but for the circuit depicted in Fig. P12.23. 12.24 Perform the same computation as in Sec. 12.3, but for the circuit depicted in Fig. P12.24. 12.25 Solve the circuit in Fig. P12.25 for the currents in each wire. Use Gauss elimination with pivoting. 12.26 An electrical engineer supervises the production of three types of electrical components. Three kinds of material—metal, plastic, and rubber—are required for production. The amounts needed to produce each component are

1 2 3

15 17 19

0.30 0.40 0.55

1.0 1.2 1.5

If totals of 3.89, 0.095, and 0.282 kg of metal, plastic, and rubber, respectively, are available each day, how many components can be produced per day? 12.27 Determine the currents for the circuit in Fig. P12.27. 12.28 Determine the currents for the circuit in Fig. P12.28. 12.29 The following system of equations was generated by applying the mesh current law to the circuit in Fig. P12.29: 55I1 2 25I4 5 2200 237I3 2 4I4 5 2250 225I1 2 4I3 1 29I4 5 100 Solve for I1, I3, and I4.

PROBLEMS 10 ⍀

5⍀

80 V + –

15 ⍀

337

I1

+ 50 V –

25 ⍀

10 A

10 ⍀

20 ⍀

20 ⍀

25 ⍀

FIGURE P12.27

100 V + –

6⍀

I2

25 ⍀

I4

4⍀

I3

8⍀

FIGURE P12.29

j2 4⍀

20 V + –

i1

Mechanical/Aerospace Engineering 12.31 Perform the same computation as in Sec. 12.4, but add a third spring between masses 1 and 2 and triple k for all springs. 12.32 Perform the same computation as in Sec. 12.4, but change the masses from 2, 3, and 2.5 kg to 10, 3.5, and 2 kg, respectively. 12.33 Idealized spring-mass systems have numerous applications throughout engineering. Figure P12.33 shows an arrangement of four springs in series being depressed with a force of 2000 kg. At equilibrium, force-balance equations can be developed defining the interrelationships between the springs,

8⍀

2⍀

i3

5⍀

FIGURE P12.28

k2 (x2 2 x1 ) 5 k1x1 k3 (x3 2 x2 ) 5 k2 (x2 2 x1 ) k4 (x4 2 x3 ) 5 k3 (x3 2 x2 )

12.30 The following system of equations was generated by applying the mesh current law to the circuit in Fig. P12.30:

F 5 k4 (x4 2 x3 )

60I1 2 40I2 5 200

where the k’s are spring constants. If k1 through k4 are 150, 50, 75, and 225 N/m, respectively, compute the x’s. 12.34 Three blocks are connected by a weightless cord and rest on an inclined plane (Fig. P12.34a). Employing a procedure similar to the one used in the analysis of the falling parachutists in Example

240I1 1 150I2 2 100I3 5 0 2100I2 1 130I3 5 230 Solve for I1, I2, and I3.

20 ⍀

FIGURE P12.30

200 V + –

I1

10 ⍀

40 ⍀

I2

80 V –+

100 ⍀

I3

30 ⍀

I4

10 A

338

CASE STUDIES: LINEAR ALGEBRAIC EQUATIONS x

9.11 yields the following set of simultaneous equations (free-body diagrams are shown in Fig. P12.34b):

F

100a 1 T 5 519.72 x4

50a 2 T 1 R 5 216.55 2 R 5 108.28

25a

k4

Solve for acceleration a and the tensions T and R in the two ropes. 12.35 Perform a computation similar to that called for in Prob. 12.34, but for the system shown in Fig. P12.35. 12.36 Perform the same computation as in Prob. 12.34, but for the system depicted in Fig. P12.36 (angles are 458). 12.37 Consider the three mass-four spring system in Fig. P12.37. Determining the equations of motion from gFx 5 ma, for each mass using its free-body diagram results in the following differential equations:

x3 k3 x2 k2 x1 k1

k1 1 k2 k2 x$1 1 a b x1 2 a b x2 5 0 m1 m1

0

k2 1 k3 k3 k2 x$2 2 a b x1 1 a b x2 2 a b x3 5 0 m2 m2 m2

FIGURE P12.33

k3 k3 1 k4 x$3 2 a b x2 1 a b x3 5 0 m3 m3

10

0

kg

50

a,

kg

ac

ce

le

ra

25

tio

n

kg

FIGURE P12.34

45⬚

692.96 100 ⫻ 9.8 = 980

T

50 ⫻ 9.8 = 490

(b)

R

.9 7 64 = 37 5 17 3.

24

⫻

24 17 3.

48 34 6.

346.48

0.

37 5 0. ⫻

48 34 6.

⫻ 96 69 2.

69 2.

96

0.

25

=

=

R

17

T

12

3.

24

9. 93

(a)

173.24

25 ⫻ 9.8 = 245

PROBLEMS

g 0k

Ta = 10

kg

Friction = 0.2

T0 = 40

g

Friction = 0.5 Friction = 0.3

4

30⬚

k 50

10

339

T5 = 200

⌬x Ta = 10

60⬚

x=0

x = 10

FIGURE P12.38 A noninsulated uniform rod positioned between two walls of constant but different temperature. The ﬁnite difference representation employs four interior nodes.

kg

FIGURE P12.35

where T 5 temperature (8C), x 5 distance along the rod (m), h9 5 a heat transfer coefficient between the rod and the ambient air (m22), and Ta 5 the temperature of the surrounding air (8C). This equation can be transformed into a set of linear algebraic equations by using a finite divided difference approximation for the second derivative (recall Section 4.1.3),

kg

10

8 kg

15

Friction = 0.2 Friction = 0.8

5 kg

FIGURE P12.36

k1

Ti11 2 2Ti 1 Ti21 d 2T 5 dx2 ¢x2 x1

m1

k2

x2 m2

k3

x3 m3

where Ti designates the temperature at node i. This approximation can be substituted into Eq. (P12.38.1) to give k4

FIGURE P12.37

where k1 5 k4 5 10 N/m, k2 5 k3 5 30 N/m, and m1 5 m2 5 m3 5 2 kg. Write the three equations in matrix form: 0 5 [Acceleration vector] 1 [k/m matrix] [displacement vector x] At a specific time when x1 5 0.05 m, x2 5 0.04 m, and x3 5 0.03 m, this forms a tridiagonal matrix. Solve for the acceleration of each mass. 12.38 Linear algebraic equations can arise in the solution of differential equations. For example, the following differential equation derives from a heat balance for a long, thin rod (Fig. P12.38): d 2T 1 h¿(Ta 2 T) 5 0 dx2

(P12.38.1)

2Ti21 1 (2 1 h¿¢x2 )Ti 2 Ti11 5 h¿¢x2Ta This equation can be written for each of the interior nodes of the rod resulting in a tridiagonal system of equations. The first and last nodes at the rod’s ends are fixed by boundary conditions. (a) Develop an analytical solution for Eq. (P12.38.1) for a 10-m rod with Ta 5 20, T(x 5 0) 5 40, T(x 5 10) 5 200, and h9 5 0.02. (b) Develop a numerical solution for the same parameter values employed in (a) using a finite-difference solution with four interior nodes as shown in Fig. P12.38 (Dx 5 2 m). 12.39 The steady-state distribution of temperature on a heated plate can be modeled by the Laplace equation, 05

0 2T 0 2T 1 2 2 0x 0y

If the plate is represented by a series of nodes (Fig. P12.39), centered finite-divided differences can be substituted for the second derivatives, which results in a system of linear algebraic equations. Use the Gauss-Seidel method to solve for the temperatures of the nodes in Fig. P12.39.

340

CASE STUDIES: LINEAR ALGEBRAIC EQUATIONS 25⬚C

200⬚C

200⬚C

25⬚C

T12

T22

T11

T21

z

0⬚C

50

2m

N

0⬚C

y

2m 75⬚C

75⬚C

2m

Ball and socket

B

FIGURE P12.39

12.40 A rod on a ball and socket joint is attached to cables A and B, as in Fig. P12.40. (a) If a 50-N force is exerted on the massless rod at G, what is the tensile force at cables A and B? (b) Solve for the reactant forces at the base of the rod. Call the base point P.

1m

FIGURE P12.40

2m

A

1m

x

27.1

CURRENT 1ST LEVEL HEAD

341

EPILOGUE: PART THREE PT3.4

TRADE-OFFS Table PT3.2 provides a summary of the trade-offs involved in solving simultaneous linear algebraic equations. Two methods—graphical and Cramer’s rule—are limited to small (# 3) numbers of equations and thus have little utility for practical problem solving. However, these techniques are useful didactic tools for understanding the behavior of linear systems in general. The numerical methods themselves are divided into two general categories: exact and approximate methods. As the name implies, the former are intended to yield exact answers. However, because they are affected by round-off errors, they sometimes yield imprecise results. The magnitude of the round-off error varies from system to system and is dependent on a number of factors. These include the system’s dimensions, its condition, and whether the matrix of coefficients is sparse or full. In addition, computer precision will affect round-off error. It is recommended that a pivoting strategy be employed in any computer program implementing exact elimination methods. The inclusion of such a strategy minimizes round-off error and avoids problems such as division by zero. All other things being equal, LU decomposition–based algorithms are the methods of choice because of their efficiency and flexibility.

TABLE PT3.2 Comparison of the characteristics of alternative methods for ﬁnding solutions of simultaneous linear algebraic equations. Method

Stability

Precision

Breadth of Application

Programming Effort Comments

Graphical

—

Poor

Limited

—

Cramer’s rule

—

Affected by Limited round-off error

—

Gauss elimination (with — partial pivoting) LU decomposition —

Affected by General round-off error Affected by General round-off error May not Excellent Appropriate only converge if not for diagonally diagonally dominant dominant systems

Moderate

Gauss-Seidel

Easy

Moderate

May take more time than the numerical method, but can be useful for visualization Excessive computational effort required for more than three equations

Preferred elimination method; allows computation of matrix inverse

341

342

EPILOGUE: PART THREE

Although elimination methods have great utility, their use of the entire matrix of coefficients can be somewhat limiting when dealing with very large, sparse systems. This is due to the fact that large portions of computer memory would be devoted to storage of meaningless zeros. For banded systems, techniques are available to implement elimination methods without having to store the entire coefficient matrix. The approximate technique described in this book is called the Gauss-Seidel method. It differs from the exact techniques in that it employs an iterative scheme to obtain progressively closer estimates of the solution. Thus, the effect of round-off is a moot point with the Gauss-Seidel method because the iterations can be continued as long as is necessary to obtain the desired precision. In addition, versions of the GaussSeidel method can be developed to efficiently utilize computer storage requirements for sparse systems. Consequently, the Gauss-Seidel technique has utility for large systems of equations where storage requirements would pose significant problems for the exact techniques. The disadvantage of the Gauss-Seidel method is that it does not always converge or sometimes converges slowly on the true solution. It is strictly reliable only for those systems that are diagonally dominant. However, relaxation methods are available that sometimes offset these disadvantages. In addition, because many sets of linear algebraic equations originating from physical systems exhibit diagonal dominance, the GaussSeidel method has great utility for engineering problem solving. In summary, a variety of factors will bear on your choice of a technique for a particular problem involving linear algebraic equations. However, as outlined above, the size and sparseness of the system are particularly important factors in determining your choice.

PT3.5

IMPORTANT RELATIONSHIPS AND FORMULAS Every part of this book includes a section that summarizes important formulas. Although Part Three does not really deal with single formulas, we have used Table PT3.3 to summarize the algorithms that were covered. The table provides an overview that should be helpful for review and in elucidating the major differences between the methods.

PT3.6

ADVANCED METHODS AND ADDITIONAL REFERENCES General references on the solution of simultaneous linear equations can be found in Fadeev and Fadeeva (1963), Stewart (1973), Varga (1962), and Young (1971). Ralston and Rabinowitz (1978) provide a general summary. Many advanced techniques are available to increase the savings in time and/or space when solving linear algebraic equations. Most of these focus on exploiting properties of the equations such as symmetry and bandedness. In particular, algorithms are available to operate on sparse matrices to convert them to a minimum banded format. Jacobs (1977) and Tewarson (1973) include information on this area. Once they are in a minimum banded format, there are a variety of efficient solution strategies that are employed such as the active column storage approach of Bathe and Wilson (1976). Aside from n 3 n sets of equations, there are other systems where the number of equations, m, and number of unknowns, n, are not equal. Systems where m , n are called underdetermined. In such cases, there can be either no solution or else more than

PT3.6

ADVANCED METHODS AND ADDITIONAL REFERENCES

343

TABLE PT3.3 Summary of important information presented in Part Three.

Method

Procedure

Gauss elimination

a11 £ a21 a31

a12 a22 a32

Potential Problems and Remedies a13 a23 a33

a11 0 c1 0 c2 § 1 £ 0 c3

a12 a'22

a13 a'23 a''33

x3 5 c''3ya''33 Z c1 Z c'2 § 1 x2 5 1c'2 2 a'23x3 2ya'22 Z c''3 x1 5 1c1 2 a12x1 2 a13x3 2ya11

Decomposition

LU decomposition

a11 £ a21 a31

a12 a22 a32

a13 1 a23 § 1 £ l21 a33 l31

0 1 l32

Back Substitution

0 d1 c1 u11 0 § • d2 ¶ 5 • c2 ¶ 1 £ 0 1 d3 c3 0

u12 u22 0

u13 x1 d1 x1 u23 § • x2 ¶ 5 • d2 ¶ 5 • x2 ¶ u33 x3 d3 x3

Forward Substitution Gauss-Seidel method

Problems: III conditioning Round-off Division by zero Remedies: Higher precision Partial pivoting

continue iteratively until x1i 5 (c1 2 a12x2i21 2 a13x3i21 )ya11 xii 2 xii21 i i i21 x2 5 (c2 2 a21x1 2 a23x3 )ya22 ¶ ` ` 100% , es xii x3i 5 (c3 2 a31x1i 2 a32x2i )ya33 for all x'i s

Problems: III conditioning Round-off Division by zero Remedies: Higher precision Partial pivoting Problems: Divergent or converges slowly Remedies: Diagonal dominance Relaxation

one. Systems where m . n are called overdetermined. For such situations, there is in general no exact solution. However, it is often possible to develop a compromise solution that attempts to determine answers that come “closest” to satisfying all the equations simultaneously. A common approach is to solve the equation in a “least-squares” sense (Lawson and Hanson, 1974; Wilkinson and Reinsch, 1971). Alternatively, linear programming methods can be used where the equations are solved in an “optimal” sense by minimizing some objective function (Dantzig, 1963; Luenberger, 1984; and Rabinowitz, 1968). We describe this approach in detail in Chap. 15.

PART FOUR

OPTIMIZATION PT4.1

MOTIVATION Root location (Part 2) and optimization are related in the sense that both involve guessing and searching for a point on a function. The fundamental difference between the two types of problems is illustrated in Fig. PT4.1. Root location involves searching for zeros of a function or functions. In contrast, optimization involves searching for either the minimum or the maximum. The optimum is the point where the curve is flat. In mathematical terms, this corresponds to the x value where the derivative f9(x) is equal to zero. Additionally, the second derivative, f 0(x), indicates whether the optimum is a minimum or a maximum: if f 0(x) , 0, the point is a maximum; if f 0(x) . 0, the point is a minimum. Now, understanding the relationship between roots and optima would suggest a possible strategy for finding the latter. That is, you can differentiate the function and locate the root (that is, the zero) of the new function. In fact, some optimization methods seek to find an optima by solving the root problem: f9(x) 5 0. It should be noted that such searches are often complicated because f9(x) is not available analytically. Thus, one must sometimes use finite-difference approximations to estimate the derivative. Beyond viewing optimization as a roots problem, it should be noted that the task of locating optima is aided by some extra mathematical structure that is not part of simple root finding. This tends to make optimization a more tractable task, particularly for multidimensional cases.

FIGURE PT4.1 A function of a single variable illustrating the difference between roots and optima.

f (x)

0

f ⬘(x) = 0 f ⬙(x) ⬍ 0 f (x) = 0

Maximum

Root Root

Minimum

Root

x

f ⬘(x) = 0 f ⬙(x) ⬎ 0

345

346

OPTIMIZATION

PT4.1.1 Noncomputer Methods and History As mentioned above, differential calculus methods are still used to determine optimum solutions. All engineering and science students recall working maxima-minima problems by determining first derivatives of functions in their calculus courses. Bernoulli, Euler, Lagrange, and others laid the foundations of the calculus of variations, which deals with the minimization of functions. The Lagrange multiplier method was developed to optimize constrained problems, that is, optimization problems where the variables are bounded in some way. The first major advances in numerical approaches occurred only with the development of digital computers after World War II. Koopmans in the United Kingdom and Kantorovich in the former Soviet Union independently worked on the general problem of least-cost distribution of supplies and products. In 1947, Koopman’s student Dantzig invented the simplex procedure for solving linear programming problems. This approach paved the way for other methods of constrained optimization by a number of investigators, notably Charnes and his coworkers. Approaches for unconstrained optimization also developed rapidly following the widespread availability of computers. PT4.1.2 Optimization and Engineering Practice Most of the mathematical models we have dealt with to this point have been descriptive models. That is, they have been derived to simulate the behavior of an engineering device or system. In contrast, optimization typically deals with finding the “best result,” or optimum solution, of a problem. Thus, in the context of modeling, they are often termed prescriptive models since they can be used to prescribe a course of action or the best design. Engineers must continuously design devices and products that perform tasks in an efficient fashion. In doing so, they are constrained by the limitations of the physical world. Further, they must keep costs down. Thus, they are always confronting optimization problems that balance performance and limitations. Some common instances are listed in Table PT4.1. The following example has been developed to help you get a feel for the way in which such problems might be formulated. TABLE PT4.1 Some common examples of optimization problems in engineering. • • • • • • • • • • • • • • • •

Design aircraft for minimum weight and maximum strength. Optimal trajectories of space vehicles. Design civil engineering structures for minimum cost. Design water-resource projects like dams to mitigate ﬂood damage while yielding maximum hydropower. Predict structural behavior by minimizing potential energy. Material-cutting strategy for minimum cost. Design pump and heat transfer equipment for maximum efﬁciency. Maximize power output of electrical networks and machinery while minimizing heat generation. Shortest route of salesperson visiting various cities during one sales trip. Optimal planning and scheduling. Statistical analysis and models with minimum error. Optimal pipeline networks. Inventory control. Maintenance planning to minimize cost. Minimize waiting and idling times. Design waste treatment systems to meet water-quality standards at least cost.

PT4.1

EXAMPLE PT4.1

MOTIVATION

347

Optimization of Parachute Cost Problem Statement. Throughout the rest of the book, we have used the falling parachutist to illustrate the basic problem areas of numerical methods. You may have noticed that none of these examples concentrate on what happens after the chute opens. In this example, we will examine a case where the chute has opened and we are interested in predicting impact velocity at the ground. You are an engineer working for an agency planning to airlift supplies to refugees in a war zone. The supplies will be dropped at low altitude (500 m) so that the drop is not detected and the supplies fall as close as possible to the refugee camp. The chutes open immediately upon leaving the plane. To reduce damage, the vertical velocity on impact must be below a critical value of yc 5 20 m/s. The parachute used for the drop is depicted in Fig. PT4.2. The cross-sectional area of the chute is that of a half sphere, A 5 2pr2

(PT4.1)

The length of each of the 16 cords connecting the chute to the mass is related to the chute radius by / 5 12r

(PT4.2)

You know that the drag force for the chute is a linear function of its cross-sectional area described by the following formula c 5 kc A

(PT4.3)

where c 5 drag coefficient (kg/s) and kc 5 a proportionality constant parameterizing the effect of area on drag [kg/(s ? m2)]. Also, you can divide the payload into as many parcels as you like. That is, the mass of each individual parcel can be calculated as m5

Mt n

FIGURE PT4.2 A deployed parachute. r

ᐉ

m

348

OPTIMIZATION

where m 5 mass of an individual parcel (kg), Mt 5 total load being dropped (kg), and n 5 total number of parcels. Finally, the cost of each chute is related to chute size in a nonlinear fashion, Cost per chute 5 c0 1 c1/ 1 c2A2

(PT4.4)

where c0, c1, and c2 5 cost coefficients. The constant term, c0, is the base price for the chutes. The nonlinear relationship between cost and area exists because larger chutes are much more difficult to construct than small chutes. Determine the size (r) and number of chutes (n) that result in minimum cost while at the same time meeting the requirement of having a sufficiently small impact velocity. Solution. The objective here is to determine the number and size of parachutes to minimize the cost of the airlift. The problem is constrained because the parcels must have an impact velocity less than a critical value. The cost can be computed by multiplying the cost of the individual parachute [Eq. (PT4.4)] by the number of parachutes (n). Thus, the function you wish to minimize, which is formally called the objective function, is written as Minimize C 5 n(c0 1 c1/ 1 c2A2 )

(PT4.5)

where C 5 cost ($) and A and / are calculated by Eqs. (PT4.1) and (PT4.2), respectively. Next, we must specify the constraints. For this problem there are two constraints. First, the impact velocity must be equal to or less than the critical velocity, y # yc

(PT4.6)

Second, the number of parcels must be an integer and greater than or equal to 1, n$1

(PT4.7)

where n is an integer. At this point, the optimization problem has been formulated. As can be seen, it is a nonlinear constrained problem. Although the problem has been broadly formulated, one more issue must be addressed: How do we determine the impact velocity y? Recall from Chap. 1 that the velocity of a falling object can be computed with y5

gm ( 1 2 e 2 (cym)t ) c

(1.10)

where y 5 velocity (m/s), g 5 acceleration of gravity (m/s2), m 5 mass (kg), and t 5 time (s). Although Eq. (1.10) provides a relationship between y and t, we need to know how long the mass falls. Therefore, we need a relationship between the drop distance z and the time of fall t. The drop distance can be calculated from the velocity in Eq. (1.10) by integration z5

#

t

gm ( 1 2 e2(c/m)t ) dt c 0

(PT4.8)

This integral can be evaluated to yield z 5 z0 2

gm gm2 t 1 2 ( 1 2 e2(c/m)t ) c c

(PT4.9)

PT4.1

MOTIVATION

349

600

400

z (m)

200

Impact v (m/s)

0

0

5

10

15

t (s)

FIGURE PT4.3 The height z and velocity v of a deployed parachute as it falls to earth (z 5 0).

where z0 5 initial height (m). This function, as plotted in Fig. PT4.3, provides a way to predict z given knowledge of t. However, we do not need z as a function of t to solve this problem. Rather, we need to compute the time required for the parcel to fall the distance z0. Thus, we recognize that we must reformulate Eq. (PT4.9) as a root-finding problem. That is, we must solve for the time at which z goes to zero, f(t) 5 0 5 z0 2

gm gm2 t 1 2 ( 1 2 e2(cym)t ) c c

(PT4.10)

Once the time to impact is computed, we can substitute it into Eq. (1.10) to solve for the impact velocity. The final specification of the problem, therefore, would be Minimize C 5 n(c0 1 c1/ 1 c2 A2 )

(PT4.11)

subject to y # yc n$1

(PT4.12) (PT4.13)

where A 5 2pr2

(PT4.14)

/ 5 12r c 5 kc A

(PT4.15)

m5

Mt n

(PT4.16) (PT4.17)

350

OPTIMIZATION

t 5 root c z0 2 y5

gm2 gm t 1 2 ( 1 2 e2(cym)t ) d c c

gm ( 1 2 e2(cym)t ) c

(PT4.18) (PT4.19)

We will solve this problem in Example 15.4 in Chap. 15. For the time being recognize that it has most of the fundamental elements of other optimization problems you will routinely confront in engineering practice. These are • The problem will involve an objective function that embodies your goal. • There will be a number of design variables. These variables can be real numbers or they can be integers. In our example, these are r (real) and n (integer). • The problem will include constraints that reflect the limitations you are working under. We should make one more point before proceeding. Although the objective function and constraints may superficially appear to be simple equations [e.g., Eq. (PT4.12)], they may in fact be the “tip of the iceberg.” That is, they may be underlain by complex dependencies and models. For instance, as in our example, they may involve other numerical methods [Eq. (PT4.18)]. This means that the functional relationships you will be using could actually represent large and complicated calculations. Thus, techniques that can find the optimal solution, while minimizing function evaluations, can be extremely valuable.

PT4.2

MATHEMATICAL BACKGROUND There are a number of mathematical concepts and operations that underlie optimization. Because we believe that they will be more relevant to you in context, we will defer discussion of specific mathematical prerequisites until they are needed. For example, we will discuss the important concepts of the gradient and Hessians at the beginning of Chap. 14 on multivariate unconstrained optimization. In the meantime, we will limit ourselves here to the more general topic of how optimization problems are classified. An optimization or mathematical programming problem generally can be stated as: Find x, which minimizes or maximizes f(x) subject to di (x) # ai ei (x) 5 bi

i 5 1, 2, p , m i 5 1, 2, p , p

(PT4.20) (PT4.21)

where x is an n-dimensional design vector, f (x) is the objective function, di(x) are inequality constraints, ei(x) are equality constraints, and ai and bi are constants. Optimization problems can be classified on the basis of the form of f(x): • If f(x) and the constraints are linear, we have linear programming. • If f(x) is quadratic and the constraints are linear, we have quadratic programming. • If f(x) is not linear or quadratic and/or the constraints are nonlinear, we have nonlinear programming.

PT4.3

ORIENTATION

351

f (x) f (x)

x*

Minimum f (x) Maximum – f (x)

y

Optimum f (x*, y*) f (x, y)

y* x

– f (x)

(a)

(b)

x*

x

FIGURE PT4.4 (a) One-dimensional optimization. This ﬁgure also illustrates how minimization of f(x) is equivalent to the maximization of 2f(x). (b) Two-dimensional optimization. Note that this ﬁgure can be taken to represent either a maximization (contours increase in elevation up to the maximum like a mountain) or a minimization (contours decrease in elevation down to the minimum like a valley).

Further, when Eqs. (PT4.20) and (PT4.21) are included, we have a constrained optimization problem; otherwise, it is an unconstrained optimization problem. Note that for constrained problems, the degrees of freedom are given by n2p2m. Generally, to obtain a solution, p 1 m must be # n. If p 1 m . n, the problem is said to be overconstrained. Another way in which optimization problems are classified is by dimensionality. This is most commonly done by dividing them into one-dimensional and multidimensional problems. As the name implies, one-dimensional problems involve functions that depend on a single dependent variable. As in Fig. PT4.4a, the search then consists of climbing or descending one-dimensional peaks and valleys. Multidimensional problems involve functions that depend on two or more dependent variables. In the same spirit, a two-dimensional optimization can again be visualized as searching out peaks and valleys (Fig. PT4.4b). However, just as in real hiking, we are not constrained to walk a single direction, instead the topography is examined to efficiently reach the goal. Finally, the process of finding a maximum versus finding a minimum is essentially identical because the same value, x*, both minimizes f(x) and maximizes 2f(x). This equivalence is illustrated graphically for a one-dimensional function in Fig. PT4.4a.

PT4.3

ORIENTATION Some orientation is helpful before proceeding to the numerical methods for optimization. The following is intended to provide an overview of the material in Part Four. In addition, some objectives have been included to help you focus your efforts when studying the material.

352

OPTIMIZATION

PT4.3.1 Scope and Preview Figure PT4.5 is a schematic representation of the organization of Part Four. Examine this figure carefully, starting at the top and working clockwise. After the present introduction, Chap. 13 is devoted to one-dimensional unconstrained optimization. Methods are presented to find the minimum or maximum of a function of a single variable. Three methods are covered: golden-section search, parabolic interpolation, and Newton’s method. An advanced hybrid approach, Brent’s method, that combines the reliability of the golden-section search with the speed of parabolic interpolation is also described. Chapter 14 covers two general types of methods to solve multidimensional unconstrained optimization problems. Direct methods such as random searches, univariate searches, and pattern searches do not require the evaluation of the function’s derivatives. On the other hand, gradient methods use either first and sometimes second derivatives to find the optimum. The chapter introduces the gradient and the Hessian, which are multidimensional representations of the first and second derivatives. The method of steepest ascent/descent is then covered in some detail. This is followed by descriptions of some advanced methods: conjugate gradient, Newton’s method, Marquardt’s method, and quasi-Newton methods. Chapter 15 is devoted to constrained optimization. Linear programming is described in detail using both a graphical representation and the simplex method. The detailed analysis of nonlinear constrained optimization is beyond this book’s scope, but we provide an overview of the major approaches. In addition, we illustrate how such problems (along with the problems covered in Chaps. 13 and 14) can be obtained with software packages such as Excel, MATLAB, and Mathcad. Chapter 16 extends the above concepts to actual engineering problems. Engineering applications are used to illustrate how optimization problems are formulated and provide insight into the application of the solution techniques in professional practice. An epilogue is included at the end of Part Four. It contains an overview of the methods discussed in Chaps. 13, 14, and 15. This overview includes a description of trade-offs related to the proper use of each technique. This section also provides references for some numerical methods that are beyond the scope of this text. PT4.3.2 Goals and Objectives Study Objectives. After completing Part Four, you should have sufficient information to successfully approach a wide variety of engineering problems dealing with optimization. In general, you should have mastered the techniques, have learned to assess their reliability, and be capable of analyzing alternative methods for any particular problem. In addition to these general goals, the specific concepts in Table PT4.2 should be assimilated for a comprehensive understanding of the material in Part Four. Computer Objectives. You should be able to write a subprogram to implement a simple one-dimensional (like golden-section search or parabolic interpolation) and multidimensional (like the random-search method) search. In addition, software packages such as Excel, MATLAB, or Mathcad have varying capabilities for optimization. You can use this part of the book to become familiar with these capabilities.

PT4.3

ORIENTATION

PT 4.1 Motivation

353

PT 4.2 Mathematical background

PT 4.3 Orientation

13.1 Golden-section search

PART 4 Optimization

13.2 Parabolic interpolation

PT 4.5 Additional references

CHAPTER 13 One-Dimensional Unconstrained Optimization

EPILOGUE PT 4.4 Trade-offs

13.4 Brent's method

16.4 Mechanical engineering

16.3 Electrical engineering

13.3 Newton's method

CHAPTER 14 Multidimensional Unconstrained Optimization

CHAPTER 16 Case Studies

16.2 Civil engineering

14.2 Gradient methods

CHAPTER 15 Constrained Optimization

16.1 Chemical engineering

15.3 Software packages

15.1 Linear programming 15.2 Nonlinear constrained

FIGURE PT4.5 Schematic of the organization of the material in Part Four: Optimization.

14.1 Direct methods

354

OPTIMIZATION TABLE PT4.2 Speciﬁc study objectives for Part Four. 1. Understand why and where optimization occurs in engineering problem solving. 2. Understand the major elements of the general optimization problem: objective function, decision variables, and constraints. 3. Be able to distinguish between linear and nonlinear optimization, and between constrained and unconstrained problems. 4. Be able to deﬁne the golden ratio and understand how it makes one-dimensional optimization efﬁcient. 5. Locate the optimum of a single variable function with the golden-section search, parabolic interpolation, and Newton’s method. Also, recognize the trade-offs among these approaches, with particular attention to initial guesses and convergence. 6. Understand how Brent’s optimization method combines the reliability of the golden-section search with the speed of parabolic interpolation. 7. Be capable of writing a program and solving for the optimum of a multivariable function using random searching. 8. Understand the ideas behind pattern searches, conjugate directions, and Powell’s method. 9. Be able to deﬁne and evaluate the gradient and Hessian of a multivariable function both analytically and numerically. 10. Compute by hand the optimum of a two-variable function using the method of steepest ascent/ descent. 11. Understand the basic ideas behind the conjugate gradient, Newton’s, Marquardt’s, and quasiNewton methods. In particular, understand the trade-offs among the approaches and recognize how each improves on the steepest ascent/descent. 12. Be capable of recognizing and setting up a linear programming problem to represent applicable engineering problems. 13. Be able to solve a two-dimensional linear programming problem with both the graphical and simplex methods. 14. Understand the four possible outcomes of a linear programming problem. 15. Be able to set up and solve nonlinear constrained optimization problems using a software package.

CHAPTER

13 One-Dimensional Unconstrained Optimization This section will describe techniques to find the minimum or maximum of a function of a single variable, f(x). A useful image in this regard is the one-dimensional, “roller coaster”– like function depicted in Fig. 13.1. Recall from Part Two that root location was complicated by the fact that several roots can occur for a single function. Similarly, both local and global optima can occur in optimization. Such cases are called multimodal. In almost all instances, we will be interested in finding the absolute highest or lowest value of a function. Thus, we must take care that we do not mistake a local result for the global optimum. Distinguishing a global from a local extremum can be a very difficult problem for the general case. There are three usual ways to approach this problem. First, insight into the behavior of low-dimensional functions can sometimes be obtained graphically. Second, finding optima based on widely varying and perhaps randomly generated starting guesses, and then selecting the largest of these as global. Finally, perturbing the starting point associated with a local optimum and seeing if the routine returns a better point or always returns to the same point. Although all these approaches can have utility, the fact is that in some problems (usually the large ones), there may be no practical way to ensure that you have located a global optimum. However, although you should always

FIGURE 13.1 A function that asymptotically approaches zero at plus and minus q and has two maximum and two minimum points in the vicinity of the origin. The two points to the right are local optima, whereas the two to the left are global. f (x) Global maximum

Global minimum

Local maximum x Local minimum

355

356

ONE-DIMENSIONAL UNCONSTRAINED OPTIMIZATION

be sensitive to the issue, it is fortunate that there are numerous engineering problems where you can locate the global optimum in an unambiguous fashion. Just as in root location, optimization in one dimension can be divided into bracketing and open methods. As described in the next section, the golden-section search is an example of a bracketing method that depends on initial guesses that bracket a single optimum. This is followed by an alternative approach, parabolic interpolation, which often converges faster than the golden-section search, but sometimes diverges. Another method described in this chapter is an open method based on the idea from calculus that the minimum or maximum can be found by solving f9(x) 5 0. This reduces the optimization problem to finding the root of f9(x) using techniques of the sort described in Part Two. We will demonstrate one version of this approach—Newton’s method. Finally, an advanced hybrid approach, Brent’s method, is described. This approach combines the reliability of the golden-section search with the speed of parabolic interpolation.

13.1

GOLDEN-SECTION SEARCH In solving for the root of a single nonlinear equation, the goal was to find the value of the variable x that yields a zero of the function f(x). Single-variable optimization has the goal of finding the value of x that yields an extremum, either a maximum or minimum of f(x). The golden-section search is a simple, general-purpose, single-variable search technique. It is similar in spirit to the bisection approach for locating roots in Chap. 5. Recall that bisection hinged on defining an interval, specified by a lower guess (xl) and an upper guess (xu), that bracketed a single root. The presence of a root between these bounds was verified by determining that f(xl) and f(xu) had different signs. The root was then estimated as the midpoint of this interval, xr 5

xl 1 xu 2

The final step in a bisection iteration involved determining a new smaller bracket. This was done by replacing whichever of the bounds xl or xu had a function value with the same sign as f(xr). One advantage of this approach was that the new value xr replaced one of the old bounds. Now we can develop a similar approach for locating the optimum of a one-dimensional function. For simplicity, we will focus on the problem of finding a maximum. When we discuss the computer algorithm, we will describe the minor modifications needed to simulate a minimum. As with bisection, we can start by defining an interval that contains a single answer. That is, the interval should contain a single maximum, and hence is called unimodal. We can adopt the same nomenclature as for bisection, where xl and xu defined the lower and upper bounds, respectively, of such an interval. However, in contrast to bisection, we need a new strategy for finding a maximum within the interval. Rather than using only two function values (which are sufficient to detect a sign change, and hence a zero), we would need three function values to detect whether a maximum occurred. Thus, an additional point within the interval has to be chosen. Next, we have to pick a fourth point.

13.1

GOLDEN-SECTION SEARCH

357

f (x)

Maximum First iteration Second iteration

xl

xu x ᐉ1

ᐉ0

ᐉ2

ᐉ2

FIGURE 13.2 The initial step of the golden-section search algorithm involves choosing two interior points according to the golden ratio.

Then the test for the maximum could be applied to discern whether the maximum occurred within the first three or the last three points. The key to making this approach efficient is the wise choice of the intermediate points. As in bisection, the goal is to minimize function evaluations by replacing old values with new values. This goal can be achieved by specifying that the following two conditions hold (Fig. 13.2): /0 5 /1 1 /2

(13.1)

/1 /2 5 /0 /1

(13.2)

The first condition specifies that the sum of the two sublengths /1 and /2 must equal the original interval length. The second says that the ratio of the lengths must be equal. Equation (13.1) can be substituted into Eq. (13.2), /1 /2 5 /1 1 /2 /1

(13.3)

If the reciprocal is taken and R 5 /2 y /1, we arrive at

11R5

1 R

(13.4)

or R2 1 R 2 1 5 0

(13.5)

which can be solved for the positive root R5

21 1 11 2 4(21) 15 2 1 5 5 0.61803p 2 2

(13.6)

358

ONE-DIMENSIONAL UNCONSTRAINED OPTIMIZATION

Box 13.1

The Golden Ratio and Fibonacci Numbers

In many cultures, certain numbers are ascribed qualities. For example, we in the West are all familiar with “Lucky 7” and “Friday the 13th.” Ancient Greeks called the following number the “golden ratio:” 15 2 1 5 0.61803 p 2 This ratio was employed for a number of purposes, including the development of the rectangle in Fig. 13.3. These proportions were considered aesthetically pleasing by the Greeks. Among other things, many of their temples followed this shape. The golden ratio is related to an important mathematical series known as the Fibonacci numbers, which are 0.61803

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, p Thus, each number after the first two represents the sum of the preceding two. This sequence pops up in many diverse areas of science and engineering. In the context of the present discussion, an interesting property of the Fibonacci sequence relates to the ratio of consecutive numbers in the sequence; that is, 0y1 5 0, 1y1 5 1, 1y2 5 0.5, 2y3 5 0.667, 3y5 5 0.6, 5y8 5 0.625, 8y13 5 0.615, and so on. As one proceeds, the ratio of consecutive numbers approaches the golden ratio!

1

FIGURE 13.3 The Parthenon in Athens, Greece, was constructed in the 5th century B.C. Its front dimensions can be ﬁt almost exactly within a golden rectangle.

This value, which has been known since antiquity, is called the golden ratio (see Box 13.1). Because it allows optima to be found efficiently, it is the key element of the golden-section method we have been developing conceptually. Now let us derive an algorithm to implement this approach on the computer. As mentioned above and as depicted in Fig. 13.4, the method starts with two initial guesses, xl and xu, that bracket one local extremum of f(x). Next, two interior points x1 and x2 are chosen according to the golden ratio, d5

15 2 1 (xu 2 xl ) 2

x1 5 xl 1 d x2 5 xu 2 d The function is evaluated at these two interior points. Two results can occur: 1. If, as is the case in Fig. 13.4, f(x1) . f(x2), then the domain of x to the left of x2, from xl to x2, can be eliminated because it does not contain the maximum. For this case, x2 becomes the new xl for the next round. 2. If f(x2) . f(x1), then the domain of x to the right of x1, from x1 to xu would have been eliminated. In this case, x1 becomes the new xu for the next round.

13.1

GOLDEN-SECTION SEARCH

f (x)

359

Extremum (maximum)

Eliminate

xl

x1

d x2

(a)

x d

xu

f (x)

xl

x2 x1

xu x

Old x2 Old x1

(b)

FIGURE 13.4 (a) The initial step of the golden-section search algorithm involves choosing two interior points according to the golden ratio. (b) The second step involves deﬁning a new interval that includes the optimum.

Now, here is the real benefit from the use of the golden ratio. Because the original x1 and x2 were chosen using the golden ratio, we do not have to recalculate all the function values for the next iteration. For example, for the case illustrated in Fig. 13.4, the old x1 becomes the new x2. This means that we already have the value for the new f(x2), since it is the same as the function value at the old x1. To complete the algorithm, we now only need to determine the new x1. This is done with the same proportionality as before, x1 5 xl 1

15 2 1 (xu 2 xl ) 2

A similar approach would be used for the alternate case where the optimum fell in the left subinterval. As the iterations are repeated, the interval containing the extremum is reduced rapidly. In fact, each round the interval is reduced by a factor of the golden ratio (about 61.8%). That means that after 10 rounds, the interval is shrunk to about 0.61810 or 0.008 or 0.8% of its initial length. After 20 rounds, it is about 0.0066%. This is not quite as good as the reduction achieved with bisection, but this is a harder problem.

360

ONE-DIMENSIONAL UNCONSTRAINED OPTIMIZATION

EXAMPLE 13.1

Golden-Section Search Problem Statement. f(x) 5 2 sin x 2

Use the golden-section search to find the maximum of x2 10

within the interval xl 5 0 and xu 5 4. Solution. First, the golden ratio is used to create the two interior points d5

15 2 1 (4 2 0) 5 2.472 2

x1 5 0 1 2.472 5 2.472 x2 5 4 2 2.472 5 1.528 The function can be evaluated at the interior points f(x2 ) 5 f(1.528) 5 2 sin(1.528) 2

1.5282 5 1.765 10

f(x1 ) 5 f(2.472) 5 0.63 Because f(x2) . f(x1), the maximum is in the interval defined by xl, x2, and x1. Thus, for the new interval, the lower bound remains xl 5 0, and x1 becomes the upper bound, that is, xu 5 2.472. In addition, the former x2 value becomes the new x1, that is, x1 5 1.528. Further, we do not have to recalculate f(x1) because it was determined on the previous iteration as f(1.528) 5 1.765. All that remains is to compute the new values of d and x2, d5

15 2 1 (2.472 2 0) 5 1.528 2

x2 5 2.4721 2 1.528 5 0.944 The function evaluation at x2 is f(0.994) 5 1.531. Since this value is less than the function value at x1, the maximum is in the interval prescribed by x2, x1, and xu. The process can be repeated, with the results tabulated below: i

xl

f(xl)

x2

f(x2)

x1

f(x1)

xu

f(xu)

d

1 2 3 4 5 6 7 8

0 0 0.9443 0.9443 1.3050 1.3050 1.3050 1.3901

0 0 1.5310 1.5310 1.7595 1.7595 1.7595 1.7742

1.5279 0.9443 1.5279 1.3050 1.5279 1.4427 1.3901 1.4427

1.7647 1.5310 1.7647 1.7595 1.7647 1.7755 1.7742 1.7755

2.4721 1.5279 1.8885 1.5279 1.6656 1.5279 1.4427 1.4752

0.6300 1.7647 1.5432 1.7647 1.7136 1.7647 1.7755 1.7732

4.0000 2.4721 2.4721 1.8885 1.8885 1.6656 1.5279 1.5279

23.1136 0.6300 0.6300 1.5432 1.5432 1.7136 1.7647 1.7647

2.4721 1.5279 0.9443 0.5836 0.3607 0.2229 0.1378 0.0851

13.1

GOLDEN-SECTION SEARCH

361

Note that the current maximum is highlighted for every iteration. After the eighth iteration, the maximum occurs at x 5 1.4427 with a function value of 1.7755. Thus, the result is converging on the true value of 1.7757 at x 5 1.4276.

Recall that for bisection (Sec. 5.2.1), an exact upper bound for the error can be calculated at each iteration. Using similar reasoning, an upper bound for golden-section search can be derived as follows: Once an iteration is complete, the optimum will either fall in one of two intervals. If x2 is the optimum function value, it will be in the lower interval (xl, x2, x1). If x1 is the optimum function value, it will be in the upper interval (x2, x1, xu). Because the interior points are symmetrical, either case can be used to define the error. Looking at the upper interval, if the true value were at the far left, the maximum distance from the estimate would be ¢xa 5 x1 2 x2 5 xl 1 R(xu 2 xl ) 2 xu 1 R(xu 2 xl ) 5 (xl 2 xu ) 1 2R(xu 2 xl ) 5 (2R 2 1) (xu 2 xl ) or 0.236(xu 2 xl). If the true value were at the far right, the maximum distance from the estimate would be ¢xb 5 xu 2 x1 5 xu 2 xl 2 R(xu 2 xl ) 5 (1 2 R)(xu 2 xl ) or 0.382(xu 2 xl). Therefore, this case would represent the maximum error. This result can then be normalized to the optimal value for that iteration, xopt, to yield ea 5 (1 2 R) `

xu 2 xl ` 100% xopt

This estimate provides a basis for terminating the iterations. Pseudocode for the golden-section-search algorithm for maximization is presented in Fig. 13.5a. The minor modifications to convert the algorithm to minimization are listed in Fig. 13.5b. In both versions the x value for the optimum is returned as the function value (gold). In addition, the value of f(x) at the optimum is returned as the variable ( fx). You may be wondering why we have stressed the reduced function evaluations of the golden-section search. Of course, for solving a single optimization, the speed savings would be negligible. However, there are two important contexts where minimizing the number of function evaluations can be important. These are 1. Many evaluations. There are cases where the golden-section-search algorithm may be a part of a much larger calculation. In such cases, it may be called many times. Therefore, keeping function evaluations to a minimum could pay great dividends for such cases.

362

ONE-DIMENSIONAL UNCONSTRAINED OPTIMIZATION

FIGURE 13.5 Algorithm for the golden-section search.

FUNCTION Gold (xlow, xhigh, maxit, es, fx) R 5 (50.5 2 1)Y2 x/ = xlow; xu 5 xhigh iter 5 1 d 5 R * (xu 2 x/) x1 5 x/ 1 d; x2 5 xu 2 d f1 5 f(x1) f2 5 f(x2) IF f1 . f2 THEN xopt 5 x1 fx 5 f1 ELSE xopt 5 x2 fx 5 f2 END IF DO d 5 R*d; xint 5 xu 2 x/ IF f1 . f2 THEN x/ 5 x2 x2 5 x1 x1 5 x/1d f2 5 f1 f1 5 f(x1) ELSE xu 5 x1 x1 5 x2 x2 5 xu2d f1 5 f2 f2 5 f(x2) END IF iter 5 iter11 IF f1 . f2 THEN xopt 5 x1 fx 5 f1 ELSE xopt 5 x2 fx 5 f2 END IF IF xopt ﬁ 0. THEN ea 5 (1.2R) *ABS(xintyxopt) * 100. END IF IF ea # es OR iter $ maxit EXIT END DO Gold 5 xopt END Gold (a) Maximization

IF f1 , f2 THEN

IF f1 , f2 THEN

IF f1 , f2 THEN

(b) Minimization

13.2

PARABOLIC INTERPOLATION

363

Parabolic approximation of maximum

True maximum f (x) True function

x0

x1

Parabolic function

x3

x2

x

FIGURE 13.6 Graphical description of parabolic interpolation.

2. Time-consuming evaluation. For pedagogical reasons, we use simple functions in most of our examples. You should understand that a function can be very complex and timeconsuming to evaluate. For example, in a later part of this book, we will describe how optimization can be used to estimate the parameters of a model consisting of a system of differential equations. For such cases, the “function” involves time-consuming model integration. Any method that minimizes such evaluations would be advantageous.

13.2

PARABOLIC INTERPOLATION Parabolic interpolation takes advantage of the fact that a second-order polynomial often provides a good approximation to the shape of f(x) near an optimum (Fig. 13.6). Just as there is only one straight line connecting two points, there is only one quadratic polynomial or parabola connecting three points. Thus, if we have three points that jointly bracket an optimum, we can fit a parabola to the points. Then we can differentiate it, set the result equal to zero, and solve for an estimate of the optimal x. It can be shown through some algebraic manipulations that the result is x3 5

f(x0 ) ( x21 2 x22 ) 1 f(x1 ) ( x22 2 x20 ) 1 f(x2 ) ( x20 2 x21 ) 2 f(x0 )(x1 2 x2 ) 1 2 f(x1 )(x2 2 x0 ) 1 2 f(x2 )(x0 2 x1 )

(13.7)

where x0, x1, and x2 are the initial guesses, and x3 is the value of x that corresponds to the maximum value of the parabolic fit to the guesses. After generating the new point, there are two strategies for selecting the points for the next iteration. The simplest approach, which is similar to the secant method, is to merely assign the new points sequentially. That is, for the new iteration, z0 5 z1, z1 5 z2, and z2 5 z3. Alternatively, as illustrated in the following example, a bracketing approach, similar to bisection or the golden-section search, can be employed.

364

ONE-DIMENSIONAL UNCONSTRAINED OPTIMIZATION

EXAMPLE 13.2

Parabolic Interpolation Problem Statement.

Use parabolic interpolation to approximate the maximum of

f(x) 5 2 sin x 2

x2 10

with initial guesses of x0 5 0, x1 5 1, and x2 5 4. Solution. The function values at the three guesses can be evaluated, x0 5 0 f(x0 ) 5 0 x1 5 1 f(x1 ) 5 1.5829 x2 5 4 f(x2 ) 5 23.1136 and substituted into Eq. (13.7) to give x3 5

0 ( 12 2 42 ) 1 1.5829 ( 42 2 02 ) 1 (23.1136) ( 02 2 12 ) 5 1.5055 2(0)(1 2 4) 1 2(1.5829)(4 2 0) 1 2(23.1136)(0 2 1)

which has a function value of f(1.5055) 5 1.7691. Next, a strategy similar to the golden-section search can be employed to determine which point should be discarded. Because the function value for the new point is higher than for the intermediate point (x1) and the new x value is to the right of the intermediate point, the lower guess (x0) is discarded. Therefore, for the next iteration, x0 5 1 f(x0 ) 5 1.5829 x1 5 1.5055 f(x1 ) 5 1.7691 x2 5 4 f(x2 ) 5 23.1136 which can be substituted into Eq. (13.7) to give 1.5829 ( 1.50552 2 42 ) 1 1.7691 ( 42 2 12 ) 1 (23.1136) ( 12 2 1.50552 ) 2(1.5829)(1.5055 2 4) 1 2(1.7691)(4 2 1) 1 2(23.1136)(1 2 1.5055) 5 1.4903

x3 5

which has a function value of f(1.4903) 5 1.7714. The process can be repeated, with the results tabulated below: i

x0

f(x0)

x1

f(x1)

x2

f(x2)

x3

f(x3)

1 2 3 4 5

0.0000 1.0000 1.0000 1.0000 1.4256

0.0000 1.5829 1.5829 1.5829 1.7757

1.0000 1.5055 1.4903 1.4256 1.4266

1.5829 1.7691 1.7714 1.7757 1.7757

4.0000 4.0000 1.5055 1.4903 1.4903

23.1136 23.1136 1.7691 1.7714 1.7714

1.5055 1.4903 1.4256 1.4266 1.4275

1.7691 1.7714 1.7757 1.7757 1.7757

Thus, within five iterations, the result is converging rapidly on the true value of 1.7757 at x 5 1.4276.

13.3

NEWTON’S METHOD

365

We should mention that just like the false-position method, parabolic interpolation can get hung up with just one end of the interval converging. Thus, convergence can be slow. For example, notice that in our example, 1.0000 was an endpoint for most of the iterations. This method, as well as others using third-order polynomials, can be formulated into algorithms that contain convergence tests, careful selection strategies for the points to retain on each iteration, and attempts to minimize round-off error accumulation.

13.3

NEWTON’S METHOD Recall that the Newton-Raphson method of Chap. 6 is an open method that finds the root x of a function such that f(x) 5 0. The method is summarized as xi11 5 xi 2

f(xi ) f ¿(xi )

A similar open approach can be used to find an optimum of f(x) by defining a new function, g(x) 5 f9(x). Thus, because the same optimal value x* satisfies both f ¿(x*) 5 g(x*) 5 0

we can use the following, f ¿(xi ) f –(xi )

xi11 5 xi 2

(13.8)

as a technique to find the minimum or maximum of f(x). It should be noted that this equation can also be derived by writing a second-order Taylor series for f(x) and setting the derivative of the series equal to zero. Newton’s method is an open method similar to Newton-Raphson because it does not require initial guesses that bracket the optimum. In addition, it also shares the disadvantage that it may be divergent. Finally, it is usually a good idea to check that the second derivative has the correct sign to confirm that the technique is converging on the result you desire. EXAMPLE 13.3

Newton’s Method Problem Statement. f(x) 5 2 sin x 2

Use Newton’s method to find the maximum of x2 10

with an initial guess of x0 5 2.5. Solution. The first and second derivatives of the function can be evaluated as f ¿(x) 5 2 cos x 2

x 5

f –(x) 5 22 sin x 2

1 5

366

ONE-DIMENSIONAL UNCONSTRAINED OPTIMIZATION

which can be substituted into Eq. (13.8) to give xi11 5 xi 2

2 cos xi 2 xi y5 22 sin xi 2 1y5

Substituting the initial guess yields x1 5 2.5 2

2 cos 2.5 2 2.5y5 5 0.99508 22 sin 2.5 2 1y5

which has a function value of 1.57859. The second iteration gives x1 5 0.995 2

2 cos 0.995 2 0.995y5 5 1.46901 22 sin 0.995 2 1y5

which has a function value of 1.77385. The process can be repeated, with the results tabulated below: i

x

f(x)

f’(x)

f’’(x)

0 1 2 3 4

2.5 0.99508 1.46901 1.42764 1.42755

0.57194 1.57859 1.77385 1.77573 1.77573

22.10229 0.88985 20.09058 20.00020 0.00000

21.39694 21.87761 22.18965 22.17954 22.17952

Thus, within four iterations, the result converges rapidly on the true value.

Although Newton’s method works well in some cases, it is impractical for cases where the derivatives cannot be conveniently evaluated. For these cases, other approaches that do not involve derivative evaluation are available. For example, a secant-like version of Newton’s method can be developed by using finite-difference approximations for the derivative evaluations. A bigger reservation regarding the approach is that it may diverge based on the nature of the function and the quality of the initial guess. Thus, it is usually employed only when we are close to the optimum. As described next, hybrid techniques that use bracketing approaches far from the optimum and open methods near the optimum attempt to exploit the strong points of both approaches.

13.4

BRENT’S METHOD Recall that in Sec. 6.4, we described Brent’s method for root location. This hybrid method combined several root-finding methods into a single algorithm that balanced reliability with efficiency. Brent also developed a similar approach for one-dimensional minimization. It combines the slow, dependable golden-section search with the faster, but possibly unreliable, parabolic interpolation. It first attempts parabolic interpolation and keeps applying it as long as acceptable results are obtained. If not, it uses the golden-section search to get matters in hand. Figure 13.7 presents pseudocode for the algorithm based on a MATLAB software M-file developed by Cleve Moler (2005). It represents a stripped-down version of the

13.4

FIGURE 13.7 Pseudocode for Brent’s minimum-ﬁnding algorithm based on a MATLAB M-ﬁle developed by Cleve Moler (2005).

BRENT’S METHOD

Function fminsimp(x1, xu) tol 5 0.000001; phi 5 (1 + 15)/2;; rho 5 2 2 phi u 5 x1 1 rho*(xu 2 x1); v 5 u; w 5 u; x 5 u fu 5 f(u); fv 5 fu; fw 5 fu; fx 5 fu xm 5 0.5*(x1 1 xu); d 5 0; e 5 0 DO IF |x 2 xm| # tol EXIT para 5 |e| . tol IF para THEN (Try parabolic fit) r 5 (x 2 w)*(fx 2 fv); q 5 (x 2 v)*(fx 2 fw) p 5 (x 2 v)*q 2 (x 2 w)*r; s 5 2*(q 2 r) IF s . 0 THEN p 5 2p s 5 |s| ' Is the parabola acceptable? para 5 |p| , |0.5*s*e| And p . s*(x1 2 x) And p , s*(xu 2 x) IF para THEN e 5 d; d 5 p/s (Parabolic interpolation step) ENDIF ENDIF IF Not para THEN IF x $ xm THEN (Golden-section search step) e 5 x1 2 x ELSE e 5 xu 2 x ENDIF d 5 rho*e ENDIF u 5 x 1 d; fu 5 f(u) IF fu # fx THEN (Update x1, xu, x, v, w, xm) IF u $ x THEN x1 5 x ELSE xu 5 x ENDIF v 5 w; fv 5 fw; w 5 x; fw 5 fx; x 5 u; fx 5 fu ELSE IF u , x THEN x1 5 u ELSE xu 5 u ENDIF IF fu # fw Or w 5 x THEN v 5 w; fv 5 fw; w 5 u; fw 5 fu ELSEIF fu # fv Or v 5 x Or v 5 w THEN v 5 u; fv 5 fu ENDIF ENDIF xm 5 0.5*(x1 1 xu) ENDDO fminsimp 5 fu END fminsimp

367

368

ONE-DIMENSIONAL UNCONSTRAINED OPTIMIZATION

fminbnd function, which is the professional minimization function employed in MATLAB. For that reason, we call the simplified version fminsimp. Note that it requires another function f that holds the equation for which the minimum is being evaluated. This concludes our treatment of methods to solve the optima of functions of a single variable. Some engineering examples are presented in Chap. 16. In addition, the techniques described here are an important element of some procedures to optimize multivariable functions, as discussed in Chap. 14.

PROBLEMS 13.1 Given the formula 2

f (x) 5 2x 1 8x 2 12 (a) Determine the maximum and the corresponding value of x for this function analytically (i.e., using differentiation). (b) Verify that Eq. (13.7) yields the same results based on initial guesses of x0 5 0, x1 5 2, and x2 5 6. 13.2 Given f (x) 5 21.5x6 2 2x4 1 12x (a) Plot the function. (b) Use analytical methods to prove that the function is concave for all values of x. (c) Differentiate the function and then use a root-location method to solve for the maximum f(x) and the corresponding value of x. 13.3 Solve for the value of x that maximizes f(x) in Prob. 13.2 using the golden-section search. Employ initial guesses of xl 5 0 and xu 5 2 and perform three iterations. 13.4 Repeat Prob. 13.3, except use parabolic interpolation in the same fashion as Example 13.2. Employ initial guesses of x0 5 0, x1 5 1, and x2 5 2 and perform three iterations. 13.5 Repeat Prob. 13.3 but use Newton’s method. Employ an initial guess of x0 5 2 and perform three iterations. 13.6 Employ the following methods to find the maximum of

13.8 Employ the following methods to find the maximum of the function from Prob. 13.7: (a) Golden-section search (xl 5 22, xu 5 1, es 5 1%). (b) Parabolic interpolation (x0 5 22, x1 5 21, x2 5 1, iterations 5 4). Select new points sequentially as in the secant method. (c) Newton’s method (x0 5 21, es 5 1%). 13.9 Consider the following function: f (x) 5 2x 1

Perform 10 iterations of parabolic interpolation to locate the minimum. Select new points in the same fashion as in Example 13.2. Comment on the convergence of your results. (x0 5 0.1, x1 5 0.5, x2 5 5) 13.10 Consider the following function: f (x) 5 3 1 6x 1 5x2 1 3x3 1 4x4 Locate the minimum by finding the root of the derivative of this function. Use bisection with initial guesses of xl 5 22 and xu 5 1. 13.11 Determine the minimum of the function from Prob. 13.10 with the following methods: (a) Newton’s method (x0 5 21, es 5 1%). (b) Newton’s method, but using a finite difference approximation for the derivative estimates.

f (x) 5 4x 2 1.8x2 1 1.2x3 2 0.3x4 (a) Golden-section search (xl 5 22, xu 5 4, es 5 1%). (b) Parabolic interpolation (x0 5 1.75, x1 5 2, x2 5 2.5, iterations 5 4). Select new points sequentially as in the secant method. (c) Newton’s method (x0 5 3, es 5 1%). 13.7 Consider the following function: f (x) 5 2 x4 2 2x3 2 8x2 2 5x Use analytical and graphical methods to show the function has a maximum for some value of x in the range 22 # x # 1.

3 x

f ¿(x) 5 f –(x) 5

f (xi 1 dxi ) 2 f (xi 2 dxi ) 2dxi f (xi 1 dxi ) 2 2f (xi ) 2 f (xi 2 dxi ) (dxi ) 2

where d 5 a perturbation fraction (5 0.01). Use an initial guess of x0 5 21 and iterate to es 5 1%. 13.12 Develop a program using a programming or macro language to implement the golden-section search algorithm. Design the program so that it is expressly designed to locate a maximum. The subroutine should have the following features:

PROBLEMS • Iterate until the relative error falls below a stopping criterion or exceeds a maximum number of iterations. • Return both the optimal x and f(x). • Minimize the number of function evaluations. Test your program with the same problem as Example 13.1. 13.13 Develop a program as described in Prob. 13.12, but make it perform minimization or maximization depending on the user’s preference. 13.14 Develop a program using a programming or macro language to implement the parabolic interpolation algorithm. Design the program so that it is expressly designed to locate a maximum and selects new points as in Example 13.2. The subroutine should have the following features: • Base it on two initial guesses, and have the program generate the third initial value at the midpoint of the interval. • Check whether the guesses bracket a maximum. If not, the subroutine should not implement the algorithm, but should return an error message. • Iterate until the relative error falls below a stopping criterion or exceeds a maximum number of iterations. • Return both the optimal x and f(x). • Minimize the number of function evaluations. Test your program with the same problem as Example 13.2. 13.15 Develop a program using a programming or macro language to implement Newton’s method. The subroutine should have the following features: • Iterate until the relative error falls below a stopping criterion or exceeds a maximum number of iterations. • Returns both the optimal x and f(x). Test your program with the same problem as Example 13.3. 13.16 Pressure measurements are taken at certain points behind an airfoil over time. These data best fit the curve y 5 6 cos x 2 1.5 sin x from x 5 0 to 6 s. Use four iterations of the golden-search method to find the minimum pressure. Set xl 5 2 and xu 5 4. 13.17 The trajectory of a ball can be computed with g y 5 (tan u0 )x 2 2 x2 1 y0 2y0 cos2 u0 where y 5 the height (m), u0 5 the initial angle (radians), y0 5 the initial velocity (m/s), g 5 the gravitational constant 5 9.81 m/s2, and y0 5 the initial height (m). Use the golden-section search to determine the maximum height given y0 5 1 m, y0 5 25 m/s and u0 5 508. Iterate until the approximate error falls below es 5 1% using initial guesses of xl 5 0 and xu 5 60 m. 13.18 The deflection of a uniform beam subject to a linearly increasing distributed load can be computed as w0 y5 (2x5 1 2L2 x3 2 L4x) 120EIL

369 Given that L 5 600 cm, E 5 50,000 kN/cm2, I 5 30,000 cm4, and w0 5 2.5 kN/cm, determine the point of maximum deflection (a) graphically, (b) using the golden-section search until the approximate error falls below es 5 1% with initial guesses of xl 5 0 and xu 5 L. 13.19 An object with a mass of 100 kg is projected upward from the surface of the earth at a velocity of 50 m/s. If the object is subject to linear drag (c 5 15 kg/s), use the golden-section search to determine the maximum height the object attains. Hint: recall Sec. PT4.1.2. 13.20 The normal distribution is a bell-shaped curve defined by 2

y 5 e2x

Use the golden-section search to determine the location of the inflection point of this curve for positive x. 13.21 An object can be projected upward at a specified velocity. If it is subject to linear drag, its altitude as a function of time can be computed as z 5 z0 1

mg mg m ay0 1 t b ( 1 2 e2(cym)t ) 2 c c c

where z 5 altitude (m) above the earth’s surface (defined as z 5 0), z0 5 the initial altitude (m), m 5 mass (kg), c 5 a linear drag coefficient (kg/s), v0 5 initial velocity (m/s), and t 5 time (s). Note that for this formulation, positive velocity is considered to be in the upward direction. Given the following parameter values: g 5 9.81 m/s2, z0 5 100 m, v0 5 55 m/s, m 5 80 kg, and c 5 15 kg/s, the equation can be used to calculate the jumper’s altitude. Determine the time and altitude of the peak elevation (a) graphically, (b) analytically, and (c) with the golden-section search until the approximate error falls below es 5 1% with initial guesses of tl 5 0 and tu 5 10 s. 13.22 Use the golden-section search to determine the length of the shortest ladder that reaches from the ground over the fence to touch the building’s wall (Fig. P13.22). Test it for the case where h 5 d 5 4 m.

h

d

FIGURE P13.22 A ladder leaning against a fence and just touching a wall.

CHAPTER

14 Multidimensional Unconstrained Optimization This chapter describes techniques to find the minimum or maximum of a function of several variables. Recall from Chap. 13 that our visual image of a one-dimensional search was like a roller coaster. For two-dimensional cases, the image becomes that of mountains and valleys (Fig. 14.1). For higher-dimensional problems, convenient images are not possible. We have chosen to limit this chapter to the two-dimensional case. We have adopted this approach because the essential features of multidimensional searches are often best communicated visually. Techniques for multidimensional unconstrained optimization can be classified in a number of ways. For purposes of the present discussion, we will divide them depending on whether they require derivative evaluation. The approaches that do not require derivative evaluation are called nongradient, or direct, methods. Those that require derivatives are called gradient, or descent (or ascent), methods.

FIGURE 14.1 The most tangible way to visualize two-dimensional searches is in the context of ascending a mountain (maximization) or descending into a valley (minimization). (a) A 2-D topographic map that corresponds to the 3-D mountain in (b).

Lines of constant f

f

x

x

y y

(a) 370

(b)

14.1

14.1

DIRECT METHODS

371

DIRECT METHODS These methods vary from simple brute force approaches to more elegant techniques that attempt to exploit the nature of the function. We will start our discussion with a brute force approach. 14.1.1 Random Search A simple example of a brute force approach is the random search method. As the name implies, this method repeatedly evaluates the function at randomly selected values of the independent variables. If a sufficient number of samples are conducted, the optimum will eventually be located.

EXAMPLE 14.1

Random Search Method Problem Statement.

Use a random number generator to locate the maximum of

f(x, y) 5 y 2 x 2 2x2 2 2xy 2 y2

(E14.1.1)

in the domain bounded by x 5 22 to 2 and y 5 1 to 3. The domain is depicted in Fig. 14.2. Notice that a single maximum of 1.5 occurs at x 5 21 and y 5 1.5. Solution. Random number generators typically generate values between 0 and 1. If we designate such a number as r, the following formula can be used to generate x values randomly within a range between xl to xu: x 5 xl 1 (xu 2 xl )r For the present application, xl 5 22 and xu 5 2, and the formula is x 5 22 1 (2 2 (22) )r 5 22 1 4r This can be tested by substituting 0 and 1 to yield 22 and 2, respectively. FIGURE 14.2 Equation (E14.1.1) showing the maximum at x 5 21 and y 5 1.5. y 3 – 10

– 20

0 2 0 1 –2

–1

0

Maximum

1

2

x

372

MULTIDIMENSIONAL UNCONSTRAINED OPTIMIZATION

Similarly for y, a formula for the present example could be developed as y 5 yl 1 ( yu 2 yl )r 5 1 1 (3 2 1)r 5 1 1 2r The following Excel VBA macrocode uses the VBA random number function Rnd, to generate (x, y) pairs. These are then substituted into Eq. (E14.1.1). The maximum value from among these random trials is stored in the variable maxf, and the corresponding x and y values in maxx and maxy, respectively. maxf = −1E9 For j = 1 To n x = −2 + 4 * Rnd y = 1 + 2 * Rnd fn = y − x − 2 * x ^ 2 − 2 * x * y − y ^ 2 If fn > maxf Then maxf = fn maxx = x maxy = y End If Next j

A number of iterations yields Iterations 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

x 20.9886 21.0040 21.0040 21.0040 21.0040 20.9837 20.9960 20.9960 20.9960 20.9978

y

f (x, y)

1.4282 1.4724 1.4724 1.4724 1.4724 1.4936 1.5079 1.5079 1.5079 1.5039

1.2462 1.2490 1.2490 1.2490 1.2490 1.2496 1.2498 1.2498 1.2498 1.2500

The results indicate that the technique homes in on the true maximum.

This simple brute force approach works even for discontinuous and nondifferentiable functions. Furthermore, it always finds the global optimum rather than a local optimum. Its major shortcoming is that as the number of independent variables grows, the implementation effort required can become onerous. In addition, it is not efficient because it takes no account of the behavior of the underlying function. The remainder of the approaches described in this chapter do take function behavior into account as well as the results of previous trials to improve the speed of convergence. Thus, although the random search can certainly prove useful in specific problem contexts, the following methods have more general utility and almost always lead to more efficient convergence.

14.1

DIRECT METHODS

373

It should be noted that more sophisticated search techniques are available. These are heuristic approaches that were developed to handle either nonlinear and/or discontinuous problems that classical optimization cannot usually handle well, if at all. Simulated annealing, tabu search, artificial neural networks, and genetic algorithms are a few. The most widely applied is the genetic algorithm, with a number of commercial packages available. Holland (1975) pioneered the genetic algorithm approach and Davis (1991) and Goldberg (1989) provide good overviews of the theory and application of the method. 14.1.2 Univariate and Pattern Searches It is very appealing to have an efficient optimization approach that does not require evaluation of derivatives. The random search method described above does not require derivative evaluation, but it is not very efficient. This section describes an approach, the univariate search method, that is more efficient and still does not require derivative evaluation. The basic strategy underlying the univariate search method is to change one variable at a time to improve the approximation while the other variables are held constant. Since only one variable is changed, the problem reduces to a sequence of one-dimensional searches that can be solved using a variety of methods (including those described in Chap. 13). Let us perform a univariate search graphically, as shown in Fig. 14.3. Start at point 1, and move along the x axis with y constant to the maximum at point 2. You can see that point 2 is a maximum by noticing that the trajectory along the x axis just touches a contour line at the point. Next, move along the y axis with x constant to point 3. Continue this process generating points 4, 5, 6, etc.

FIGURE 14.3 A graphical depiction of how a univariate search is conducted. y

5 6

3

4

1 2 x

374

MULTIDIMENSIONAL UNCONSTRAINED OPTIMIZATION

y

2 1

x

FIGURE 14.4 Conjugate directions.

Although we are gradually moving toward the maximum, the search becomes less efficient as we move along the narrow ridge toward the maximum. However, also note that lines joining alternate points such as 1-3, 3-5 or 2-4, 4-6 point in the general direction of the maximum. These trajectories present an opportunity to shoot directly along the ridge toward the maximum. Such trajectories are called pattern directions. Formal algorithms are available that capitalize on the idea of pattern directions to find optimum values efficiently. The best known of these algorithms is called Powell’s method. It is based on the observation (see Fig. 14.4) that if points 1 and 2 are obtained by one-dimensional searches in the same direction but from different starting points, then the line formed by 1 and 2 will be directed toward the maximum. Such lines are called conjugate directions. In fact, it can be proved that if f(x, y) is a quadratic function, sequential searches along conjugate directions will converge exactly in a finite number of steps regardless of the starting point. Since a general nonlinear function can often be reasonably approximated by a quadratic function, methods based on conjugate directions are usually quite efficient and are in fact quadratically convergent as they approach the optimum. Let us graphically implement a simplified version of Powell’s method to find the maximum of f(x, y) 5 c 2 (x 2 a) 2 2 (y 2 b) 2 where a, b, and c are positive constants. This equation results in circular contours in the x, y plane, as shown in Fig. 14.5. Initiate the search at point 0 with starting directions h1 and h2. Note that h1 and h2 are not necessarily conjugate directions. From zero, move along h1 until a maximum is located

14.2

GRADIENT METHODS

375

y

3

h1

h3

h2

h4 h2

5 h3

4

2 h2

0 1

x

FIGURE 14.5 Powell’s method.

at point 1. Then search from point 1 along direction h2 to find point 2. Next, form a new search direction h3 through points 0 and 2. Search along this direction until the maximum at point 3 is located. Then search from point 3 in the h2 direction until the maximum at point 4 is located. From point 4 arrive at point 5 by again searching along h3. Now, observe that both points 5 and 3 have been located by searching in the h3 direction from two different points. Powell has shown that h4 (formed by points 3 and 5) and h3 are conjugate directions. Thus, searching from point 5 along h4 brings us directly to the maximum. Powell’s method can be refined to make it more efficient, but the formal algorithms are beyond the scope of this text. However, it is an efficient method that is quadratically convergent without requiring derivative evaluation.

14.2

GRADIENT METHODS As the name implies, gradient methods explicitly use derivative information to generate efficient algorithms to locate optima. Before describing specific approaches, we must first review some key mathematical concepts and operations. 14.2.1 Gradients and Hessians Recall from calculus that the first derivative of a one-dimensional function provides a slope or tangent to the function being differentiated. From the standpoint of optimization, this is useful information. For example, if the slope is positive, it tells us that increasing the independent variable will lead to a higher value of the function we are exploring. From calculus, also recall that the first derivative may tell us when we have reached an optimal value since this is the point that the derivative goes to zero. Further, the sign of the second derivative can tell us whether we have reached a minimum (positive second derivative) or a maximum (negative second derivative).

376

MULTIDIMENSIONAL UNCONSTRAINED OPTIMIZATION x

x=a y=b h=0

h y

FIGURE 14.6 The directional gradient is deﬁned along an axis h that forms an angle u with the x axis.

These ideas were useful to us in the one-dimensional search algorithms we explored in Chap. 13. However, to fully understand multidimensional searches, we must first understand how the first and second derivatives are expressed in a multidimensional context. The Gradient. Suppose we have a two-dimensional function f(x, y). An example might be your elevation on a mountain as a function of your position. Suppose that you are at a specific location on the mountain (a, b) and you want to know the slope in an arbitrary direction. One way to define the direction is along a new axis h that forms an angle u with the x axis (Fig. 14.6). The elevation along this new axis can be thought of as a new function g(h). If you define your position as being the origin of this axis (that is, h 5 0), the slope in this direction would be designated as g9(0). This slope, which is called the directional derivative, can be calculated from the partial derivatives along the x and y axis by g¿(0) 5

0f 0f cos u 1 sin u 0x 0y

(14.1)

where the partial derivatives are evaluated at x 5 a and y 5 b. Assuming that your goal is to gain the most elevation with the next step, the next logical question would be: what direction is the steepest ascent? The answer to this question is provided very neatly by what is referred to mathematically as the gradient, which is defined as §f 5

0f 0f i1 j 0x 0y

(14.2)

This vector is also referred to as “del f.” It represents the directional derivative of f(x, y) at point x 5 a and y 5 b.

14.2

GRADIENT METHODS

377

Vector notation provides a concise means to generalize the gradient to n dimensions, as 0f (x) 0x1 0f (x) 0x2 i . y §f(x) 5 . . 0f (x) 0xn How do we use the gradient? For the mountain-climbing problem, if we are interested in gaining elevation as quickly as possible, the gradient tells us what direction to move locally and how much we will gain by taking it. Note, however, that this strategy does not necessarily take us on a direct path to the summit! We will discuss these ideas in more depth later in this chapter. EXAMPLE 14.2

Using the Gradient to Evaluate the Path of Steepest Ascent Problem Statement. the function

Employ the gradient to evaluate the steepest ascent direction for

f(x, y) 5 xy2 at the point (2, 2). Assume that positive x is pointed east and positive y is pointed north. Solution. First, our elevation can be determined as f(2, 2) 5 2(2) 2 5 8 Next, the partial derivatives can be evaluated, 0f 5 y2 5 22 5 4 0x 0f 5 2xy 5 2(2)(2) 5 8 0y which can be used to determine the gradient as §f 5 4i 1 8j This vector can be sketched on a topographical map of the function, as in Fig. 14.7. This immediately tells us that the direction we must take is 8 u 5 tan21 a b 5 1.107 radians (563.4°) 4 relative to the x axis. The slope in this direction, which is the magnitude of = f, can be calculated as 242 1 82 5 8.944

378

MULTIDIMENSIONAL UNCONSTRAINED OPTIMIZATION

y 4 8

24

40

3

2

1

0

0

1

2

3

4

x

FIGURE 14.7 The arrow follows the direction of steepest ascent calculated with the gradient.

Thus, during our first step, we will initially gain 8.944 units of elevation rise for a unit distance advanced along this steepest path. Observe that Eq. (14.1) yields the same result, g¿(0) 5 4 cos(1.107) 1 8 sin(1.107) 5 8.944 Note that for any other direction, say u 5 1.107y2 5 0.5235, g9(0) 5 4 cos(0.5235) 1 8 sin(0.5235) 5 7.608, which is smaller. As we move forward, both the direction and magnitude of the steepest path will change. These changes can be quantified at each step using the gradient, and your climbing direction modified accordingly. A final insight can be gained by inspecting Fig. 14.7. As indicated, the direction of steepest ascent is perpendicular, or orthogonal, to the elevation contour at the coordinate (2, 2). This is a general characteristic of the gradient.

Aside from defining a steepest path, the first derivative can also be used to discern whether an optimum has been reached. As is the case for a one-dimensional function, if the partial derivatives with respect to both x and y are zero, a two-dimensional optimum has been reached. The Hessian. For one-dimensional problems, both the first and second derivatives provide valuable information for searching out optima. The first derivative (a) provides a steepest trajectory of the function and (b) tells us that we have reached an optimum. Once at an optimum, the second derivative tells us whether we are a maximum [negative

14.2

GRADIENT METHODS

379

f (x, y) (a, b)

x

y

y=x

FIGURE 14.8 A saddle point (x 5 a and y 5 b). Notice that when the curve is viewed along the x and y directions, the function appears to go through a minimum (positive second derivative), whereas when viewed along an axis x 5 y, it is concave downward (negative second derivative).

f 0(x)] or a minimum [positive f 0(x)]. In the previous paragraphs, we illustrated how the gradient provides best local trajectories for multidimensional problems. Now, we will examine how the second derivative is used in such contexts. You might expect that if the partial second derivatives with respect to both x and y are both negative, then you have reached a maximum. Figure 14.8 shows a function where this is not true. The point (a, b) of this graph appears to be a minimum when observed along either the x dimension or the y dimension. In both instances, the second partial derivatives are positive. However, if the function is observed along the line y 5 x, it can be seen that a maximum occurs at the same point. This shape is called a saddle, and clearly, neither a maximum or a minimum occurs at the point. Whether a maximum or a minimum occurs involves not only the partials with respect to x and y but also the second partial with respect to x and y. Assuming that the partial derivatives are continuous at and near the point being evaluated, the following quantity can be computed: ZH Z 5

02 f 02 f 0 x 2 0 y2

2a

02 f 2 b 0x0y

Three cases can occur If Z H Z . 0 and 0 2 fy0x 2 . 0, then f(x, y) has a local minimum. If Z H Z . 0 and 0 2fy0x 2 , 0, then f(x, y) has a local maximum. If Z H Z , 0, then f(x, y) has a saddle point.

(14.3)

380

MULTIDIMENSIONAL UNCONSTRAINED OPTIMIZATION

The quantity Z H Z is equal to the determinant of a matrix made up of the second derivatives,1 02 f 0 x2 H5 ≥ 2 0 f 0y0x

02 f 0x0y 02 f ¥

(14.4)

0 y2

where this matrix is formally referred to as the Hessian of f. Besides providing a way to discern whether a multidimensional function has reached an optimum, the Hessian has other uses in optimization (for example, for the multidimensional form of Newton’s method). In particular, it allows searches to include secondorder curvature to attain superior results. Finite-Difference Approximations. It should be mentioned that, for cases where they are difficult or inconvenient to compute analytically, both the gradient and the determinant of the Hessian can be evaluated numerically. In most cases, the approach introduced in Sec. 6.3.3 for the modified secant method is employed. That is, the independent variables can be perturbed slightly to generate the required partial derivatives. For example, if a centered-difference approach is adopted, they can be computed as 0f f(x 1 dx, y) 2 f(x 2 dx, y) (14.5) 5 0x 2dx f(x, y 1 dy) 2 f(x, y 2 dy) 0f 5 (14.6) 0y 2dy f(x 1 dx, y) 2 2f(x, y) 1 f(x 2 dx, y) 02 f 5 (14.7) 2 0x dx2 f(x, y 1 dy) 2 2f(x, y) 1 f(x, y 2 dy) 02 f 5 (14.8) 2 0y dy2 02 f 5 0x0y f(x 1 dx, y 1 dy) 2 f(x 1 dx, y 2 dy) 2 f(x 2 dx, y 1 dy) 1 f(x 2 dx, y 2 dy) 4dx dy (14.9)

where d is some small fractional value. Note that the methods employed in commercial software packages also use forward differences. In addition, they are usually more complicated than the approximations listed in Eqs. (14.5) through (14.9). Dennis and Schnabel (1996) provide more detail on such approaches. Regardless of how the approximation is implemented, the important point is that you may have the option of evaluating the gradient and/or the Hessian analytically. This can sometimes be an arduous task, but the performance of the algorithm may benefit Note that 0 2fy( 0x0y) 5 0 2fy( 0y0x).

1

14.2

GRADIENT METHODS

381

enough to make your effort worthwhile. The closed-form derivatives will be exact, but more importantly, you will reduce the number of function evaluations. This latter point can have a critical impact on the execution time. On the other hand, you will often exercise the option of having the quantities computed internally using numerical approaches. In many cases, the performance will be quite adequate and you will be saved the difficulty of numerous partial differentiations. Such would be the case on the optimizers used in certain spreadsheets and mathematical software packages (for example, Excel). In such cases, you may not even be given the option of entering an analytically derived gradient and Hessian. However, for small to moderately sized problems, this is usually not a major shortcoming. 14.2.2 Steepest Ascent Method An obvious strategy for climbing a hill would be to determine the maximum slope at your starting position and then start walking in that direction. But clearly, another problem arises almost immediately. Unless you were really lucky and started on a ridge that pointed directly to the summit, as soon as you moved, your path would diverge from the steepest ascent direction. Recognizing this fact, you might adopt the following strategy. You could walk a short distance along the gradient direction. Then you could stop, reevaluate the gradient and walk another short distance. By repeating the process you would eventually get to the top of the hill. Although this strategy sounds superficially sound, it is not very practical. In particular, the continuous reevaluation of the gradient can be computationally demanding. A preferred approach involves moving in a fixed path along the initial gradient until f(x, y) stops increasing, that is, becomes level along your direction of travel. This stopping point becomes the starting point where §f is reevaluated and a new direction followed. The process is repeated until the summit is reached. This approach is called the steepest ascent method.2 It is the most straightforward of the gradient search techniques. The basic idea behind the approach is depicted in Fig. 14.9. We start at an initial point (x0, y0) labeled “0” in the figure. At this point, we determine the direction of steepest ascent, that is, the gradient. We then search along the direction of the gradient, h0, until we find a maximum, which is labeled “1” in the figure. The process is then repeated. Thus, the problem boils down to two parts: (1) determining the “best” direction to search and (2) determining the “best value” along that search direction. As we will see, the effectiveness of the various algorithms described in the coming pages depends on how clever we are at both parts. For the time being, the steepest ascent method uses the gradient approach as its choice for the “best” direction. We have already shown how the gradient is evaluated in Example 14.1. Now, before examining how the algorithm goes about locating the maximum along the steepest direction, we must pause to explore how to transform a function of x and y into a function of h along the gradient direction. 2

Because of our emphasis on maximization here, we use the terminology steepest ascent. The same approach can also be used for minimization, in which case the terminology steepest descent is used.

382

MULTIDIMENSIONAL UNCONSTRAINED OPTIMIZATION

y h2 h0 h1 2 1

0 x

FIGURE 14.9 A graphical depiction of the method of steepest ascent.

y

ⵜf = 3i + 4j

10

h

6 h

2 h

1

=

=

=

2

1

0

4

7

x

FIGURE 14.10 The relationship between an arbitrary direction h and x and y coordinates.

Starting at x0, y0 the coordinates of any point in the gradient direction can be expressed as 0f h 0x 0f y 5 y0 1 h 0y x 5 x0 1

(14.10) (14.11)

14.2

GRADIENT METHODS

383

where h is distance along the h axis. For example, suppose x0 5 1 and y0 5 2 and §f 5 3i 1 4j, as shown in Fig. 14.10. The coordinates of any point along the h axis are given by x 5 1 1 3h y 5 2 1 4h

(14.12) (14.13)

The following example illustrates how we can use these transformations to convert a two-dimensional function of x and y into a one-dimensional function in h. EXAMPLE 14.3

Developing a 1-D Function Along the Gradient Direction Problem Statement. Suppose we have the following two-dimensional function: f(x, y) 5 2xy 1 2x 2 x2 2 2y2 Develop a one-dimensional version of this equation along the gradient direction at point x 5 21 and y 5 1. Solution. The partial derivatives can be evaluated at (21, 1), 0f 5 2y 1 2 2 2x 5 2(1) 1 2 2 2(21) 5 6 0x 0f 5 2x 2 4y 5 2(21) 2 4(1) 5 26 0y Therefore, the gradient vector is §f 5 6i 2 6j To find the maximum, we could search along the gradient direction, that is, along an h axis running along the direction of this vector. The function can be expressed along this axis as f ax0 1

0f 0f h, y0 1 hb 5 f(21 1 6h, 1 2 6h) 0x 0y

5 2(21 1 6h)(1 2 6h) 1 2(21 1 6h) 2 (21 1 6h) 2 2 2(1 2 6h) 2 where the partial derivatives are evaluated at x 5 21 and y 5 1. By combining terms, we develop a one-dimensional function g(h) that maps f(x, y) along the h axis, g(h) 5 2180h2 1 72h 2 7

Now that we have developed a function along the path of steepest ascent, we can explore how to answer the second question. That is, how far along this path do we travel? One approach might be to move along this path until we find the maximum of this function. We will call the location of this maximum h*. This is the value of the step that maximizes g (and hence, f ) in the gradient direction. This problem is equivalent to finding the maximum of a function of a single variable h. This can be done using different one-dimensional search techniques like the ones we discussed in Chap. 13. Thus, we

384

MULTIDIMENSIONAL UNCONSTRAINED OPTIMIZATION

convert from finding the optimum of a two-dimensional function to performing a onedimensional search along the gradient direction. This method is called steepest ascent when an arbitrary step size h is used. If a value of a single step h* is found that brings us directly to the maximum along the gradient direction, the method is called the optimal steepest ascent. EXAMPLE 14.4

Optimal Steepest Ascent Problem Statement.

Maximize the following function:

f(x, y) 5 2xy 1 2x 2 x2 2 2y2 using initial guesses, x 5 21 and y 5 1. Solution. Because this function is so simple, we can first generate an analytical solution. To do this, the partial derivatives can be evaluated as 0f 5 2y 1 2 2 2x 5 0 0x 0f 5 2x 2 4y 5 0 0y This pair of equations can be solved for the optimum, x 5 2 and y 5 1. The second partial derivatives can also be determined and evaluated at the optimum, 02 f 0x 2 02 f

5 22

5 24 0y 2 02f 02f 5 52 0x0y 0y0x and the determinant of the Hessian is computed [Eq. (14.3)], ZH Z 5 22(24) 2 22 5 4 Therefore, because Z H Z . 0 and 0 2fy0x2 , 0, function value f(2, 1) is a maximum. Now let us implement steepest ascent. Recall that, at the end of Example 14.3, we had already implemented the initial steps of the problem by generating g(h) 5 2180h2 1 72h 2 7 Now, because this is a simple parabola, we can directly locate the maximum (that is, h 5 h*) by solving the problem, g¿(h*) 5 0 2360h* 1 72 5 0 h* 5 0.2 This means that if we travel along the h axis, g(h) reaches a minimum value when h 5 h* 5 0.2. This result can be placed back into Eqs. (14.10) and (14.11) to solve for the

14.2

GRADIENT METHODS

385

y 3

Maximum

2 1

2

0

0 –1 –2

1 0

2

4

x

FIGURE 14.11 The method of optimal steepest ascent.

(x, y) coordinates corresponding to this point, x 5 21 1 6(0.2) 5 0.2 y 5 1 2 6(0.2) 5 20.2 This step is depicted in Fig. 14.11 as the move from point 0 to 1. The second step is merely implemented by repeating the procedure. First, the partial derivatives can be evaluated at the new starting point (0.2, 20.2) to give 0f 5 2(20.2) 1 2 2 2(0.2) 5 1.2 0x 0f 5 2(0.2) 2 4(20.2) 5 1.2 0y Therefore, the gradient vector is §f 5 1.2 i 1 1.2 j This means that the steepest direction is now pointed up and to the right at a 458 angle with the x axis (see Fig. 14.11). The coordinates along this new h axis can now be expressed as x 5 0.2 1 1.2h y 5 20.2 1 1.2h Substituting these values into the function yields f(0.2 1 1.2h, 20.2 1 1.2h) 5 g(h) 5 21.44h2 1 2.88h 1 0.2 The step h* to take us to the maximum along the search direction can then be directly computed as g¿(h*) 5 22.88h* 1 2.88 5 0 h* 5 1

386

MULTIDIMENSIONAL UNCONSTRAINED OPTIMIZATION

This result can be placed back into Eqs. (14.10) and (14.11) to solve for the (x, y) coordinates corresponding to this new point, x 5 0.2 1 1.2(1) 5 1.4 y 5 20.2 1 1.2(1) 5 1 As depicted in Fig. 14.11, we move to the new coordinates, labeled point 2 in the plot, and in so doing move closer to the maximum. The approach can be repeated with the final result converging on the analytical solution, x 5 2 and y 5 1.

It can be shown that the method of steepest descent is linearly convergent. Further, it tends to move very slowly along long, narrow ridges. This is because the new gradient at each maximum point will be perpendicular to the original direction. Thus, the technique takes many small steps criss-crossing the direct route to the summit. Hence, although it is reliable, there are other approaches that converge much more rapidly, particularly in the vicinity of an optimum. The remainder of the section is devoted to such methods. 14.2.3 Advanced Gradient Approaches Conjugate Gradient Method (Fletcher-Reeves). In Sec. 14.1.2, we have seen how conjugate directions in Powell’s method greatly improved the efficiency of a univariate search. In a similar manner, we can also improve the linearly convergent steepest ascent using conjugate gradients. In fact, an optimization method that makes use of conjugate gradients to define search directions can be shown to be quadratically convergent. This also ensures that the method will optimize a quadratic function exactly in a finite number of steps regardless of the starting point. Since most well-behaved functions can be approximated reasonably well by a quadratic in the vicinity of an optimum, quadratically convergent approaches are often very efficient near an optimum. We have seen how starting with two arbitrary search directions, Powell’s method produced new conjugate search directions. This method is quadratically convergent and does not require gradient information. On the other hand, if evaluation of derivatives is practical, we can devise algorithms that combine the ideas of steepest descent and conjugate directions to achieve robust initial performance and rapid convergence as the technique gravitates toward the optimum. The Fletcher-Reeves conjugate gradient algorithm modifies the steepest-ascent method by imposing the condition that successive gradient search directions be mutually conjugate. The proof and algorithm are beyond the scope of the text but are described by Rao (1996). Newton’s Method. Newton’s method for a single variable (recall Sec. 13.3) can be extended to multivariate cases. Write a second-order Taylor series for f(x) near x 5 xi, f(x) 5 f(xi ) 1 §f T (xi )(x 2 xi ) 1

1 (x 2 xi ) T Hi (x 2 xi ) 2

where Hi is the Hessian matrix. At the minimum, 0f(x) 5 0 for j 5 1, 2, p , n 0xj

14.2

GRADIENT METHODS

387 x

y

FIGURE 14.12 When the starting point is close to the optimal point, following the gradient can be inefﬁcient. Newton methods attempt to search along a direct path to the optimum (solid line).

Thus, §f 5 §f(xi ) 1 Hi (x 2 xi ) 5 0 If H is nonsingular, xi11 5 xi 2 Hi21 §f

(14.14)

which can be shown to converge quadratically near the optimum. This method again performs better than the steepest ascent method (see Fig. 14.12). However, note that the method requires both the computation of second derivatives and matrix inversion at each iteration. Thus, the method is not very useful in practice for functions with large numbers of variables. Furthermore, Newton’s method may not converge if the starting point is not close to the optimum. Marquardt Method. We know that the method of steepest ascent increases the function value even if the starting point is far from an optimum. On the other hand, we have just described Newton’s method, which converges rapidly near the maximum. Marquardt’s method uses the steepest descent method when x is far from x*, and Newton’s method when x closes in on an optimum. This is accomplished by modifying the diagonal of the Hessian in Eq. (14.14), H˜ i 5 Hi 1 ai I where ai is a positive constant and I is the identity matrix. At the start of the procedure, ai is assumed to be large and H˜ i21

77 10(4.9) 1 8(3.9) > 80 4.9 # 9 3.9 # 6 Consequently, as is also clear from the plot, producing at the optimal amount of each product brings us right to the point where we just meet the resource (1) and time constraints (2). Such constraints are said to be binding. Further, as is also evident graphically, neither of the storage constraints [(3) and (4)] acts as a limitation. Such constraints are called nonbinding. This leads to the practical conclusion that, for this case, we can increase profits by either increasing our resource supply (the raw gas) or increasing our production time. Further, it indicates that increasing storage would have no impact on profit. The result obtained in the previous example is one of four possible outcomes that can be generally obtained in a linear programming problem. These are 1. Unique solution. As in the example, the maximum objective function intersects a single point. 2. Alternate solutions. Suppose that the objective function in the example had coefficients so that it was precisely parallel to one of the constraints. In our example problem,

15.1

LINEAR PROGRAMMING

x2

395

x2

x2

Z 0

x1

(a)

0

x1

0

(b)

x1

(c)

FIGURE 15.2 Aside from a single optimal solution (for example, Fig. 15.1b), there are three other possible outcomes of a linear programming problem: (a) alternative optima, (b) no feasible solution, and (c) an unbounded result.

one way in which this would occur would be if the profits were changed to $140/ tonne and $220/tonne. Then, rather than a single point, the problem would have an infinite number of optima corresponding to a line segment (Fig. 15.2a). 3. No feasible solution. As in Fig. 15.2b, it is possible that the problem is set up so that there is no feasible solution. This can be due to dealing with an unsolvable problem or due to errors in setting up the problem. The latter can result if the problem is over-constrained to the point that no solution can satisfy all the constraints. 4. Unbounded problems. As in Fig. 15.2c, this usually means that the problem is underconstrained and therefore open-ended. As with the no-feasible-solution case, it can often arise from errors committed during problem specification. Now let us suppose that our problem involves a unique solution. The graphical approach might suggest an enumerative strategy for hunting down the maximum. From Fig. 15.1, it should be clear that the optimum always occurs at one of the corner points where two constraints meet. Such a point is known formally as an extreme point. Thus, out of the infinite number of possibilities in the decision space, focusing on extreme points clearly narrows down the possible options. Further, we can recognize that not every extreme point is feasible, that is, satisfying all constraints. For example, notice that point F in Fig. 15.1a is an extreme point but is not feasible. Limiting ourselves to feasible extreme points narrows the field down still further. Finally, once all feasible extreme points are identified, the one yielding the best value of the objective function represents the optimum solution. Finding this optimal solution could be done by exhaustively (and inefficiently) evaluating the value of the objective function at every feasible extreme point. The following section discusses the simplex method, which offers a preferable strategy that charts a selective course through a sequence of feasible extreme points to arrive at the optimum in an extremely efficient manner.

396

CONSTRAINED OPTIMIZATION

15.1.3 The Simplex Method The simplex method is predicated on the assumption that the optimal solution will be an extreme point. Thus, the approach must be able to discern whether during problem solution an extreme point occurs. To do this, the constraint equations are reformulated as equalities by introducing what are called slack variables. Slack Variables. As the name implies, a slack variable measures how much of a constrained resource is available, that is, how much “slack” of the resource is available. For example, recall the resource constraint used in Examples 15.1 and 15.2, 7x1 1 11x2 # 77 We can define a slack variable S1 as the amount of raw gas that is not used for a particular production level (x1, x2). If this quantity is added to the left side of the constraint, it makes the relationship exact, 7x1 1 11x2 1 S1 5 77 Now recognize what the slack variable tells us. If it is positive, it means that we have some “slack” for this constraint. That is, we have some surplus resource that is not being fully utilized. If it is negative, it tells us that we have exceeded the constraint. Finally, if it is zero, we exactly meet the constraint. That is, we have used up all the allowable resource. Since this is exactly the condition where constraint lines intersect, the slack variable provides a means to detect extreme points. A different slack variable is developed for each constraint equation, resulting in what is called the fully augmented version, Maximize Z 5 150x1 1 175x2 subject to 7x1 1 11x2 1 S1 10x1 1 8x2 1 S2 x1 1 S3 x2 1 S4 x1, x2, S1, S2, S3, S4 $ 0

5 77 5 80 59 56

(15.4a) (15.4b) (15.4c) (15.4d)

Notice how we have set up the four equality equations so that the unknowns are aligned in columns. We did this to underscore that we are now dealing with a system of linear algebraic equations (recall Part Three). In the following section, we will show how these equations can be used to determine extreme points algebraically. Algebraic Solution. In contrast to Part Three, where we had n equations with n unknowns, our example system [Eqs. (15.4)] is underspecified or underdetermined, that is, it has more unknowns than equations. In general terms, there are n structural variables (that is, the original unknowns), m surplus or slack variables (one per constraint), and n 1 m total variables (structural plus surplus). For the gas production problem we have 2 structural variables, 4 slack variables, and 6 total variables. Thus, the problem involves solving 4 equations with 6 unknowns.

15.1

LINEAR PROGRAMMING

397

The difference between the number of unknowns and the number of equations (equal to 2 for our problem) is directly related to how we can distinguish a feasible extreme point. Specifically, every feasible point has 2 variables out of 6 equal to zero. For example, the five corner points of the area ABCDE have the following zero values: Extreme Point A B C D E

Zero Variables x1, x 2, S1, S1, x 1,

x2 S2 S2 S4 S4

This observation leads to the conclusion that the extreme points can be determined from the standard form by setting two of the variables equal to zero. In our example, this reduces the problem to a solvable form of 4 equations with 4 unknowns. For example, for point E, setting x1 5 S4 5 0 reduces the standard form to 11x2 1 S1 8x2 1 S2 1 S3 x2

5 77 5 80 59 56

which can be solved for x2 5 6, S1 5 11, S2 5 32, and S3 5 9. Together with x1 5 S4 5 0, these values define point E. To generalize, a basic solution for m linear equations with n unknowns is developed by setting n 2 m variables to zero, and solving the m equations for the m remaining unknowns. The zero variables are formally referred to as nonbasic variables, whereas the remaining m variables are called basic variables. If all the basic variables are nonnegative, the result is called a basic feasible solution. The optimum will be one of these. Now a direct approach to determining the optimal solution would be to calculate all the basic solutions, determine which were feasible, and among those, which had the highest value of Z. There are two reasons why this is not a wise approach. First, for even moderately sized problems, the approach can involve solving a great number of equations. For m equations with n unknowns, this results in solving Cnm 5

n! m!(n 2 m)!

simultaneous equations. For example, if there are 10 equations (m 5 10) with 16 unknowns (n 5 16), you would have 8008 [5 16!y(10! 6!)] 10 3 10 systems of equations to solve! Second, a significant portion of these may be infeasible. For example, in the present problem, out of C46 5 15 extreme points, only 5 are feasible. Clearly, if we could avoid solving all these unnecessary systems, a more efficient algorithm would be developed. Such an approach is described next.

398

CONSTRAINED OPTIMIZATION

Simplex Method Implementation. The simplex method avoids inefficiencies outlined in the previous section. It does this by starting with a basic feasible solution. Then it moves through a sequence of other basic feasible solutions that successively improve the value of the objective function. Eventually, the optimal value is reached and the method is terminated. We will illustrate the approach using the gas-processing problem from Examples 15.1 and 15.2. The first step is to start at a basic feasible solution (that is, at an extreme corner point of the feasible space). For cases like ours, an obvious starting point would be point A; that is, x1 5 x2 5 0. The original 6 equations with 4 unknowns become S1 S2 S3

5 77 5 80 59 S4 5 6

Thus, the starting values for the basic variables are given automatically as being equal to the right-hand sides of the constraints. Before proceeding to the next step, the beginning information can now be summarized in a convenient tabular format called a tableau. As shown below, the tableau provides a concise summary of the key information constituting the linear programming problem. Basic

Z

x1

x2

S1

S2

S3

S4

Solution

Z S1 S2 S3 S4

1 0 0 0 0

2150 7 10 1 0

2175 11 8 0 1

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

0 77 80 9 6

Intercept 11 8 9 `

Notice that for the purposes of the tableau, the objective function is expressed as Z 2 150x1 2 175x2 2 0S1 2 0S2 2 0S3 2 0S4 5 0

(15.5)

The next step involves moving to a new basic feasible solution that leads to an improvement of the objective function. This is accomplished by increasing a current nonbasic variable (at this point, x1 or x2) above zero so that Z increases. Recall that, for the present example, extreme points must have 2 zero values. Therefore, one of the current basic variables (S1, S2, S3, or S4) must also be set to zero. To summarize this important step: one of the current nonbasic variables must be made basic (nonzero). This variable is called the entering variable. In the process, one of the current basic variables is made nonbasic (zero). This variable is called the leaving variable. Now, let us develop a mathematical approach for choosing the entering and leaving variables. Because of the convention by which the objective function is written [(Eq. (15.5)], the entering variable can be any variable in the objective function having a negative coefficient (because this will make Z bigger). The variable with the largest negative value is conventionally chosen because it usually leads to the largest increase

15.1

LINEAR PROGRAMMING

399

x2

2

8

E

4 D

4

3

C 1 A

B

0 4

8

F

x1

FIGURE 15.3 Graphical depiction of how the simplex method successively moves through feasible basic solutions to arrive at the optimum in an efﬁcient manner.

in Z. For our case, x2 would be the entering variable since its coefficient, 2175, is more negative than the coefficient of x1, 2150. At this point the graphical solution can be consulted for insight. As in Fig. 15.3, we start at the initial point A. Based on its coefficient, x2 should be chosen to enter. However, to keep the present example brief, we choose x1 since we can see from the graph that this will bring us to the maximum quicker. Next, we must choose the leaving variable from among the current basic variables— S1, S2, S3, or S4. Graphically, we can see that there are two possibilities. Moving to point B will drive S2 to zero, whereas moving to point F will drive S1 to zero. However, the graph also makes it clear that F is not possible because it lies outside the feasible solution space. Thus, we decide to move from A to B. How is the same result detected mathematically? One way is to calculate the values at which the constraint lines intersect the axis or line corresponding to the entering variable (in our case, the x1 axis). We can calculate this value as the ratio of the righthand side of the constraint (the “Solution” column of the tableau) to the corresponding coefficient of x1. For example, for the first constraints slack variable S1, the result is Intercept 5

77 5 11 7

The remaining intercepts can be calculated and listed as the last column of the tableau. Because 8 is the smallest positive intercept, it means that the second constraint line will be reached first as x1 is increased. Hence, S2 should be the leaving variable.

400

CONSTRAINED OPTIMIZATION

At this point, we have moved to point B (x2 5 S2 5 0), and the new basic solution becomes 7x1 1 S1 5 77 10x1 5 80 x1 1 S3 5 9 S4 5 6 The solution of this system of equations effectively defines the values of the basic variables at point B: x1 5 8, S1 5 21, S3 5 1, and S4 5 6. The tableau can be used to make the same calculation by employing the GaussJordan method. Recall that the basic strategy behind Gauss-Jordan involved converting the pivot element to 1 and then eliminating the coefficients in the same column above and below the pivot element (recall Sec. 9.7). For this example, the pivot row is S2 (the leaving variable) and the pivot element is 10 (the coefficient of the entering variable, x1). Dividing the row by 10 and replacing S2 by x1 gives Basic

Z

Z S1 x1 S3 S4

1 0 0 0 0

x1 2150 7 1 1 0

x2

S1

S2

S3

S4

Solution

2175 11 0.8 0 1

0 1 0 0 0

0 0 0.1 0 0

0 0 0 1 0

0 0 0 0 1

0 77 8 9 6

Intercept

Next, the x1 coefficients in the other rows can be eliminated. For example, for the objective function row, the pivot row is multiplied by 2150 and the result subtracted from the first row to give Z

x1

x2

S1

S2

S3

S4

Solution

1 20

2150 2(2150)

2175 2(2120)

0 20

0 2(215)

0 0

0 0

0 2(21200)

0

255

15

0

0

1200

1

0

Similar operations can be performed on the remaining rows to give the new tableau, Basic

Z

x1

x2

S1

S2

S3

S4

Solution

Intercept

Z S1 x1 S3 S4

1 0 0 0 0

0 0 1 0 0

255 5.4 0.8 20.8 1

0 1 0 0 0

15 20.7 0.1 20.1 0

0 0 0 1 0

0 0 0 0 1

1200 21 8 1 6

3.889 10 21.25 6

15.2

NONLINEAR CONSTRAINED OPTIMIZATION

401

Thus, the new tableau summarizes all the information for point B. This includes the fact that the move has increased the objective function to Z 5 1200. This tableau can then be used to chart our next, and in this case final, step. Only one more variable, x2, has a negative value in the objective function, and it is therefore chosen as the entering variable. According to the intercept values (now calculated as the solution column over the coefficients in the x2 column), the first constraint has the smallest positive value, and therefore, S1 is selected as the leaving variable. Thus, the simplex method moves us from points B to C in Fig. 15.3. Finally, the Gauss-Jordan elimination can be implemented to solve the simultaneous equations. The result is the final tableau, Basic

Z

x1

x2

S1

Z x2 x1 S3 S4

1 0 0 0 0

0 0 1 0 0

0 1 0 0 0

10.1852 0.1852 20.1481 0.1481 20.1852

S2 7.8704 20.1296 0.2037 20.2037 0.1296

S3

S4

Solution

0 0 0 1 0

0 0 0 0 1

1413.889 3.889 4.889 4.111 2.111

We know that the result is final because there are no negative coefficients remaining in the objective function row. The final solution is tabulated as x1 5 3.889 and x2 5 4.889, which give a maximum objective function of Z 5 1413.889. Further, because S3 and S4 are still in the basis, we know that the solution is limited by the first and second constraints.

15.2

NONLINEAR CONSTRAINED OPTIMIZATION There are a number of approaches for handling nonlinear optimization problems in the presence of constraints. These can generally be divided into indirect and direct approaches (Rao, 1996). A typical indirect approach uses so-called penalty functions. These involve placing additional expressions to make the objective function less optimal as the solution approaches a constraint. Thus, the solution will be discouraged from violating constraints. Although such methods can be useful in some problems, they can become arduous when the problem involves many constraints. The generalized reduced gradient (GRG) search method is one of the more popular of the direct methods (for details, see Fylstra et al., 1998; Lasdon et al., 1978; Lasdon and Smith, 1992). It is, in fact, the nonlinear method used within the Excel Solver. It first “reduces” the problem to an unconstrained optimization problem. It does this by solving a set of nonlinear equations for the basic variables in terms of the nonbasic variables. Then, the unconstrained problem is solved using approaches similar to those described in Chap. 14. First, a search direction is chosen along which an improvement in the objective function is sought. The default choice is a quasi-Newton approach (BFGS) that, as described in Chap. 14, requires storage of an approximation of the Hessian matrix. This approach performs very well for most cases. The conjugate gradient approach is also available in Excel as an alternative for large problems. The Excel Solver has the nice feature that it automatically switches to the conjugate gradient method, depending on available storage. Once the search direction is established, a one-dimensional search is carried out along that direction using a variable step-size approach.

SOFTWARE

402

CONSTRAINED OPTIMIZATION

15.3

OPTIMIZATION WITH SOFTWARE PACKAGES Software packages have great capabilities for optimization. In this section, we will give you an introduction to some of the more useful ones. 15.3.1 Excel for Linear Programming There are a variety of software packages expressly designed to implement linear programming. However, because of its broad availability, we will focus on the Excel spreadsheet. This involves using the Solver option previously employed in Chap. 7 for root location. The manner in which Solver is used for linear programming is similar to our previous applications in that these data are entered into spreadsheet cells. The basic strategy is to arrive at a single cell that is to be optimized as a function of variations of other cells on the spreadsheet. The following example illustrates how this can be done for the gas-processing problem.

EXAMPLE 15.3

Using Excel’s Solver for a Linear Programming Problem Problem Statement. Use Excel to solve the gas-processing problem we have been examining in this chapter. Solution. An Excel worksheet set up to calculate the pertinent values in the gasprocessing problem is shown in Fig. 15.4. The unshaded cells are those containing numeric and labeling data. The shaded cells involve quantities that are calculated based on other cells. Recognize that the cell to be maximized is D12, which contains the total profit. The cells to be varied are B4:C4, which hold the amounts of regular and premium gas produced.

FIGURE 15.4 Excel spreadsheet set up to use the Solver for linear programming.

15.3

OPTIMIZATION WITH SOFTWARE PACKAGES

403

Once the spreadsheet is created, Solver is chosen from the Data tab (recall Sec. 7.7.1). At this point a dialogue box will be displayed, querying you for pertinent information. The pertinent cells of the Solver dialogue box are filled out as

The constraints must be added one by one by selecting the “Add” button. This will open up a dialogue box that looks like

As shown, the constraint that the total raw gas (cell D6) must be less than or equal to the available supply (E6) can be added as shown. After adding each constraint, the “Add” button can be selected. When all four constraints have been entered, the OK button is selected to return to the Solver dialogue box. Now, before execution, the Solver options button should be selected and the box labeled “Assume linear model” should be checked off. This will make Excel employ a version of the simplex algorithm (rather than the more general nonlinear solver it usually uses) that will speed up your application. After selecting this option, return to the Solver menu. When the OK button is selected, a dialogue box will open with a report on the success of the operation. For the present case, the Solver obtains the correct solution (Fig. 15.5)

CONSTRAINED OPTIMIZATION

SOFTWARE

404

FIGURE 15.5 Excel spreadsheet showing solution to linear programming problem.

Beyond obtaining the solution, the Solver also provides some useful summary reports. We will explore these in the engineering application described in Sec. 16.2. 15.3.2 Excel for Nonlinear Optimization The manner in which Solver is used for nonlinear optimization is similar to our previous applications in that these data are entered into spreadsheet cells. Once again, the basic strategy is to arrive at a single cell that is to be optimized as a function of variations of other cells on the spreadsheet. The following example illustrates how this can be done for the parachutist problem we set up in the introduction to this part of the book (recall Example PT4.1). EXAMPLE 15.4

Using Excel‘s Solver for Nonlinear Constrained Optimization Problem Statement. Recall from Example PT4.1 that we developed a nonlinear constrained optimization to minimize the cost for a parachute drop into a refugee camp. Parameters for this problem are Parameter Total mass Acceleration of gravity Cost coefﬁcient (constant) Cost coefﬁcient (length) Cost coefﬁcient (area) Critical impact velocity Area effect on drag Initial drop height

Symbol

Value

Unit

Mt g c0 c1 c2 vc kc z0

2000 9.8 200 56 0.1 20 3 500

kg m/s2 $ $/m $/m2 m/s kg/(s ? m2) m

15.3

OPTIMIZATION WITH SOFTWARE PACKAGES

405

Substituting these values into Eqs. (PT4.11) through (PT4.19) gives Minimize C 5 n(200 1 56/ 1 0.1A2 ) subject to y # 20 n$1 where n is an integer and all other variables are real. In addition, the following quantities are defined as A 5 2pr2 / 5 12r c 5 3A Mt m5 n

(E15.4.1)

t 5 root c500 2 y5

9.8m 9.8m2 t1 ( 1 2 e2(cym)t )d c c2

(E15.4.2)

9.8m ( 1 2 e2(cym)t ) c

Use Excel to solve this problem for the design variables r and n that minimize cost C. Solution. Before implementation of this problem on Excel, we must first deal with the problem of determining the root in the above formulation [Eq. (E15.4.2)]. One method might be to develop a macro to implement a root-location method such as bisection or the secant method. (Note that we will illustrate how this is done in the next chapter in Sec. 16.3.) For the time being, an easier approach is possible by developing the following fixedpoint iteration solution to Eq. (E15.4.2), ti11 5 c500 1

9.8m2 c ( 1 2 e2(cym)ti )d 2 9.8m c

(E15.4.3)

Thus, t can be adjusted until Eq. (E15.4.3) is satisfied. It can be shown that for the range of parameters used in the present problem, this formula always converges. Now, how can this equation be solved on a spreadsheet? As shown below, two cells can be set up to hold a value for t and for the right-hand side of Eq. (E15.4.3) [that is, f(t)].

SOFTWARE

406

CONSTRAINED OPTIMIZATION

You can type Eq. (E15.4.3) into cell B21 so that it gets its time value from cell B20 and the other parameter values from cells elsewhere on the sheet (see below for how we set up the whole sheet). Then go to cell B20 and point its value to cell B21. Once you enter these formulations, you will immediately get the error message: “Cannot resolve circular references” because B20 depends on B21 and vice versa. Now, go to the Tools/Options selections from the menu and select calculation. From the calculation dialogue box, check off “iteration” and hit “OK.” Immediately the spreadsheet will iterate these cells and the result will come out as

Thus, the cells will converge on the root. If you want to make it more precise, just strike the F9 key to make it iterate some more (the default is 100 iterations, which you can change if you wish). An Excel worksheet to calculate the pertinent values can then be set up as shown in Fig. 15.6. The unshaded cells are those containing numeric and labeling data. The

FIGURE 15.6 Excel spreadsheet set up for the nonlinear parachute optimization problem.

15.3

OPTIMIZATION WITH SOFTWARE PACKAGES

407

shaded cells involve quantities that are calculated based on other cells. For example, the mass in B17 was computed with Eq. (E15.4.1) based on the values for Mt (B4) and n (E5). Note also that some cells are redundant. For example, cell E11 points back to cell E5. The information is repeated in cell E11 so that the structure of the constraints is evident from the sheet. Finally, recognize that the cell to be minimized is E15, which contains the total cost. The cells to be varied are E4:E5, which hold the radius and the number of parachutes. Once the spreadsheet is created, the selection Solver is chosen from the Data tab. At this point a dialogue box will be displayed, querying you for pertinent information. The pertinent cells of the Solver dialogue box would be filled out as

The constraints must be added one by one by selecting the “Add” button. This will open up a dialogue box that looks like

As shown, the constraint that the actual impact velocity (cell E10) must be less than or equal to the required velocity (G10) can be added as shown. After adding each constraint, the “Add” button can be selected. Note that the down arrow allows you to choose among several types of constraints (,5, .5, 5, and integer). Thus, we can force the number of parachutes (E5) to be an integer. When all three constraints have been entered, the “OK” button is selected to return to the Solver dialogue box. After selecting this option return to the Solver menu. When the “OK” button is selected, a dialogue box will open with a report on the success of the operation. For the present case, the Solver obtains the correct solution as in Fig. 15.7.

408

CONSTRAINED OPTIMIZATION

FIGURE 15.7 Excel spreadsheet showing the solution for the nonlinear parachute optimization problem.

Thus, we determine that the minimum cost of $4377.26 will occur if we break the load up into six parcels with a chute radius of 2.944 m. Beyond obtaining the solution, the Solver also provides some useful summary reports. We will explore these in the engineering application described in Sec. 16.2.

15.3.3 MATLAB As summarized in Table 15.1, MATLAB software has a variety of built-in functions to perform optimization. The following examples illustrates how they can be used. TABLE 15.1 MATLAB functions to implement optimization. Function

Description

fminbnd fminsearch

Minimize function of one variable with bound constraints Minimize function of several variables

15.3

EXAMPLE 15.5

OPTIMIZATION WITH SOFTWARE PACKAGES

409

Using MATLAB for One-Dimensional Optimization Problem Statement. Use the MATLAB fminbnd function to find the maximum of f(x) 5 2 sin x 2

x2 2

within the interval xl 5 0 and xu 5 4. Recall that in Chap. 13, we used several methods to solve this problem for x 5 1.7757 and f(x) 5 1.4276. Solution. First, we must create an M-file to hold the function. function f=fx(x) f = −(2*sin(x)−x^2/10)

Because we are interested in maximization, we enter the negative of the function. Then, we invoke the fminbnd function with >> x=fminbnd('fx',0,4)

The result is f =

−1.7757

x =

1.4275

Note that additional arguments can be included. One useful addition is to set optimization options such as error tolerance or maximum iterations. This is done with the optimset function, which was used previously in Example 7.6 and has the general format, optimset('param1',value1,'param2',value2,...)

where parami is a parameter specifying the type of option and valuei is the value assigned to that option. For example, if you wanted to set the tolerance at 1 31022, optimset('TolX',le–2)

Thus, solving the present problem to a tolerance of 1 3 1022 can be generated with >> fminbnd('fx',0,4,optimset('TolX',le–2))

with the result f =

−1.7757

ans = 1.4270

410

CONSTRAINED OPTIMIZATION

SOFTWARE

A complete set of parameters can be found by invoking Help as in >> Help optimset

MATLAB has a variety of capabilities for dealing with multidimensional functions. Recall from Chap. 13 that our visual image of a one-dimensional search was like a roller coaster. For two-dimensional cases, the image becomes that of mountains and valleys. As in the following example, MATLAB’s graphic capabilities provide a handy means to visualize such functions.

EXAMPLE 15.6

Visualizing a Two-Dimensional Function Problem Statement. Use MATLAB’s graphical capabilities to display the following function and visually estimate its minimum in the range 22 # x1 # 0 and 0 # x2 # 3: f(x1, x2 ) 5 2 1 x1 2 x2 1 2x21 1 2x1x2 1 x22 Solution. The following script generates contour and mesh plots of the function: x=linspace(−2,0,40);y=linspace(0,3,40); [X,Y] = meshgrid(x,y); Z=2+X−Y+2*X.^2+2*X.*Y+Y.^2; subplot(1,2,1); cs=contour(X,Y,Z);clabel(cs); xlabel('x_1');ylabel('x_2'); title('(a) Contour plot');grid; subplot(1,2,2); cs=surfc(X,Y,Z); zmin=floor(min(Z)); zmax=ceil(max(Z)); xlabel('x_1');ylabel('x_2');zlabel('f(x_1,x_2)'); title('(b) Mesh plot');

As displayed in Fig. 15.8, both plots indicate that function has a minimum value of about f(x1, x2) 5 0 to 1 located at about x1 5 21 and x2 5 1.5. Standard MATLAB has a function fminsearch that can be used to determine the minimum of a multidimensional function. It is based on the Nelder-Mead method, which is a direct-search method that uses only function values (does not require derivatives) and handles nonsmooth objective functions. A simple expression of its syntax is [xmin, fval] = fminsearch(function,x1,x2)

where xmin and fval are the location and value of the minimum, function is the name of the function being evaluated, and x1 and x2 are the bounds of the interval being searched.

15.3

OPTIMIZATION WITH SOFTWARE PACKAGES

411

FIGURE 15.8 (a) Contour and (b) mesh plots of a two-dimensional function. (b) Mesh plot

x2

(x1, x2 )

(a) Contour plot

x2

x1

EXAMPLE 15.7

x1

Using MATLAB for Multidimensional Optimization Problem Statement. Use the MATLAB fminsearch function to find the maximum for the simple function we just graphed in Example 15.6. f(x1, x2 ) 5 2 1 x1 2 x2 1 2x21 1 2x1x2 1 x22

Employ initial guesses of x 5 20.5 and y 5 0.5. Solution. We can invoke the fminsearch function with >> f=@(x) 2+x(1)−x(2)+2*x(1)^2+2*x(1)*x(2)+x(2)^2; >> [x,fval]=fminsearch(f,[−0.5,0.5]) x =

−1.0000 fval = 0.7500

1.5000

Just as with fminbnd, arguments can be included in order to specify additional parameters of the optimization process. For example, the optimset function can be used to limit the maximum number of iterations >> [x,fval]=fminsearch(f,[−0.5,0.5],optimset('MaxIter',2))

412

CONSTRAINED OPTIMIZATION

with the result Exiting: Maximum number of iterations has been exceeded − increase MaxIter option. Current function value: 1.225625 x = −0.5000 0.5250 fval = 1.2256

Thus, because we have set a very stringent limit on the iterations, the optimization terminates well before the maximum is reached.

15.3.4 Mathcad Mathcad contains a numeric mode function called Find that can be used to solve up to 50 simultaneous nonlinear algebraic equations with inequality constraints. The use of this function for unconstrained applications was described in Part Two. If Find fails to locate a solution that satisfies the equations and constraints, it returns the error message “did not find solution.” However, Mathcad also contains a similar function called Minerr. This function gives solution results that minimize the errors in the constraints even when exact solutions cannot be found. This function solves equations and accommodates several constraints using the Levenberg-Marquardt method taken from the public-domain MINPACK algorithms developed and published by the Argonne National Laboratory. Let’s develop an example where Find is used to solve a system of nonlinear equations with constraints. Initial guesses of x 5 21 and y 5 1 are input using the definition symbol as shown in Fig. 15.9. The word Given then alerts Mathcad that what follows is a system of equations. Then we can enter the equations and the inequality constraint. Note that for this application, Mathcad requires the use of a symbolic equal sign (typed as [Ctrl]5) and . to separate the left and right sides of an equation. Now the vector consisting of xval and yval is computed using Find (x,y) and the values are shown using an equal sign. A graph that displays the equations and constraints as well as the solution can be placed on the worksheet by clicking to the desired location. This places a red crosshair at that location. Then use the Insert/Graph/X-Y Plot pull-down menu to place an empty plot on the worksheet with placeholders for the expressions to be graphed and for the ranges of the x and y axes. Four variables are plotted on the y axis as shown: the top and bottom halves of the equation for the circle, the linear function, and a vertical line to represent the x . 2 constraint. In addition, the solution is included as a point. Once the graph has been created, you can use the Format/Graph/X-Y Plot pull-down menu to vary the type of graph; change the color, type, and weight of the trace of the function; and add titles, labels, and other features. The graph and the numerical values for xval and yval nicely portray the solution as the intersection of the circle and the line in the region where x . 2.

PROBLEMS

413

FIGURE 15.9 Mathcad screen for a nonlinear constrained optimization problem.

PROBLEMS 15.1 A company makes two types of products, A and B. These products are produced during a 40-hr work week and then shipped out at the end of the week. They require 20 and 5 kg of raw material per kg of product, respectively, and the company has access to 9500 kg of raw material per week. Only one product can be created at a time with production times for each of 0.04 and 0.12 hr, respectively. The plant can only store 550 kg of total product per week. Finally, the company makes profits of $45 and $20 on each unit of A and B, respectively. Each unit of product is equivalent to a kg. (a) Set up the linear programming problem to maximize profit. (b) Solve the linear programming problem graphically. (c) Solve the linear programming problem with the simplex method. (d) Solve the problem with a software package. (e) Evaluate which of the following options will raise profits the most: increasing raw material, storage, or production time. 15.2 Suppose that for Example 15.1, the gas-processing plant decides to produce a third grade of product with the following characteristics:

Supreme Raw gas Production time Storage Proﬁt

15 m3/tonne 12 hr/tonne 5 tonnes $250/tonne

In addition, suppose that a new source of raw gas has been discovered so that the total available is doubled to 154 m3/week. (a) Set up the linear programming problem to maximize profit. (b) Solve the linear programming problem with the simplex method. (c) Solve the problem with a software package. (d) Evaluate which of the following options will raise profits the most: increasing raw material, storage, or production time. 15.3 Consider the linear programming problem: Maximize f (x, y) 5 1.75x 1 1.25y

414

CONSTRAINED OPTIMIZATION

subject to 1.2x 1 2.25y # 14 x 1 1.1y # 8 2.5x 1 y # 9 x$0 y$0 Obtain the solution: (a) Graphically. (b) Using the simplex method. (c) Using an appropriate software package (for example, Excel, MATLAB, or Mathcad). 15.4 Consider the linear programming problem: Maximize f (x, y) 5 6x 1 8y subject to 5x 1 2y # 40 6x 1 6y # 60 2x 1 4y # 32 x$0 y$0 Obtain the solution: (a) Graphically. (b) Using the simplex method. (c) Using an appropriate software package (for example, Excel). 15.5 Use a software package (for example, Excel, MATLAB, Mathcad) to solve the following constrained nonlinear optimization problem: Maximize f (x, y) 5 1.2x 1 2y 2 y3 subject to 2x 1 y # 2 x$0 y$0 15.6 Use a software package (for example, Excel, MATLAB, Mathcad) to solve the following constrained nonlinear optimization problem: Maximize f (x, y) 5 15x 1 15y subject to

15.7 Consider the following constrained nonlinear optimization problem: Minimize f (x, y) 5 (x 2 3) 2 1 (y 2 3) 2 subject to x 1 2y 5 4 (a) Use a graphical approach to estimate the solution. (b) Use a software package (for example, Excel) to obtain a more accurate estimate. 15.8 Use a software package to determine the maximum of f (x, y) 5 2.25xy 1 1.75y 2 1.5x2 2 2y2 15.9 Use a software package to determine the maximum of f (x, y) 5 4x 1 2y 1 x2 2 2x4 1 2xy 2 3y2 15.10 Given the following function, f (x, y) 5 28x 1 x2 1 12y 1 4y2 2 2xy use a software package to determine the minimum: (a) Graphically. (b) Numerically. (c) Substitute the result of (b) back into the function to determine the minimum f(x, y). (d) Determine the Hessian and its determinant, and substitute the result of part (b) back into the latter to verify that a minimum has been detected. 15.11 You are asked to design a covered conical pit to store 50 m3 of waste liquid. Assume excavation costs at $100ym3, side lining costs at $50ym2, and cover cost at 25ym2. Determine the dimensions of the pit that minimize cost (a) if the side slope is unconstrained and (b) if the side slope must be less than 458. 15.12 An automobile company has two versions of the same model car for sale, a two-door coupe and the full-size four door. (a) Graphically solve how many cars of each design should be produced to maximize profit and what that profit is. (b) Solve the same problem with Excel.

Proﬁt Production time Storage Consumer demand

Two Door

Four Door

$13,500/car 15 h/car 400 cars 700/car

$15,000/car 20 h/car 350 cars 500/car

Availability 8000 h/year 240,000 cars

x2 1 y2 # 1 x 1 2y # 2.1 x$0 y$0

15.13 Og is the leader of the surprisingly mathematically advanced, though technologically run-of-the-mill, Calm Waters caveman tribe. He must decide on the number of stone clubs and stone axes to be produced for the upcoming battle against the neighboring

PROBLEMS Peaceful Sunset tribe. Experience has taught him that each club is good for, on the average, 0.45 kills and 0.65 maims, while each axe produces 0.70 kills and 0.35 maims. Production of a club requires 5.1 lb of stone and 2.1 man-hours of labor while an axe requires 3.2 lb of stone and 4.3 man-hours of labor. Og’s tribe has 240 lb of stone available for weapons production, and a total of 200 man-hours of labor available before the expected time of this battle (that Og is sure will end war for all time). Og values a kill as worth two maims in quantifying the damage inflicted on the enemy, and he wishes to produce that mix of weapons that will maximize damage. (a) Formulate this as a linear programming problem. Make sure to define your decision variables. (b) Represent this problem graphically, making sure to identify all the feasible corner points and the infeasible corner points. (c) Solve the problem graphically. (d) Solve the problem using the computer. 15.14 Develop an M-file that is expressly designed to locate a maximum with the golden-section search algorithm. In other words, set it up so that it directly finds the maximum rather than finding the minimum of 2f(x). Test your program with the same problem as Example 13.1. The function should have the following features:

415 • Check whether the guesses bracket a maximum. If not, the function should not implement the algorithm, but should return an error message. • Iterate until the relative error falls below a stopping criterion or exceeds a maximum number of iterations. • Return both the optimal x and f(x). • Use a bracketing approach (as in Example 13.2) to replace old values with new values. 15.17 The length of the longest ladder that can negotiate the corner depicted in Fig. P15.17 can be determined by computing the value of u that minimizes the following function: L(u) 5

w1 w2 1 sin u sin(p 2 a 2 u)

For the case where w1 5 w2 5 2 m, use a numerical method (including software) to develop a plot of L versus a range of a’s from 458 to 1358. FIGURE P15.17 A ladder negotiating a corner formed by two hallways.

• Iterate until the relative error falls below a stopping criterion or exceeds a maximum number of iterations. • Return both the optimal x and f(x). 15.15 Develop an M-file to locate a minimum with the goldensection search. Rather than using the standard stopping criteria (as in Fig. 13.5), determine the number of iterations needed to attain a desired tolerance. 15.16 Develop an M-file to implement parabolic interpolation to locate a minimum. Test your program with the same problem as Example 13.2. The function should have the following features: • Base it on two initial guesses, and have the program generate the third initial value at the midpoint of the interval.

L

w1

q a w2

16

CHAPTER

Case Studies: Optimization

The purpose of this chapter is to use the numerical procedures discussed in Chaps. 13 through 15 to solve actual engineering problems involving optimization. These problems are important because engineers are often called upon to come up with the “best” solution to a problem. Because many of these cases involve complex systems and interactions, numerical methods and computers are often a necessity for developing optimal solutions. The following applications are typical of those that are routinely encountered during upper-class and graduate studies. Furthermore, they are representative of problems you will address professionally. The problems are drawn from the major discipline areas of engineering: chemical/bio, civil/environmental, electrical, and mechanical/aerospace. The first application, taken from chemical/bio engineering, deals with using nonlinear constrained optimization to design an optimal cylindrical tank. The Excel Solver is used to develop the solution. Next, we use linear programming to assess a problem from civil/environmental engineering: minimizing the cost of waste treatment to meet water-quality objectives in a river. In this example, we introduce the notion of shadow prices and their use in assessing the sensitivity of a linear programming solution. The third application, taken from electrical engineering, involves maximizing the power across a potentiometer in an electric circuit. The solution involves one-dimensional unconstrained optimization. Aside from solving the problem, we illustrate how the Visual Basic macro language allows access to the golden-section search algorithm within the context of the Excel environment. Finally, the fourth application, taken from mechanical/aerospace engineering, involves determining the equilibrium position of a multi-spring system based on the minimum potential energy.

16.1

LEAST-COST DESIGN OF A TANK (CHEMICAL/BIO ENGINEERING) Background. Chemical engineers (as well as other specialists such as mechanical and civil engineers) often encounter the general problem of designing containers to transport liquids and gases. Suppose that you are asked to determine the dimensions of a small

416

16.1

LEAST-COST DESIGN OF A TANK

417

L

t

t D

Lmax

Dmax

FIGURE 16.1 Parameters for determining the optimal dimensions of a cylindrical tank.

TABLE 16.1 Parameters for determining the optimal dimensions of a cylindrical tank used to transport toxic wastes. Parameter Required volume Thickness Density Bed length Bed width Material cost Welding cost

Symbol

Value

Units

Vo t r Lmax Dmax cm cw

0.8 3 8000 2 1 4.5 20

m3 cm kg/m3 m m $/kg $/m

cylindrical tank to transport toxic waste that is to be mounted on the back of a pickup truck. Your overall objective will be to minimize the cost of the tank. However, aside from cost, you must ensure that it holds the required amount of liquid and that it does not exceed the dimensions of the truck’s bed. Note that because the tank will be carrying toxic waste, the tank thickness is specified by regulations. A schematic of the tank and bed are shown in Fig. 16.1. As can be seen, the tank consists of a cylinder with two plates welded on each end. The cost of the tank involves two components: (1) material expense, which is based on weight, and (2) welding expense based on length of weld. Note that the latter involves welding both the interior and the exterior seams where the plates connect with the cylinder. The data needed for the problem are summarized in Table 16.1. Solution. The objective here is to construct a tank for a minimum cost. The cost is related to the design variables (length and diameter) as they effect the mass of the tank and the welding lengths. Further, the problem is constrained because the tank must (1) fit within the truck bed and (2) carry the required volume of material.

418

CASE STUDIES: OPTIMIZATION

The cost consists of tank material and welding costs. Therefore, the objective function can be formulated as minimizing C 5 cm m 1 cw /w

(16.1)

where C 5 cost ($), m 5 mass (kg), /w 5 weld length (m), and cm and cw 5 cost factors for mass ($/kg) and weld length ($/m), respectively. Next, we will formulate how the mass and weld lengths are related to the dimensions of the drum. First, the mass can be calculated as the volume of material times its density. The volume of the material used to create the side walls (that is, the cylinder) can be computed as Vcylinder 5 Lp c a

2 D D 2 1 tb 2 a b d 2 2

For each circular end plate, it is Vplate 5 p a

2 D 1 tb t 2

Thus, the mass is computed by m 5 r e Lp c a

2 2 D 2 D D 1 tb 2 a b d 1 2p a 1 tb t f 2 2 2

(16.2)

where r 5 density (kg/m3). The weld length for attaching each plate is equal to the cylinder’s inside and outside circumference. For the two plates, the total weld length would be /w 5 2 c 2p a

D D 1 tb 1 2p d 5 4p(D 1 t) 2 2

(16.3)

Given values for D and L (remember, thickness t is fixed by regulations), Eqs. (16.1) through (16.3) provide a means to compute cost. Also recognize that when Eqs. (16.2) and (16.3) are substituted into Eq. (16.1), the resulting objective function is nonlinear in the unknowns. Next, we can formulate the constraints. First, we must compute how much volume can be held within the finished tank, V5

pD2 L 4

This value must be equal to the desired volume. Thus, one constraint is pD2L 5 Vo 4 where Vo is the desired volume (m3). The remaining constraints deal with ensuring that the tank will fit within the dimensions of the truck bed, L # Lmax D # Dmax

16.1

LEAST-COST DESIGN OF A TANK

419

The problem is now specified. Substituting the values from Table 16.1, it can be summarized as Maximize C 5 4.5m 1 20/w subject to pD2L 5 0.8 4 L#2 D#1 where m 5 8000 e Lp c a

2 2 D D 2 D 1 0.03b 2 a b d 1 2p a 1 0.03b 0.03 f 2 2 2

and /w 5 4p(D 1 0.03) The problem can now be solved in a number of ways. However, the simplest approach for a problem of this magnitude is to use a tool like the Excel Solver. The spreadsheet to accomplish this is shown in Fig. 16.2. For the case shown, we enter the upper limits for D and L. For this case, the volume is more than required (1.57 . 0.8).

FIGURE 16.2 Excel spreadsheet set up to evaluate the cost of a tank subject to a volume requirement and size constraints.

420

CASE STUDIES: OPTIMIZATION

Once the spreadsheet is created, the selection Solver is chosen from the Data tab. At this point a dialogue box will be displayed, querying you for pertinent information. The pertinent cells of the Solver dialogue box would be filled out as

When the OK button is selected, a dialogue box will open with a report on the success of the operation. For the present case, the Solver obtains the correct solution, which is shown in Fig. 16.3. Notice that the optimal diameter is nudging up against the constraint of 1 m. Thus, if the required capacity of the tank were increased, we would run up against this constraint and the problem would reduce to a one-dimensional search for length.

FIGURE 16.3 Results of minimization. The price is reduced from $9154 to $5723 because of the smaller volume using dimensions of D 5 0.98 m and L 5 1.05 m.

16.2

16.2

LEAST-COST TREATMENT OF WASTEWATER

421

LEAST-COST TREATMENT OF WASTEWATER (CIVIL/ENVIRONMENTAL ENGINEERING) Background. Wastewater discharges from big cities are often a major cause of river pollution. Figure 16.4 illustrates the type of system that an environmental engineer might confront. Several cities are located on a river and its tributary. Each generates pollution at a loading rate P that has units of milligrams per day (mg/d). The pollution loading is subject to waste treatment that results in a fractional removal x. Thus, the amount discharged to the river is the excess not removed by treatment, Wi 5 (1 2 xi )Pi

(16.4)

where Wi 5 waste discharge from the ith city. When the waste discharge enters the stream, it mixes with pollution from upstream sources. If complete mixing is assumed at the discharge point, the resulting concentration at the discharge point can be calculated by a simple mass balance, ci 5

Wi 1 Qu cu Qi

(16.5)

where Qu 5 flow (L/d), cu 5 concentration (mg/L) in the river immediately upstream of the discharge, and Qi 5 flow downstream of the discharge point (L/d). After the concentration at the mixing point is established, chemical and biological decomposition processes can remove some of the pollution as it flows downstream. For the present case, we will assume that this removal can be represented by a simple fractional reduction factor R. Assuming that the headwaters (that is, the river above cities 1 and 2) are pollutionfree, the concentrations at the four nodes can be computed as (1 2 x1 )P1 Q13 (1 2 x2 )P2 c2 5 Q23 R13 Q13 c1 1 R23 Q23 c2 1 (1 2 x3 )P3 c3 5 Q34 c1 5

c4 5

FIGURE 16.4 Four wastewater treatment plants discharging pollution to a river system. The river segments between the cities are labeled with circled numbers.

R34 Q34 c3 1 (1 2 x4 )P4 Q45 P2

P1

WWTP2

W2 2

WWTP1 W1 1

23 13

WWTP4

3 W3

P3

P4

WWTP3

34

W4 4

45

(16.6)

422

CASE STUDIES: OPTIMIZATION

TABLE 16.2 Parameters for four wastewater treatment plants discharging pollution to a river system, along with the resulting concentration (ci) for zero treatment. Flow, removal, and standards for the river segments are also listed. City 1 2 3 4

Pi (mg/d) 1.00 2.00 4.00 2.50

3 3 3 3

9

10 109 109 109

di ($1026/mg) 2 2 4 4

ci (mg/L)

Segment

100 40 47.3 22.5

1–3 2–3 3–4 4–5

Q (L/d) 1.00 5.00 1.10 2.50

3 3 3 3

7

10 107 108 108

R

cs (mg/L)

0.5 0.35 0.6

20 20 20 20

Next, it is recognized that the waste treatment costs a different amount, di ($1000/mg removed), at each of the facilities. Thus, the total cost of treatment (on a daily basis) can be calculated as Z 5 d1 P1 x1 1 d2 P2 x2 1 d3 P3 x3 1 d4 P4 x4

(16.7)

where Z is total daily cost of treatment ($1000/d). The final piece in the “decision puzzle” involves environmental regulations. To protect the beneficial uses of the river (for example, boating, fisheries, bathing), regulations say that the river concentration must not exceed a water-quality standard of cs. Parameters for the river system in Fig. 16.4 are summarized in Table 16.2. Notice that there is a difference in treatment cost between the upstream (1 and 2) and the downstream cities (3 and 4) because of the outmoded nature of the downstream plants. The concentration can be calculated with Eq. (16.6) and the result listed in the shaded column for the case where no waste treatment is implemented (that is, all the x’s 5 0). Notice that the standard of 20 mg/L is being violated at all mixing points. Use linear programming to determine the treatment levels that meet the water-quality standards for the minimum cost. Also, evaluate the impact of making the standard more stringent below city 3. That is, redo the exercise, but with the standards for segments 3–4 and 4–5 lowered to 10 mg/L. Solution. All the factors outlined above can be combined into the following linear programming problem: Minimize Z 5 d1P1x1 1 d2P2x2 1 d3P3x3 1 d4P4x4

(16.8)

subject to the following constraints (1 2 x1 )P1 # cs1 Q13 (1 2 x2 )P2 # cs2 Q23 R13Q13c1 1 R23Q23c2 1 (1 2 x3 )P3 # cs3 Q34

(16.9)

R34Q34c3 1 (1 2 x4 )P4 # cs4 Q45 0 # x1, x2, x3, x4 # 1

(16.10)

16.2

LEAST-COST TREATMENT OF WASTEWATER

423

FIGURE 16.5 Excel spreadsheet set up to evaluate the cost of waste treatment on a regulated river system. Column F contains the calculation of concentration according to Eq. (16.6). Cells F4 and H4 are highlighted to show the formulas used to calculate c1 and treatment cost for city 1. In addition, highlighted cell H9 shows the formula (Eq. 16.8) for total cost that is to be minimized.

Thus, the objective function is to minimize treatment cost [Eq. (16.8)] subject to the constraint that water-quality standards must be met for all parts of the system [Eq. (16.9)]. In addition, treatment cannot be negative or greater than 100% removal [Eq. (16.10)]. The problem can be solved using a variety of packages. For the present application, we use the Excel spreadsheet. As seen in Fig. 16.5, these data along with the concentration calculations can be set up nicely in the spreadsheet cells. Once the spreadsheet is created, the selection Solver is chosen from the Data tab. At this point a dialogue box will be displayed, querying you for pertinent information. The pertinent cells of the Solver dialogue box would be filled out as

Notice that not all the constraints are shown, because the dialogue box displays only six constraints at a time.

424

CASE STUDIES: OPTIMIZATION

FIGURE 16.6 Results of minimization. The water-quality standards are met at a cost of $12,600/day. Notice that despite the fact that no treatment is required for city 4, the concentration at its mixing point actually exceeds the standard.

When the OK button is selected, a dialogue box will open with a report on the success of the operation. For the present case, the Solver obtains the correct solution, which is shown in Fig. 16.6. Before accepting the solution (by selecting the OK button on the Solver Reports box), notice that three reports can be generated: Answer, Sensitivity, and Limits. Select the Sensitivity Report and then hit the OK button to accept the solution. The Solver will automatically generate a Sensitivity Report, as in Fig. 16.7. Now let us examine the solution (Fig. 16.6). Notice that the standard will be met at all the mixing points. In fact, the concentration at city 4 will actually be less than the standard (16.28 mg/L), even though no treatment would be required for city 4. As a final exercise, we can lower the standards for reaches 3–4 and 4–5 to 10 mg/L. Before doing this, we can examine the Sensitivity Report. For the present case, the key column of Fig. 16.7 is the Lagrange Multiplier (aka the “shadow price”). The shadow price is a value that expresses the sensitivity of the objective function (in our case, cost) to a unit change of one of the constraints (water-quality standards). It therefore represents the additional cost that will be incurred by making the standards more stringent. For our example, it is revealing that the largest shadow price, 2$440yDcs3, occurs for one of the standard changes (that is, downstream from city 3) that we are contemplating. This tips us off that our modification will be costly. This is confirmed when we rerun Solver with the new standards (that is, we lower cells G6 and G7 to 10). As seen in Table 16.3, the result is that treatment cost is increased from $12,600/day to $19,640/day. In addition, reducing the standard concentrations for the lower reaches means that city 4 must begin to treat its waste and city 3 must upgrade its treatment. Notice also that the treatment of the upstream cities is unaffected.

16.3

MAXIMUM POWER TRANSFER FOR A CIRCUIT

425

FIGURE 16.7 Sensitivity Report for spreadsheet set up to evaluate the cost of waste treatment on a regulated river system.

TABLE 16.3 Comparison of two scenarios involving the impact of different regulations on treatment costs. Scenario 1: All cs 5 20 City

x

c

City

x

c

1 2 3 4

0.8 0.5 0.5625 0

20 20 20 15.28

1 2 3 4

0.8 0.5 0.8375 0.264

20 20 10 10

Cost 5 $12,600

16.3

Scenario 2: Downstream cs 5 10

Cost 5 $19,640

MAXIMUM POWER TRANSFER FOR A CIRCUIT (ELECTRICAL ENGINEERING) Background. The simple resistor circuit in Fig. 16.8 contains three fixed resistors and one adjustable resistor. Adjustable resistors are called potentiometers. The values for the parameters are V 5 80 V, R1 5 8 V, R2 5 12 V, and R3 5 10 V. (a) Find the value of the adjustable resistance Ra that maximizes the power transfer across terminals 1 and 2. (b) Perform a sensitivity analysis to determine how the maximum power and the corresponding setting of the potentiometer (Ra) varies as V is varied over the range from 45 to 105 V.

426

CASE STUDIES: OPTIMIZATION R1

R2 1

FIGURE 16.8 A resistor circuit with an adjustable resistor, or potentiometer.

⫹ V

R3

⫺

Ra 2

40 Maximum power

P(Ra) 20

FIGURE 16.9 A plot of power transfer across terminals 1-2 from Fig. 16.8 as a function of the potentiometer resistance Ra.

0 0

100 Ra

50

Solution. An expression for power for the circuit can be derived from Kirchhoff’s laws as 2 VR3 Ra d R1 (Ra 1 R2 1 R3 ) 1 R3 Ra 1 R3 R2 P(Ra ) 5 Ra

c

(16.11)

Substituting the parameter values gives the plot shown in Fig. 16.9. Notice that a maximum power transfer occurs at a resistance of about 16 V. We will solve this problem in two ways with the Excel spreadsheet. First, we will employ trial-and-error and the Solver option. Then, we will develop a Visual Basic macro program to perform the sensitivity analysis. (a) An Excel spreadsheet to implement Eq. (16.11) is shown in Fig. 16.10. As indicated, Eq. (16.11) can be entered into cell B9. Then the value of Ra (cell B8) can be varied in a trial-and-error fashion until a minimum drag was determined. For this example, the result is a power of 30.03 W and a potentiometer setting of Ra 5 16.44 V. A superior approach involves using the Solver option from the spreadsheet’s Data tab. At this point a dialogue box will be displayed, querying you for pertinent information. The pertinent cells of the Solver dialogue box would be filled out as

Set target cell:

B9

Equal to ● max ❍ min ❍ equal to By changing cells

B8

0

16.3

MAXIMUM POWER TRANSFER FOR A CIRCUIT

427

FIGURE 16.10 Excel determination of maximum power across a potentiometer using trial-and-error.

When the OK button is selected, a dialogue box will open with a report on the success of the operation. For the present case, the Solver obtains the same correct solution shown in Fig. 16.10. (b) Now, although the foregoing approach is excellent for a single evaluation, it is not convenient for cases where multiple optimizations would be employed. Such would be the case for the second part of this application, where we are interested in determining how the maximum power varies for different voltage settings. Of course, the Solver could be invoked multiple times for different parameter values, but this would be inefficient. A preferable course would involve developing a macro function to come up with the optimum. Such a function is listed in Fig. 16.11. Notice how closely it resembles the goldensection-search pseudocode previously presented in Fig. 13.5. In addition, notice that a function must also be defined to compute power according to Eq. (16.11). An Excel spreadsheet utilizing this macro to evaluate the sensitivity of the solution to voltage is given in Fig. 16.12. A column of values is set up that spans the range of V’s (that is, from 45 to 105 V). A function call to the macro is written in cell B9 that references the adjacent value of V (the 45 in A9). In addition, the other parameters in the function argument are also included. Notice that, whereas the reference to V is relative, the references to the lower and upper guesses and the resistances are absolute (that is, including leading $). This was done so that when the formula is copied down, the absolute references stay fixed, whereas the relative reference corresponds to the voltage in the same row. A similar strategy is used to place Eq. (16.11) in cell C9. When the formulas are copied downward, the result is as shown in Fig. 16.12. The maximum power can be plotted to visualize the impact of voltage variations. As seen in Fig. 16.13, the power increases with V. The results for the corresponding potentiometer settings (Ra) are more interesting. The spreadsheet indicates that the same setting, 16.44 V, results in maximum power. Such a result might be difficult to intuit based on casual inspection of Eq. (16.11).

428

CASE STUDIES: OPTIMIZATION

FIGURE 16.11 Excel macro written in Visual Basic to determine a maximum with the golden-section search.

Option Explicit Function Golden(xlow, xhigh, R1, R2, R3, V) Dim iter As Integer, maxit As Integer, ea As Double, es As Double Dim fx As Double, xL As Double, xU As Double, d As Double, x1 as Double Dim x2 As Double, f1 As Double, f2 As Double, xopt As Double Const R As Double = (5 ^ 0.5 – 1) / 2 maxit = 50 es = 0.001 xL = xlow xU = xhigh iter = 1 d = R * (xU – xL) x1 = xL + d x2 = xU – d f1 = f(x1, R1, R2, R3, V) f2 = f(x2, R1, R2, R3, V) If f1 > f2 Then xopt = x1 fx = f1 Else xopt = x2 fx = f2 End If Do d = R * d If f1 > f2 Then xL = x2 x2 = x1 x1 = xL + d f2 = f1 f1 = f(x1, R1, R2, R3, V) Else xU = x1 x1 = x2 x2 = xU – d f1 = f2 f2 = f(x2, R1, R2, R3, V) End If iter = iter + 1 If f1 > f2 Then xopt = x1 fx =f1 Else xopt = x2 fx = f2 End If If xopt 0 Then ea = (1 – R) * Abs((xU – xL) / xopt) * 100 If ea = maxit Then Exit Do Loop Golden = xopt End Function Function f(Ra, R1, R2, R3, V) f = (V * R3 * Ra / (R1 * (Ra + R2 + R3) + R3 * Ra + R3 * R2)) ^ 2 / Ra END FUNCTION

16.4

1 2 3 4 5 6 7 8 9 10 11 12 13

EQUILIBRIUM AND MINIMUM POTENTIAL ENERGY

A B C Maximum Power Transfer R1 8 R2 12 R3 10 Rmin 0.1 Rmax 100 V Ra 45 16.44444 60 16.44444 75 16.44444 90 16.44444 105 16.44444

429

Call to Visual Basic macro function

D

= Golden($B$6,$B$7,$B$3,$B$4,$B$5,A9)

P(Ra) 9.501689 16.89189 26.39358 38.00676 51.73142

Power calculation

=(A9*$B$5*B9/($B$3*(B9+$B$4+$B$5)+$B$5*B9+$B$3*$B$4))^2/B9

FIGURE 16.12 Excel spreadsheet to implement a sensitivity analysis of the maximum power to variations of voltage. This routine accesses the macro program for golden-section search from Fig. 16.11.

60 P (W) 40 Ra (⍀)

20 0 45

75 V (V)

105

FIGURE 16.13 Results of sensitivity analysis of the effect of voltage variations on maximum power.

16.4

EQUILIBRIUM AND MINIMUM POTENTIAL ENERGY (MECHANICAL/AEROSPACE ENGINEERING) Background. As in Figure 16.14a, an unloaded spring can be attached to a wall mount. When a horizontal force is applied the spring stretches. The displacement is related to the force by Hooke’s law, F 5 kx. The potential energy of the deformed state consists of the difference between the strain energy of the spring and the work done by the force, PE(x) 5 0.5k x2 2 F x

(16.12)

Equation (16.12) defines a parabola. Since the potential energy will be at a minimum at equilibrium, the solution for displacement can be viewed as a one-dimensional optimization

430

(a)

CASE STUDIES: OPTIMIZATION k F2 x

(b)

La

ka x2

F

FIGURE 16.14 (a) An unloaded spring attached to a wall mount. (b) Application of a horizontal force stretches the spring where the relationship between force and displacement is described by Hooke’s law.

F1

x1 Lb

kb

(a)

(b)

FIGURE 16.15 A two-spring system: (a) unloaded, and (b) loaded.

problem. Because this equation is so easy to differentiate, we can solve for the displacement as x 5 Fyk. For example, if k 5 2 N/cm and F 5 5 N, x 5 5Ny(2 N/cm)y5 5 2.5 cm. A more interesting two-dimensional case is shown in Figure 16.15. In this system, there are two degrees of freedom in that the system can move both horizontally and vertically. In the same way that we approached the one-dimensional system, the equilibrium deformations are the values of x1 and x2 that minimize the potential energy, PE(x1, x2 ) 5 0.5ka ( 2x21 1 (La 2 x2 ) 2 2 La ) 2

(16.13)

1 0.5kb ( 2x21 1 (Lb 1 x2 ) 2 2 Lb ) 2 2 F1x1 2 F2x2 If the parameters are ka 5 9 N/cm, kb 5 2 N/cm, La 5 10 cm, Lb 5 10 cm, F1 5 2 N, and F2 5 4 N, solve for the displacements and the potential energy. Background. We can use a variety of software tools to solve this problem. For example, using MATLAB, an M-file can be developed to hold the potential energy function, function p=PE(x,ka,kb,La,Lb,F1,F2) PEa=0.5*ka*(sqrt(x(1)^2+(La-x(2))^2)-La)^2; PEb=0.5*kb*(sqrt(x(1)^2+(Lb+x(2))^2)-Lb)^2; W=F1*x(1)+F2*x(2); p=PEa+PEb-W;

The solution can then be obtained with the fminsearch function, >> ka=9;kb=2;La=10;Lb=10;F1=2;F2=4; >> [x,f]=fminsearch(@PE,[—0.5,0.5],[],ka,kb,La,Lb,F1,F2) x = 4.9523 1.2769 f = -9.6422

Thus, at equilibrium, the potential energy is 29.6422 N?cm. The connecting point is located 4.9523 cm to the right and 1.2759 cm above its original position.

PROBLEMS

431

PROBLEMS Chemical/Bio Engineering 16.1 Design the optimal cylindrical container (Fig. P16.1) that is open at one end and has walls of negligible thickness. The container is to hold 0.5 m3. Design it so that the areas of its bottom and sides are minimized. 16.2 (a) Design the optimal conical container (Fig. P16.2) that has a cover and has walls of negligible thickness. The container is to hold 0.5 m3. Design it so that the areas of its top and sides are minimized. (b) Repeat (a) but for a conical container without a cover. 16.3 Design the optimal cylindrical tank with dished ends (Fig. P16.3). The container is to hold 0.5 m3 and has walls of negligible thickness. Note that the area and volume of each of the dished ends can be computed with

L

r h

FIGURE P16.3

ph(h2 1 3r2 ) A 5 p(h2 1 r2 ) V 5 6 (a) Design the tank so that its surface area is minimized. Interpret the result. (b) Repeat part (a), but add the constraint L $ 2h.

0.4 g (d⫺1) 0.2

FIGURE P16.1 A cylindrical container with no lid. 0 Open

r

h

0

5 c (mg/L)

10

FIGURE P16.4 The speciﬁc growth rate of a yeast that produces an antibiotic versus the food concentration.

16.4 The specific growth rate of a yeast that produces an antibiotic is a function of the food concentration c, g5

FIGURE P16.2 A conical container with a lid. Lid

r

h

2c 4 1 0.8c 1 c2 1 0.2c3

As depicted in Fig. P16.4, growth goes to zero at very low concentrations due to food limitation. It also goes to zero at high concentrations due to toxicity effects. Find the value of c at which growth is a maximum. 16.5 A chemical plant makes three major products on a weekly basis. Each of these products requires a certain quantity of raw chemical and different production times, and yields different profits. The pertinent information is in Table P16.5. Note that there is sufficient warehouse space at the plant to store a total of 450 kg/week.

432

CASE STUDIES: OPTIMIZATION TABLE P16.5

Raw chemical Production time Proﬁt

Product 1

Product 2

Product 3

7 kg/kg 0.05 hr/kg $30/kg

5 kg/kg 0.1 hr/kg $30/kg

13 kg/kg 0.2 hr/kg $35/kg

(a) (b) (c) (d)

Set up a linear programming problem to maximize profit. Solve the linear programming problem with the simplex method. Solve the problem with a software package. Evaluate which of the following options will raise profits the most: increasing raw chemical, production time, or storage. 16.6 Recently chemical engineers have become involved in the area known as waste minimization. This involves the operation of a chemical plant so that impacts on the environment are minimized. Suppose a refinery develops a product Z1 made from two raw materials X and Y. The production of 1 metric tonne of the product involves 1 tonne of X and 2.5 tonnes of Y and produces 1 tonne of a liquid waste W. The engineers have come up with three alternative ways to handle the waste: • Produce a tonne of a secondary product Z2 by adding an additional tonne of X to each tonne of W. • Produce a tonne of another secondary product Z3 by adding an additional tonne of Y to each tonne of W. • Treat the waste so that it is permissible to discharge it. The products yield profits of $2500, 2$50, and $200/tonne for Z1, Z2, and Z3, respectively. Note that producing Z2 actually creates a loss. The treatment process costs $300/tonne. In addition, the company has access to a limit of 7500 and 10,000 tonnes of X and Y, respectively, during the production period. Determine how much of the products and waste must be created in order to maximize profit. 16.7 A mixture of benzene and toluene are to be separated in a flash tank. At what temperature should the tank be operated to get the highest purity toluene in the liquid phase (maximizing xT)? The pressure in the flash tank is 800 mm Hg. The units for Antoine’s equation are mm Hg and 8C for pressure and temperature, respectively. xB PsatB 1 xT PsatT 5 P log10 (PsatB ) 5 6.905 2

1211 T 1 221

log10 (PsatT ) 5 6.953 2

1344 T 1 219

16.8 A compound A will be converted into B in a stirred tank reactor. The product B and unreacted A are purified in a separation unit. Unreacted A is recycled to the reactor. A process engineer has found

Resource Availability 3000 kg 55 hr/week

that the initial cost of the system is a function of the conversion xA. Find the conversion that will result in the lowest cost system. C is a proportionality constant. Cost 5 C c a

0.6 1 1 0.6 b 1 5a b d 2 xA (1 2 xA )

16.9 In problem 16.8, only one reactor is used. If two reactors are used in series, the governing equation for the system changes. Find the conversions for both reactors (xA1 and xA2) such that the total cost of the system is minimized. Cost 5 Cca

0.6

xA1 xA2 (1 2 xA1 )

b 2

1a

12

( xx )

0.6

A1 A2

(1 2 xA2 )

b 2

1 5a

1 0.6 b d xA2

16.10 For the reaction: 2A 1 B 3 C equilibrium can be expressed as: K5

[C] [A] 2 [B]

5

[C] [A0 2 2C] 2 [B0 2 C]

If K 5 2 M21, the initial concentration of A (A0) can be varied. The initial concentration of B is fixed by the process, B0 5 100. A costs $1/M and C sells for $10/M. What would be the optimum initial concentration of A to use such that the profits would be maximized? 16.11 A chemical plant requires 106 L/day of a solution. Three sources are available at different prices and supply rates. Each source also has a different concentration of an impurity that must be kept below a minimum level to prevent interference with the chemical. The data for the three sources are summarized in the following table. Determine the amount from each source to meet the requirements at the least cost. Source 1 Source 2 Source 3 Required Cost ($yL) Supply (105 Lyday) Concentration (mgyL)

0.50 20 135

1.00 10 100

1.20 5 75

minimize $10 #100

PROBLEMS

433 x

w

y d

FIGURE P16.15 A cantilever beam.

FIGURE P16.12

P

w

d

t

H

FIGURE P16.13

16.12 You must design a triangular open channel to carry a waste stream from a chemical plant to a waste stabilization pond (Fig. P16.12). The mean velocity increases with the hydraulic radius Rh 5 Ayp, where A is the cross-sectional area and p equals the wetted perimeter. Because the maximum flow rate corresponds to the maximum velocity, the optimal design amounts to minimizing the wetted perimeter. Determine the dimensions to minimize the wetted perimeter for a given cross-sectional area. Are the relative dimensions universal? 16.13 As an agricultural engineer, you must design a trapezoidal open channel to carry irrigation water (Fig. P16.13). Determine the optimal dimensions to minimize the wetted perimeter for a cross-sectional area of 100 m2. Are the relative dimensions universal? 16.14 Find the optimal dimensions for a heated cylindrical tank designed to hold 10 m3 of fluid. The ends and sides cost $200/m2 and $100/m2, respectively. In addition, a coating is applied to the entire tank area at a cost of $50/m2. Civil/Environmental Engineering 16.15 A finite-element model of a cantilever beam subject to loading and moments (Fig. P16.15) is given by optimizing f(x, y) 5 5x2 2 5xy 1 2.5y2 2 x 2 1.5y where x 5 end displacement and y 5 end moment. Find the values of x and y that minimize f(x, y).

d

(a)

(b)

FIGURE P16.16 (a) A column supporting a compressive load P. (b) The column has a cross section shaped as a thin-walled pipe.

16.16 Suppose that you are asked to design a column to support a compressive load P, as shown in Fig. P16.16a. The column has a cross-section shaped as a thin-walled pipe as shown in Fig. P16.16b. The design variables are the mean pipe diameter d and the wall thickness t. The cost of the pipe is computed by Cost 5 f (t, d) 5 c1W 1 c2d where c1 5 4 and c2 5 2 are cost factors and W 5 weight of the pipe, W 5 p dt Hr where r 5 density of the pipe material 5 0.0025 kg/cm3. The column must support the load under compressive stress and not buckle. Therefore, Actual stress (s) # maximum compressive yield stress 5 sy 5 550 kg/cm2 Actual stress # buckling stress

434

CASE STUDIES: OPTIMIZATION below the point discharge. This point is called “critical” because it represents the location where biota that depend on oxygen (like fish) would be the most stressed. Determine the critical travel time and concentration, given the following values:

The actual stress is given by P P 5 A pdt

s5

The buckling stress can be shown to be sb 5

os 5 10 mg/L kd 5 0.1 d21 ka 5 0.6 d21

pEI H2dt

ks 5 0.05 d21 Lo 5 50 mg/L Sb 5 1 mg/L/d

where E 5 modulus of elasticity and I 5 second moment of the area of the cross section. Calculus can be used to show that

c(x, y) 5 7.9 1 0.13x 1 0.21y 2 0.05x2 2 0.016y2 2 0.007x y

p I 5 dt(d2 1 t2 ) 8 Finally, diameters of available pipes are between d1 and d2 and thicknesses between t1 and t2. Develop and solve this problem by determining the values of d and t that minimize the cost. Note that H 5 275 cm, P 5 2000 kg, E 5 900,000 kg/cm2, d1 5 1 cm, d2 5 10 cm, t1 5 0.1 cm, and t2 5 1 cm. 16.17 The Streeter-Phelps model can be used to compute the dissolved oxygen concentration in a river below a point discharge of sewage (Fig. P16.17), o 5 os 2

16.18 The two-dimensional distribution of pollutant concentration in a channel can be described by

kd Lo Sb (e2ka t 2 e2(kd 1ks)t ) 2 (1 2 e2ka t ) kd 1 ks 2 ka ka (P16.17)

where o 5 dissolved oxygen concentration (mg/L), os 5 oxygen saturation concentration (mg/L), t 5 travel time (d), Lo 5 biochemical oxygen demand (BOD) concentration at the mixing point (mg/L), kd 5 rate of decomposition of BOD (d21), ks 5 rate of settling of BOD (d21), ka 5 reaeration rate (d21), and Sb 5 sediment oxygen demand (mg/L/d). As indicated in Fig. P16.17, Eq. (P16.17) produces an oxygen “sag” that reaches a critical minimum level oc some travel time tc

FIGURE P16.17 A dissolved oxygen “sag” below a point discharge of sewage into a river.

Determine the exact location of the peak concentration given the function and the knowledge that the peak lies within the bounds 210 # x # 10 and 0 # y # 20. 16.19 The flow Q (m3/s) in an open channel can be predicted with the Manning equation Q5

1 Ac R2y3 S 1y2 n

where n 5 Manning roughness coefficient (a dimensionless number used to parameterize the channel friction), Ac 5 cross-sectional area of the channel (m2), S 5 channel slope (dimensionless, meters drop per meter length), and R 5 hydraulic radius (m), which is related to more fundamental parameters by R 5 AcyP, where P 5 wetted perimeter (m). As the name implies, the wetted perimeter is the length of the channel sides and bottom that is under water. For example, for a rectangular channel, it is defined as P 5 B 1 2H, where H 5 depth (m). Suppose that you are using this formula to design a lined canal (note that farmers line canals to minimize leakage losses). (a) Given the parameters n 5 0.035, S 5 0.003, and Q 5 1 m3/s, determine the values of B and H that minimize the wetted perimeter. Note that such a calculation would minimize cost if lining costs were much larger than excavation costs. (b) Repeat part (a), but include the cost of excavation. To do this minimize the following cost function, C 5 c 1 Ac 1 c 2 P

12 o (mg/L) oc

where c1 is a cost factor for excavation 5 $100/m2 and c2 is a cost factor for lining $50/m. (c) Discuss the implications of your results. 16.20 A cylindrical beam carries a compression load P 5 3000 kN. To prevent the beam from buckling, this load must be less than a critical load,

os

8 o 4 0

5

0 tc

10 t (d)

15

20

Pc 5

p2EI L2

PROBLEMS where E 5 Young’s modulus 5 200 3 109 N/m2, I 5 pr4y4 (the area moment of inertia for a cylindrical beam of radius r), and L is the beam length. If the volume of beam V cannot exceed 0.075 m3, find the largest height L that can be utilized and the corresponding radius. 16.21 The Splash River has a flow rate of 2 3 106 m3/d, of which up to 70% can be diverted into two channels where it flows through Splish County. These channels are used for transportation, irrigation, and electric power generation, with the latter two being sources of revenue. The transportation use requires a minimum diverted flow rate of 0.3 3 106 m3/d for Channel 1 and 0.2 3 106 m3/d for Channel 2. For political reasons it has been decided that the absolute difference between the flow rates in the two channels cannot exceed 40% of the total flow diverted into the channels. The Splish County Water Management Board has also limited maintenance costs for the channel system to be no more than $1.8 3 106 per year. Annual maintenance costs are estimated based on the daily flow rate. Channel 1 costs per year are estimated by multiplying $1.1 times the m3/d of flow; while for Channel 2 the multiplication factor is $1.4 per m3/d. Electric power production revenue is also estimated based on daily flow rate. For Channel 1 this is $4.0 per m3/d, while for Channel 2 it is $3.0 per m3/d. Annual revenue from irrigation is also estimated based on daily flow rate, but the flow rates must first be corrected for water loss in the channels previous to delivery for irrigation. This loss is 30% in Channel 1 and 20% in Channel 2. In both channels the revenue is $3.2 per m3/d. Determine the flows in the channels that maximize profit. 16.22 Determine the beam cross-sectional areas that result in the minimum weight for the truss we studied in Sec. 12.2 (Fig. 12.4). The critical buckling and maximum tensile strengths of compression and tension members are 10 and 20 ksi, respectively. The truss is to be constructed of steel (density 5 3.5 lb/ft-in2). Note that the length of the horizontal member (2) is 50 ft. Also, recall that the stress in each member is equal to the force divided by cross-sectional area. Set up the problem as a linear programming problem. Obtain the solution graphically and with the Excel Solver. Electrical Engineering 16.23 A total charge Q is uniformly distributed around a ringshaped conductor with radius a. A charge q is located at a distance x from the center of the ring (Fig. P16.23). The force exerted on the charge by the ring is given by F5

q Qx 1 4pe0 (x2 1 a2 ) 3y2

where e0 5 8.85 3 10212 C2y(N m2), q 5 Q 5 2 3 1025 C, and a 5 0.9 m. Determine the distance x where the force is a maximum.

435

a x

q

Q

FIGURE P16.23

16.24 A system consists of two power plants that must deliver loads over a transmission network. The costs of generating power at plants 1 and 2 are given by F1 5 2p1 1 2 F2 5 10p2 where p1 and p2 5 power produced by each plant. The losses of power due to transmission L are given by L1 5 0.2p1 1 0.1p2 L2 5 0.2p1 1 0.5p2 The total demand for power is 30 and p1 must not exceed 42. Determine the power generation needed to meet demands while minimizing cost using an optimization routine such as those found in, for example, Excel, MATLAB, or Mathcad software. 16.25 The torque transmitted to an induction motor is a function of the slip between the rotation of the stator field and the rotor speed s where slip is defined as s5

n 2 nR n

where n 5 revolutions per second of rotating stator speed and nR 5 rotor speed. Kirchhoff’s laws can be used to show that the torque (expressed in dimensionless form) and slip are related by T5

15(s 2 s2 ) (1 2 s) (4s2 2 3s 1 4)

Figure P16.25 shows this function. Use a numerical method to determine the slip at which the maximum torque occurs. 16.26 (a) A computer equipment manufacturer produces scanners and printers. The resources needed for producing these devices and the corresponding profits are Device Capital ($/unit) Labor (hr/unit) Proﬁt ($/unit) Scanner Printer

300 400

20 10

500 400

436

CASE STUDIES: OPTIMIZATION

T 4

D 20,000

3

Minimum

2

10,000

Total

1 0

Friction

Lift 0

2

6

4

8

0

10 s

FIGURE P16.25 Torque transmitted to an inductor as a function of slip.

400

0

800

1,200 V

FIGURE P16.28 Plot of drag versus velocity for an airfoil.

If there are $127,000 worth of capital and 4270 hr of labor available each day, how many of each device should be produced per day to maximize profit? (b) Repeat the problem, but now assume that the profit for each printer sold Pp depends on the number of printers produced Xp, as in

F

Pp 5 400 2 Xp x

16.27 A manufacturer provides specialized microchips. During the next 3 months, its sales, costs, and available time are Month 1

Month 2

Month 3

1000 100 110 2400 720

2500 100 120 2400 720

2200 120 130 2400 720

Chips required Cost regular time ($/chip) Cost overtime ($/chip) Regular operation time (hr) Overtime (hr)

There are no chips in stock at the beginning of the first month. It takes 1.5 hr of production time to produce a chip and costs $5 to store a chip from one month to the next. Determine a production schedule that meets the demand requirements, does not exceed the monthly production time limitations, and minimizes cost. Note that no chips should be in stock at the end of the 3 months. Mechanical/Aerospace Engineering 16.28 The total drag on an airfoil can be estimated by D 5 0.01s V 2 1 friction

0.95 W 2 a b s V lift

F

FIGURE P16.29 Roller bearings.

where D 5 drag, s 5 ratio of air density between the flight altitude and sea level, W 5 weight, and V 5 velocity. As seen in Fig. P16.28, the two factors contributing to drag are affected differently as velocity increases. Whereas friction drag increases with velocity, the drag due to lift decreases. The combination of the two factors leads to a minimum drag. (a) If s 5 0.5 and W 5 15,000, determine the minimum drag and the velocity at which it occurs. (b) In addition, develop a sensitivity analysis to determine how this optimum varies in response to a range of W 5 12,000 to 18,000 with s 5 0.5. 16.29 Roller bearings are subject to fatigue failure caused by large contact loads F (Fig. P16.29).

PROBLEMS

437

The problem of finding the location of the maximum stress along the x axis can be shown to be equivalent to maximizing the function 0.4

ka

0.4 f (x) 5 2 21 1 x a1 2 b1x 2 1 1 x2 21 1 x

A

B

C

D

2.5 3.5 375

1.5 3 275

2.75 3 475

2 2 325

The capacities of the fabrication and finishing shops over the next month are 640 and 960 hours, respectively. Determine how many of each part should be produced in order to maximize profit. 16.32 In a similar fashion to the case study described in Sec. 16.4, develop the potential energy function for the system depicted in Fig. P16.32. Develop contour and surface plots in MATLAB. Minimize the potential energy function in order to determine the equilibrium displacements x1 and x2 given the forcing function F 5 100 N, and the parameter ka 5 20 and kb 5 15 N/m. 16.33 Recent interest in competitive and recreational cycling has meant that engineers have directed their skills toward the design

F

2

FIGURE P16.32 Two frictionless masses connected to a wall by a pair of linear elastic springs.

w

h

ᐉ x y

(a)

Part

x2 kb

1

2

Find the x that maximizes f(x). 16.30 An aerospace company is developing a new fuel additive for commercial airliners. The additive is composed of three ingredients: X, Y, and Z. For peak performance, the total amount of additive must be at least 6 mL/L of fuel. For safety reasons, the sum of the highly flammable X and Y ingredients must not exceed 2.5 mL/L. In addition, the amount of the X ingredient must always be equal to or greater than the Y, and the Z must be greater than half the Y. If the cost per mL for the ingredients X, Y, and Z is 0.05, 0.025, and 0.15, respectively, determine the minimum cost mixture for each liter of fuel. 16.31 A manufacturing firm produces four types of automobile parts. Each is first fabricated and then finished. The required worker hours and profit for each part are

Fabrication time (hr/100 units) Finishing time (hr/100 units) Proﬁt ($/100 units)

x1

F

(b)

FIGURE P16.33 (a) A mountain bike along with (b) a free-body diagram for a part of the frame.

and testing of mountain bikes (Fig. P16.33a). Suppose that you are given the task of predicting the horizontal and vertical displacement of a bike bracketing system in response to a force. Assume the forces you must analyze can be simplified as depicted in Fig. P16.33b. You are interested in testing the response of the truss to a force exerted in any number of directions designated by the angle u. The parameters for the problem are E 5 Young’s modulus 5 231011 Pa, A 5 cross-sectional area 5 0.0001 m2, w 5 width 5 0.44 m, / 5 length 5 0.56 m, and h 5 height 5 0.5 m. The displacements x and y can be solved by determining the values that yield a minimum potential energy. Determine the displacements for a force of 10,000 N and a range of u’s from 08 (horizontal) to 908 (vertical).

EPILOGUE: PART FOUR The epilogues of other parts of this book contain a discussion and a tabular summary of the trade-offs among various methods as well as important formulas and relationships. Most of the methods of this part are quite complicated and, consequently, cannot be summarized with simple formulas and tabular summaries. Therefore, we deviate somewhat here by providing the following narrative discussion of trade-offs and further references.

PT4.4

TRADE-OFFS Chapter 13 dealt with finding the optimum of an unconstrained function of a single variable. The golden-section search method is a bracketing method requiring that an interval containing a single optimum be known. It has the advantage that it minimizes function evaluations and always converges. Parabolic interpolation also works best when implemented as a bracketing method, although it can also be programmed as an open method. However, in such cases, it may diverge. Both the golden-section search and parabolic interpolation do not require derivative evaluations. Thus, they are both appropriate methods when the bracket can be readily defined and function evaluations are costly. Newton’s method is an open method not requiring that an optimum be bracketed. It can be implemented in a closed-form representation when first and second derivatives can be determined analytically. It can also be implemented in a fashion similar to the secant method with finite-difference representations of the derivatives. Although Newton’s method converges rapidly near the optimum, it is often divergent for poor guesses. Convergence is also dependent on the nature of the function. Finally, hybrid approaches are available that orchestrate various methods to attain both reliability and efficiency. Brent’s method does this by combining the reliable goldensection search with speedy parabolic interpolation. Chapter 14 covered two general types of methods to solve multidimensional unconstrained optimization problems. Direct methods such as random searches and univariate searches do not require the evaluation of the function’s derivatives and are often inefficient. However they also provide a tool to find global rather than local optima. Pattern search methods like Powell’s method can be very efficient and also do not require derivative evaluation. Gradient methods use either first and sometimes second derivatives to find the optimum. The method of steepest ascent/descent provides a reliable but sometimes slow approach. In contrast, Newton’s method often converges rapidly when in the vicinity of a root, but sometimes suffers from divergence. The Marquardt method uses the steepest descent method at the starting location far away from the optimum and switches to Newton’s method near the optimum in an attempt to take advantage of the strengths of each method.

438

PT4.5

ADDITIONAL REFERENCES

439

The Newton method can be computationally costly because it requires computation of both the gradient vector and the Hessian matrix. Quasi-Newton approaches attempt to circumvent these problems by using approximations to reduce the number of matrix evaluations (particularly the evaluation, storage, and inversion of the Hessian). Research investigations continue today that explore the characteristics and respective strengths of various hybrid and tandem methods. Some examples are the Fletcher-Reeves conjugate gradient method and Davidon-Fletcher-Powell quasi-Newton methods. Chapter 15 was devoted to constrained optimization. For linear problems, linear programming based on the simplex method provides an efficient means to obtain solutions. Approaches such as the GRG method are available to solve nonlinear constrained problems. Software packages include a wide variety of optimization capabilities. As described in Chap. 15, Excel, MATLAB software, and Mathcad all have built-in search capabilities that can be used for both one-dimensional and multidimensional problems routinely encountered in engineering and science.

PT4.5

ADDITIONAL REFERENCES General overviews of optimization including some algorithms can be found in Press et al. (2007) and Moler (2004). For multidimensional problems, additional information can be found in Dennis and Schnabel (1996), Fletcher (1980, 1981), Gill et al. (1981), and Luenberger (1984). In addition, there are a number of advanced methods that are well suited for specific problem contexts. For example, genetic algorithms use strategies inspired by evolutionary biology such as inheritance, mutation, and selection. Because they do not make assumptions regarding the underlying search space, such evolutionary algorithms are often useful for large problems with many local optima. Related techniques include simulated annealing and Tabu search. Hillier and Lieberman (2005) provide overviews of these and a number of other advanced techniques.

PART FIVE

CURVE FITTING PT5.1

MOTIVATION Data are often given for discrete values along a continuum. However, you may require estimates at points between the discrete values. The present part of this book describes techniques to fit curves to such data to obtain intermediate estimates. In addition, you may require a simplified version of a complicated function. One way to do this is to compute values of the function at a number of discrete values along the range of interest. Then, a simpler function may be derived to fit these values. Both of these applications are known as curve fitting. There are two general approaches for curve fitting that are distinguished from each other on the basis of the amount of error associated with these data. First, where these data exhibit a significant degree of error or “noise,” the strategy is to derive a single curve that represents the general trend of these data. Because any individual data point may be incorrect, we make no effort to intersect every point. Rather, the curve is designed to follow the pattern of the points taken as a group. One approach of this nature is called least-squares regression (Fig. PT5.1a). Second, where these data are known to be very precise, the basic approach is to fit a curve or a series of curves that pass directly through each of the points. Such data usually originate from tables. Examples are values for the density of water or for the heat capacity of gases as a function of temperature. The estimation of values between well-known discrete points is called interpolation (Fig. PT5.1b and c). PT5.1.1 Noncomputer Methods for Curve Fitting The simplest method for fitting a curve to data is to plot the points and then sketch a line that visually conforms to these data. Although this is a valid option when quick estimates are required, the results are dependent on the subjective viewpoint of the person sketching the curve. For example, Fig. PT5.1 shows sketches developed from the same set of data by three engineers. The first did not attempt to connect the points, but rather, characterized the general upward trend of these data with a straight line (Fig. PT5.1a). The second engineer used straight-line segments or linear interpolation to connect the points (Fig. PT5.1b). This is a very common practice in engineering. If the values are truly close to being linear or are spaced closely, such an approximation provides estimates that are adequate for many engineering calculations. However, where the underlying relationship is highly curvilinear or these data are widely spaced, significant errors can be introduced by such linear interpolation. The third engineer used curves to try to capture the meanderings suggested by these data (Fig. PT5.1c). A fourth or fifth engineer would likely develop alternative fits. Obviously, our goal here is to develop systematic and objective methods for the purpose of deriving such curves. 441

442

CURVE FITTING

f (x)

(a)

x

(b)

x

(c)

x

f (x)

f (x)

FIGURE PT5.1 Three attempts to ﬁt a “best” curve through ﬁve data points. (a) Least-squares regression, (b) linear interpolation, and (c) curvilinear interpolation.

PT5.1.2 Curve Fitting and Engineering Practice Your first exposure to curve fitting may have been to determine intermediate values from tabulated data—for instance, from interest tables for engineering economics or from steam tables for thermodynamics. Throughout the remainder of your career, you will have frequent occasion to estimate intermediate values from such tables. Although many of the widely used engineering properties have been tabulated, there are a great many more that are not available in this convenient form. Special cases and new problem contexts often require that you measure your own data and develop your own predictive relationships. Two types of applications are generally encountered when fitting experimental data: trend analysis and hypothesis testing. Trend analysis represents the process of using the pattern of these data to make predictions. For cases where these data are measured with high precision, you might

PT5.2

MATHEMATICAL BACKGROUND

443

utilize interpolating polynomials. Imprecise data are often analyzed with least-squares regression. Trend analysis may be used to predict or forecast values of the dependent variable. This can involve extrapolation beyond the limits of the observed data or interpolation within the range of the data. All fields of engineering commonly involve problems of this type. A second engineering application of experimental curve fitting is hypothesis testing. Here, an existing mathematical model is compared with measured data. If the model coefficients are unknown, it may be necessary to determine values that best fit the observed data. On the other hand, if estimates of the model coefficients are already available, it may be appropriate to compare predicted values of the model with observed values to test the adequacy of the model. Often, alternative models are compared and the “best” one is selected on the basis of empirical observations. In addition to the above engineering applications, curve fitting is important in other numerical methods such as integration and the approximate solution of differential equations. Finally, curve-fitting techniques can be used to derive simple functions to approximate complicated functions.

PT5.2

MATHEMATICAL BACKGROUND The prerequisite mathematical background for interpolation is found in the material on Taylor series expansions and finite divided differences introduced in Chap. 4. Least-squares regression requires additional information from the field of statistics. If you are familiar with the concepts of the mean, standard deviation, residual sum of the squares, normal distribution, and confidence intervals, feel free to skip the following pages and proceed directly to PT5.3. If you are unfamiliar with these concepts or are in need of a review, the following material is designed as a brief introduction to these topics. PT5.2.1 Simple Statistics Suppose that in the course of an engineering study, several measurements were made of a particular quantity. For example, Table PT5.1 contains 24 readings of the coefficient of thermal expansion of a structural steel. Taken at face value, these data provide a limited amount of information—that is, that the values range from a minimum of 6.395 to a maximum of 6.775. Additional insight can be gained by summarizing these data in one or more well-chosen statistics that convey as much information as possible about specific characteristics of the data set. These descriptive statistics are most often selected TABLE PT5.1 Measurements of the coefﬁcient of thermal expansion of structural steel [3 1026 in/(in ? 8F)]. 6.495 6.665 6.755 6.565

6.595 6.505 6.625 6.515

6.615 6.435 6.715 6.555

6.635 6.625 6.575 6.395

6.485 6.715 6.655 6.775

6.555 6.655 6.605 6.685

444

CURVE FITTING

to represent (1) the location of the center of the distribution of these data and (2) the degree of spread of the data set. The most common location statistic is the arithmetic mean. The arithmetic mean (y) of a sample is defined as the sum of the individual data points (yi) divided by the number of points (n), or y5

g yi n

(PT5.1)

where the summation (and all the succeeding summations in this introduction) is from i 5 1 through n. The most common measure of spread for a sample is the standard deviation (sy) about the mean, sy 5

St Bn 2 1

(PT5.2)

where St is the total sum of the squares of the residuals between the data points and the mean, or St 5 g (yi 2 y) 2

(PT5.3)

Thus, if the individual measurements are spread out widely around the mean, St (and, consequently, sy) will be large. If they are grouped tightly, the standard deviation will be small. The spread can also be represented by the square of the standard deviation, which is called the variance: s2y 5

g (yi 2 y) 2 n21

(PT5.4)

Note that the denominator in both Eqs. (PT5.2) and (PT5.4) is n 2 1. The quantity n 2 1 is referred to as the degrees of freedom. Hence St and sy are said to be based on n 2 1 degrees of freedom. This nomenclature derives from the fact that the sum of the quantities upon which St is based (that is, y 2 y1, y 2 y2, p , y 2 yn) is zero. Consequently, if y is known and n 2 1 of the values are specified, the remaining value is fixed. Thus, only n 2 1 of the values are said to be freely determined. Another justification for dividing by n 2 1 is the fact that there is no such thing as the spread of a single data point. For the case where n 5 1, Eqs. (PT5.2) and (PT5.4) yield a meaningless result of infinity. It should be noted that an alternative, more convenient formula is available to compute the standard deviation, s2y 5

g y2i 2 ( g yi ) 2 yn n21

This version does not require precomputation of y and yields an identical result as Eq. (PT5.4).

PT5.2

MATHEMATICAL BACKGROUND

445

A final statistic that has utility in quantifying the spread of data is the coefficient of variation (c.v.). This statistic is the ratio of the standard deviation to the mean. As such, it provides a normalized measure of the spread. It is often multiplied by 100 so that it can be expressed in the form of a percent: c.v. 5

sy y

100%

(PT5.5)

Notice that the coefficient of variation is similar in spirit to the percent relative error (t) discussed in Sec. 3.3. That is, it is the ratio of a measure of error (sy) to an estimate of the true value (y). EXAMPLE PT5.1

Simple Statistics of a Sample Problem Statement. Compute the mean, variance, standard deviation, and coefficient of variation for the data in Table PT5.1. TABLE PT5.2 Computations for statistics for the readings of the coefﬁcient of thermal expansion. The frequencies and bounds are developed to construct the histogram in Fig. PT5.2. Interval i

yi

2 (yi 2 y w)

Frequency

Lower Bound

Upper Bound

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

6.395 6.435 6.485 6.495 6.505 6.515 6.555 6.555 6.565 6.575 6.595 6.605 6.615 6.625 6.625 6.635 6.655 6.655 6.665 6.685 6.715 6.715 6.755 6.775

0.042025 0.027225 0.013225 0.011025 0.009025 0.007225 0.002025 0.002025 0.001225 0.000625 0.000025 0.000025 0.000225 0.000625 0.000625 0.001225 0.003025 0.003025 0.004225 0.007225 0.013225 0.013225 0.024025 0.030625

1 1

6.36 6.40

6.40 6.44

4

6.48

6.52

2

6.52

6.56

3

6.56

6.60

5

6.60

6.64

3

6.64

6.68

3

6.68

6.72

1 1

6.72 6.76

6.76 6.80

S

158.4

0.217000

446

CURVE FITTING

Solution. These data are added (Table PT5.2), and the results are used to compute [Eq. (PT5.1)] y5

158.4 5 6.6 24

As in Table PT5.2, the sum of the squares of the residuals is 0.217000, which can be used to compute the standard deviation [Eq. (PT5.2)]: sy 5

0.217000 5 0.097133 B 24 2 1

the variance [Eq. (PT5.4)]: s2y 5 0.009435 and the coefficient of variation [Eq. (PT5.5)]: c.v. 5

0.097133 100% 5 1.47% 6.6

PT5.2.2 The Normal Distribution Another characteristic that bears on the present discussion is the data distribution—that is, the shape with which these data are spread around the mean. A histogram provides a simple visual representation of the distribution. As seen in Table PT5.2, the histogram is constructed by sorting the measurements into intervals. The units of measurement are plotted on the abscissa and the frequency of occurrence of each interval is plotted on the ordinate. Thus, five of the measurements fall between 6.60 and 6.64. As in Fig. PT5.2, the histogram suggests that most of these data are grouped close to the mean value of 6.6. If we have a very large set of data, the histogram often can be approximated by a smooth curve. The symmetric, bell-shaped curve superimposed on Fig. PT5.2 is one such characteristic shape—the normal distribution. Given enough additional measurements, the histogram for this particular case could eventually approach the normal distribution. The concepts of the mean, standard deviation, residual sum of the squares, and normal distribution all have great relevance to engineering practice. A very simple example is their use to quantify the confidence that can be ascribed to a particular measurement. If a quantity is normally distributed, the range defined by y 2 sy to y 1 sy will encompass approximately 68 percent of the total measurements. Similarly, the range defined by y 2 2sy to y 1 2sy will encompass approximately 95 percent. For example, for the data in Table PT5.1 (y 5 6.6 and sy 5 0.097133), we can make the statement that approximately 95 percent of the readings should fall between 6.405734 and 6.794266. If someone told us that they had measured a value of 7.35, we would suspect that the measurement might be erroneous. The following section elaborates on such evaluations. PT5.2.3 Estimation of Conﬁdence Intervals As should be clear from the previous sections, one of the primary aims of statistics is to estimate the properties of a population based on a limited sample drawn from that

PT5.2

MATHEMATICAL BACKGROUND

447

5

Frequency

4 3 2 1 0

6.4

6.6

6.8

FIGURE PT5.2 A histogram used to depict the distribution of data. As the number of data points increases, the histogram could approach the smooth, bell-shaped curve called the normal distribution.

population. Clearly, it is impossible to measure the coefficient of thermal expansion for every piece of structural steel that has ever been produced. Consequently, as seen in Tables PT5.1 and PT5.2, we can randomly make a number of measurements and, on the basis of the sample, attempt to characterize the pro