Numerical methods 6th edition chapra

Knowledge and understanding are prerequisites for the effective implementation of any tool. No matter how impressive your tool chest, you will be hardpressed 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 twopronged 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 problemsolving 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 MATHEMATI

Sixth Edition Features include: which are based on exciting new areas such as bioengineering and differential equations students using this text will be able to apply their new skills to their chosen field Electronic Textbook Options an online resource where students can purchase the complete text in a digital format at almost half the cost of the traditional textbook Students can access the text online for one year learning, which include full text search, notes and highlighting, and email tools for sharing contact your sales representative or visit www.CourseSmart.com Sixth Edition Numerical Methods for Engineers Chapra Canale Steven C Chapra Raymond P Canale MD DALIM #1009815 03/12/09 CYAN MAG YELO BLK For more information, please visit www.mhhe.com/chapra for Engineers adaptive quadrature Numerical Methods The sixth edition of Numerical Methods for Engineers offers an innovative and accessible presentation of numerical methods; the book has earned the Meriam-Wiley award, which is given by the American Society for Engineering Education for the best textbook Because software packages are now regularly used for numerical analysis, this eagerly anticipated revision maintains its strong focus on appropriate use of computational tools cha01064_fm.qxd 3/25/09 10:51 AM Page i Numerical Methods for Engineers SIXTH EDITION Steven C Chapra Berger Chair in Computing and Engineering Tufts University Raymond P Canale Professor Emeritus of Civil Engineering University of Michigan cha01064_fm.qxd 3/25/09 10:51 AM Page ii NUMERICAL METHODS FOR ENGINEERS, SIXTH EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020 Copyright © 2010 by The McGraw-Hill Companies, Inc All rights reserved Previous editions © 2006, 2002, and 1998 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 The McGraw-Hill Companies, Inc., 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 VNH/VNH ISBN 978–0–07–340106–5 MHID 0–07–340106–4 Global Publisher: Raghothaman Srinivasan Sponsoring Editor: Debra B Hash Director of Development: Kristine Tibbetts Developmental Editor: Lorraine K Buczek Senior Marketing Manager: Curt Reynolds Project Manager: Joyce Watters Lead Production Supervisor: Sandy Ludovissy Associate Design Coordinator: Brenda A Rolwes Cover Designer: Studio Montage, St Louis, Missouri (USE) Cover Image: © BrandX/JupiterImages Compositor: Macmillan Publishing Solutions Typeface: 10/12 Times Roman Printer: R R Donnelley Jefferson City, MO All credits appearing on page or at the end of the book are considered to be an extension of the copyright page MATLAB™ is a registered trademark of The MathWorks, Inc Library of Congress Cataloging-in-Publication Data Chapra, Steven C Numerical methods for engineers / Steven C Chapra, Raymond P Canale — 6th ed p cm Includes bibliographical references and index ISBN 978–0–07–340106–5 — ISBN 0–07–340106–4 (hard copy : alk paper) Engineering mathematics—Data processing Numerical calculations—Data processing Microcomputers— Programming I Canale, Raymond P II Title TA345.C47 2010 518.02462—dc22 2008054296 www.mhhe.com cha01064_fm.qxd 3/25/09 10:51 AM Page iii To Margaret and Gabriel Chapra Helen and Chester Canale cha01064_fm.qxd 3/26/09 5:25 PM Page iv CONTENTS PREFACE xiv GUIDED TOUR xvi ABOUT THE AUTHORS xviii PART ONE MODELING, COMPUTERS, AND ERROR ANALYSIS PT1.1 Motivation PT1.2 Mathematical Background PT1.3 Orientation CHAPTER Mathematical Modeling and Engineering Problem Solving 11 1.1 A Simple Mathematical Model 11 1.2 Conservation Laws and Engineering 18 Problems 21 CHAPTER Programming and Software 25 2.1 Packages and Programming 25 2.2 Structured Programming 26 2.3 Modular Programming 35 2.4 Excel 37 2.5 MATLAB 41 2.6 Mathcad 45 2.7 Other Languages and Libraries 46 Problems 47 CHAPTER Approximations and Round-Off Errors 52 3.1 Significant Figures 53 3.2 Accuracy and Precision 55 3.3 Error Definitions 56 3.4 Round-Off Errors 62 Problems 76 iv cha01064_fm.qxd 3/25/09 10:51 AM Page v CONTENTS v CHAPTER Truncation Errors and the Taylor Series 78 4.1 The Taylor Series 78 4.2 Error Propagation 94 4.3 Total Numerical Error 98 4.4 Blunders, Formulation Errors, and Data Uncertainty 103 Problems 105 EPILOGUE: PART ONE 107 PT1.4 Trade-Offs 107 PT1.5 Important Relationships and Formulas 110 PT1.6 Advanced Methods and Additional References 110 PART TWO ROOTS OF EQUATIONS 113 PT2.1 Motivation 113 PT2.2 Mathematical Background 115 PT2.3 Orientation 116 CHAPTER Bracketing Methods 120 5.1 Graphical Methods 120 5.2 The Bisection Method 124 5.3 The False-Position Method 132 5.4 Incremental Searches and Determining Initial Guesses 138 Problems 139 CHAPTER Open Methods 142 6.1 Simple Fixed-Point Iteration 143 6.2 The Newton-Raphson Method 148 6.3 The Secant Method 154 6.4 Brent’s Method 159 6.5 Multiple Roots 164 6.6 Systems of Nonlinear Equations 167 Problems 171 CHAPTER Roots of Polynomials 174 7.1 Polynomials in Engineering and Science 174 7.2 Computing with Polynomials 177 7.3 Conventional Methods 180 cha01064_fm.qxd 3/25/09 10:51 AM vi Page vi CONTENTS 7.4 Müller’s Method 181 7.5 Bairstow’s Method 185 7.6 Other Methods 190 7.7 Root Location with Software Packages 190 Problems 200 CHAPTER Case Studies: Roots of Equations 202 8.1 Ideal and Nonideal Gas Laws (Chemical/Bio Engineering) 202 8.2 Greenhouse Gases and Rainwater (Civil/Environmental Engineering) 205 8.3 Design of an Electric Circuit (Electrical Engineering) 207 8.4 Pipe Friction (Mechanical/Aerospace Engineering) 209 Problems 213 EPILOGUE: PART TWO 223 PT2.4 Trade-Offs 223 PT2.5 Important Relationships and Formulas 224 PT2.6 Advanced Methods and Additional References 224 PART THREE LINEAR ALGEBRAIC EQUATIONS 227 PT3.1 Motivation 227 PT3.2 Mathematical Background 229 PT3.3 Orientation 237 CHAPTER Gauss Elimination 241 9.1 Solving Small Numbers of Equations 241 9.2 Naive Gauss Elimination 248 9.3 Pitfalls of Elimination Methods 254 9.4 Techniques for Improving Solutions 260 9.5 Complex Systems 267 9.6 Nonlinear Systems of Equations 267 9.7 Gauss-Jordan 269 9.8 Summary 271 Problems 271 CHAPTER 10 LU Decomposition and Matrix Inversion 274 10.1 LU Decomposition 274 10.2 The Matrix Inverse 283 10.3 Error Analysis and System Condition 287 Problems 293 cha01064_fm.qxd 3/25/09 10:51 AM Page vii CONTENTS CHAPTER 11 Special Matrices and Gauss-Seidel 296 11.1 Special Matrices 296 11.2 Gauss-Seidel 300 11.3 Linear Algebraic Equations with Software Packages 307 Problems 312 CHAPTER 12 Case Studies: Linear Algebraic Equations 315 12.1 Steady-State Analysis of a System of Reactors (Chemical/Bio Engineering) 315 12.2 Analysis of a Statically Determinate Truss (Civil/Environmental Engineering) 318 12.3 Currents and Voltages in Resistor Circuits (Electrical Engineering) 322 12.4 Spring-Mass Systems (Mechanical/Aerospace Engineering) 324 Problems 327 EPILOGUE: PART THREE 337 PT3.4 Trade-Offs 337 PT3.5 Important Relationships and Formulas 338 PT3.6 Advanced Methods and Additional References 338 PART FOUR OPTIMIZATION 341 PT4.1 Motivation 341 PT4.2 Mathematical Background 346 PT4.3 Orientation 347 CHAPTER 13 One-Dimensional Unconstrained Optimization 351 13.1 Golden-Section Search 352 13.2 Parabolic Interpolation 359 13.3 Newton’s Method 361 13.4 Brent’s Method 364 Problems 364 CHAPTER 14 Multidimensional Unconstrained Optimization 367 14.1 Direct Methods 368 14.2 Gradient Methods 372 Problems 385 vii cha01064_fm.qxd 3/26/09 5:25 PM viii Page viii CONTENTS CHAPTER 15 Constrained Optimization 387 15.1 Linear Programming 387 15.2 Nonlinear Constrained Optimization 398 15.3 Optimization with Software Packages 399 Problems 410 CHAPTER 16 Case Studies: Optimization 413 16.1 Least-Cost Design of a Tank (Chemical/Bio Engineering) 413 16.2 Least-Cost Treatment of Wastewater (Civil/Environmental Engineering) 418 16.3 Maximum Power Transfer for a Circuit (Electrical Engineering) 422 16.4 Equilibrium and Minimum Potential Energy (Mechanical/Aerospace Engineering) 426 Problems 428 EPILOGUE: PART FOUR 436 PT4.4 Trade-Offs 436 PT4.5 Additional References 437 PART FIVE CURVE FITTING 439 PT5.1 Motivation 439 PT5.2 Mathematical Background 441 PT5.3 Orientation 450 CHAPTER 17 Least-Squares Regression 454 17.1 Linear Regression 454 17.2 Polynomial Regression 470 17.3 Multiple Linear Regression 474 17.4 General Linear Least Squares 477 17.5 Nonlinear Regression 481 Problems 484 CHAPTER 18 Interpolation 488 18.1 Newton’s Divided-Difference Interpolating Polynomials 18.2 Lagrange Interpolating Polynomials 500 18.3 Coefficients of an Interpolating Polynomial 505 18.4 Inverse Interpolation 505 18.5 Additional Comments 506 18.6 Spline Interpolation 509 18.7 Multidimensional Interpolation 519 Problems 522 489 cha01064_fm.qxd 3/25/09 10:51 AM Page ix CONTENTS CHAPTER 19 Fourier Approximation 524 19.1 Curve Fitting with Sinusoidal Functions 525 19.2 Continuous Fourier Series 531 19.3 Frequency and Time Domains 534 19.4 Fourier Integral and Transform 538 19.5 Discrete Fourier Transform (DFT) 540 19.6 Fast Fourier Transform (FFT) 542 19.7 The Power Spectrum 549 19.8 Curve Fitting with Software Packages 550 Problems 559 CHAPTER 20 Case Studies: Curve Fitting 561 20.1 Linear Regression and Population Models (Chemical/Bio Engineering) 561 20.2 Use of Splines to Estimate Heat Transfer (Civil/Environmental Engineering) 565 20.3 Fourier Analysis (Electrical Engineering) 567 20.4 Analysis of Experimental Data (Mechanical/Aerospace Engineering) 568 Problems 570 EPILOGUE: PART FIVE 580 PT5.4 Trade-Offs 580 PT5.5 Important Relationships and Formulas 581 PT5.6 Advanced Methods and Additional References 583 PART SIX NUMERICAL DIFFERENTIATION AND INTEGRATION 585 PT6.1 Motivation 585 PT6.2 Mathematical Background 595 PT6.3 Orientation 597 CHAPTER 21 Newton-Cotes Integration Formulas 601 21.1 The Trapezoidal Rule 603 21.2 Simpson’s Rules 613 21.3 Integration with Unequal Segments 622 21.4 Open Integration Formulas 625 21.5 Multiple Integrals 625 Problems 627 ix cha01064_Ind.qxd 3/26/09 10:55 AM Page 961 INDEX case studies, 315–336 Cholesky decomposition, 298–300 computer objectives, 239–240 Cramer’s rule, 245–246 Crout decomposition, 281–282 distributed variable problems, 228, 229 elimination methods, 254–260, 338 elimination of unknowns, 246–247 engineering practice, 228–229 exact methods, 337 Excel, 307–308 Gauss elimination See Gauss elimination Gauss-Jordan, 267–269 Gauss-Seidel See Gauss-Seidel general form, 227 graphical method, 241–243 important relationships/formulas, 339 iterative refinement, 292–293 LU decomposition See LU decomposition lumped variable problems, 228, 229 Mathcad, 310–312 mathematical background, 229–237 MATLAB, 308–310 matrix See Matrix matrix condition number, 290–292 matrix inverse See Matrix inverse matrix norms, 288–290 methods, compared, 337 noncomputer methods, 227–228 numerical methods, 337–338 overview, 237–239 reactors, 315–318 references, 338–339 resistor circuits, 322–324 scope/preview, 237–239 spring-mass system, 324–326 study objectives, 239 Thomas algorithm, 297 trade-offs, 337–338 tridiagonal system, 297–298 truss, 318–322 vector norms, 288–290 Linear convergence, 144 Linear equation, 698 Linear interpolation, 132, 440, 489–490 Linear-interpolation formula, 489 Linear interpolation method, 159 Linear least squares, 477–481 Linear least-squares fit, 457–459, 462 Linear ordinary differential equation, 698 Linear programming, 346, 387–398 graphical solution, 389–391 possible outcomes, 391–392 961 simplex method, 393–398 standard form, 387–389 Linear regression, 454–470 algorithm, 463 best fit, 456–457 computer program, 462–466 general comments, 470 least-squares fit of straight line, 457–459 linearization of nonlinear relationships, 466–470 pseudocode, 463 quantification of error, 459–462 statistical assumptions, 470 Linear regression and population models, 561–565 Linear splines, 509–512 Linear vs nonlinear, 21 Linearization, 698 Linearization of nonlinear relationships, 466–470 Linearization of power equation, 468–470 Little, John N., 41 Local truncation error, 711 Logical representation, 28–32 Loops, 29–32 Lorenz, Edward, 816 Lorenz equation, 816 Lotka, Alfred J., 815 Lotka-Volterra equation, 803, 815 Lower Colorado River, 327 Lower triangular matrix, 231 LR method, 798 LU decomposition, 274–282 algorithm, 281, 282 Crout decomposition, 281–282 decomposition phase, 278 forward-substitution steps, 279 Gauss elimination, 276–281 overview, 275, 276 pseudocode, 278, 280, 282 steps in process, 275, 276 substitution steps, 278–280 Lumped-parameter systems, 913 Lumped variable problems, 228, 229 M M-files, 41, 42 MacCormack’s method, 876 Machine epsilon, 68 Maclaurin series expansion, 58, 105 Macro, 38 See also Excel Main diagonal, 231 Maintenance, 110 Manning equation, 217 Manning roughness coefficient, 217 Manning’s formula, 105 Mantissa, 64 Marksmanship, 55 Marquardt’s method, 384–385 Mass balance, 20, 114, 315 Mass balance of reactor, 913–917 Mass-spring system, 787–789 Mathcad, 941–951 condition number, 311, 312 constrained nonlinear optimization, 410 cubic spline interpolation, 557 curve fitting, 556–558 eigenvalues, 804, 805 entering text, 942 FFT, 558 graphs, 947–949 help, 951 linear equations, 310–312 main menu, 941 math palette, 942 mathematical functions and variables, 943–946 mathematical operations, 942–943 matrix computations and operations, 945 matrix functions, 944 matrix inverse, 312 multiline procedures/subprograms, 947 nonlinear system of equations, 199 numeric mode, 45 numerical integration/differentiation, 667–668 numerical methods function, 946 ODEs, 803–805 online help, 951 optimization, 409, 410 PDEs, 909–910 Poisson’s equation, 909 probability distributions, 944 QuickSheets, 951 range variables, 944–945 resource center, 951 roots of equations, 197–199 roots of polynomial, 199 standard tool bar, 941–942 stiff systems, 803 symbolic mathematics, 949–951 symbolic mode, 45 ToolTips, 951 trig and logs, 944 units, 945–946 what is it, 45 cha01064_Ind.qxd 962 3/26/09 10:55 AM Page 962 INDEX Mathematical model, 11–18 Mathematical programming problem, 346 Mathsoft, 45 MATLAB, 41–45 array operations, 936 assignment, 934–935 built-in functions, 937 condition number, 309 curve fitting, 553–556 diary file, 940 differentiation, 662 double precision, 70 editor/debugger, 41 eigenvalues, 802–803, 812, 822–823 Euler’s method, 44 extended precision, 70 factorial, 60, 61 FFT, 554 graphics, 937–938 humps function, 523 integration, 662 interpolation, 554–556 iterative calculation, 61 linear equations, 308–310 M-files, 41, 42 mathematical operations, 935–937 matrix, 41, 823 matrix analysis, 309 multidimensional optimization, 408–409 name, 308 numerical integration/differentiation, 661–667 ODEs, 799–803 one-dimensional optimization, 406–407 optimization, 405–409 PDEs, 908–909 pipe friction, 211, 212 polynomial manipulation, 193 polynomials, 938 potential energy function, 428 predator-prey equations, 799–800 primary features, 933 regression, 554, 555 root location, 193 roots of polynomials, 193–196 round-off/truncation errors in numerical differentiation, 101 save, 940 spline, 555–556 statistical analysis, 938–939 stiff systems, 800–802 two-dimensional function, 428 versions, 933 Matrix, 230–237 addition, 232 augmentation, 235 defined, 230 division, 234 inverse, 234, 284–286 linear least squares, 477–478 MATLAB, 41 multiplication, 232–234 notation, 230–231 representing linear algebraic equations, 236–237 special, 296 square, 231 subtraction, 232 trace, 235 transpose, 235 Matrix condition evaluation, 291–292 Matrix condition number, 290–292 Matrix inverse, 283–287 calculating the inverse, 284–286 ill-conditioned systems, 287–288 MATLAB, 311, 312 pseudocode, 286 stimulus-response computations, 286–287 system condition, 287–288 Matrix multiplication, 232–234 Matrix norms, 288–290 Maximum attainable growth rate, 562 Maximum likelihood principle, 459 Maximum-magnitude norm, 289 Maximum power transfer for circuit, 422–426 Mean of continuous data, 593 Mean of discrete points, 593 Mean value, 526 Method of false position, 132 Method of lines, 875 Method of optimal steepest ascent, 381–383 Method of steepest ascent, 378–383 Method of undetermined coefficients, 641–643 Method of weighted residuals (MWR), 891, 896–899 Microsoft, 37 Midpoint method, 625, 724–726, 727, 730, 760, 768 Midtest loop, 31 Milne’s method, 773–774, 775–776 Minimax principles, 457, 583 Minimum potential energy, 426–428 Minor, 244 Mixed partial derivative, 661 Model error, 104 Modified Euler, 724 Modified false position, 137–138 Modified Newton-Raphson, 165–167, 223 Modified Newton-Raphson method, 165–167, 223 Modified secant method, 157–159 Modular programming, 35–37 Modules, 36 Modulus of toughness, 688 Molal volume, 203, 204 Moler, Cleve, 41, 162, 364, 639 Müller’s method, 181–185 Multidimensional interpolation, 519–521 Multidimensional problems, 347 Multidimensional unconstrained optimization, 367–386 BFGS, 385 conjugate gradient method, 383 DFP, 385 direct methods, 368–372 finite-difference approximations, 377–378 gradient, 373–375 gradient methods, 372–385 Hessian, 375–377 Marquardt’s method, 384–385 Newton’s method, 383–384 pattern searches, 371–372 Powell’s method, 371, 372 quasi-Newton methods, 385 random search, 368–370 steepest ascent method, 378–383 trade-offs, 436–437 univariate search method, 370 Multimodal, 351 Multiple-application Simpson’s 1/3 rule, 616–618 Multiple-application trapezoidal rule, 607–610 Multiple integrals, 625–627 Multiple linear regression, 474–477 Multiple root, 122, 164–167 Multiplication, 71 Multistep methods Adams formulas, 768–772 fourth-order Adams method, 774, 775–776 higher-order methods, 772–776 integration formulas, 765–772 Milne’s method, 773–774, 775–776 Newton-Cotes formulas, 767–768 non-self-starting Heun method, 756 stability, 775–776 step size, 765 Multivariate power equation, 568 MWR, 896–899 cha01064_Ind.qxd 3/26/09 10:55 AM Page 963 INDEX N Naive Gauss elimination, 248–254 Neumann boundary condition, 784, 859 Newton-Cotes closed integration formulas, 620, 621 Newton-Cotes integration formulas, 601–630 Adams formulas, contrasted, 766, 767 closed forms, 602, 767–768 closed integration formulas, 620, 621 higher-order formulas, 620–622, 692, 693 integration of equations, 631–632 integration with unequal segments, 622–625 multiple integrals, 625–627 open forms, 602, 768 open integration formulas, 625 Simpson’s rules See Simpson’s rules trapezoidal rule See Trapezoidal rule unequal segments, 622–625 unequally spaced data, 624–625 Newton-Cotes open integration formulas, 625 Newton-Gregory backward formula, 508 Newton-Gregory central formula, 508 Newton-Gregory forward formula, 508 Newton-Raphson formula, 149 Newton-Raphson method, 148–154 additional features, 154 algorithm, 152 error estimates, 149–151 evaluate function and derivative, 178 formula, 149 Gauss elimination, 268, 269 graphical depiction, 148 ideal/nonideal gas laws, 204–205 modified method, 165–167, 223 multiple roots, 164–167 nonlinear equations, 169–171 pitfalls, 151–152 slowly converging function, 151–152, 153 termination criteria, 149 two-equation approach, 170, 171 Newtonian fluid, 578 Newton’s divided-difference interpolating polynomial, 494 Newton’s formula, 508 Newton’s interpolating polynomials, 489–500 algorithm, 498 computer applications, 497 errors, 495–497, 499–500 general form, 493 linear interpolation, 489–490 Newton’s divided-difference interpolating polynomial, 494 963 pseudocode, 498 quadratic interpolation, 491–493 Newton’s law of cooling, 688 Newton’s laws of motion, 114 Newton’s method, 361–362, 383–384 Newton’s second law of motion, 12, 700 Nodal lines, 889 Node, 889 Non-Newtonian fluid, 578 Non-self-starting Heun method, 756–764 derivation, 759–761 equations, 757 errors, 759–762 Heun approach, 756–757 modifiers, 762–764 per-step truncation error, 762 sequence of formulas, 764 step size, 765 Nonbasic variables, 394 Nonbinding constraints, 391 Noncomputer methods, Nongradient methods, 367 Nonhomogeneous system, 786 Nonideal gas laws, 202–205 Nonideal vs ideal, 21 Nonlinear boundary-value problem, 781–783 Nonlinear constrained optimization, 398 Nonlinear equations, 167 Nonlinear programming, 346 Nonlinear regression, 269, 481–484 Nonlinear system of equations, 267–269 Nonlinear vs linear, 21 Norm, 288 Normal distribution, 444, 445 Normal equation, 458 Normalization, 65 Normalized standard deviate, 649 nth finite divided difference, 493 Number systems, 62 Numerical differentiation, 90–94 Numerical differentiation and integration, 584–694 advanced methods/references, 693 antidifferentiation, 597 case studies, 671–691 commonly used derivatives, 596 computer objectives, 599–600 derivatives and integrals for data with errors, 659–660 derivatives of unequally spaced data, 658, 659 differentiation/integration, contrasted, 587, 588 engineering practice, 591–595 equal-area graphical differentiation, 588, 589 heat calculations, 671–673 high-accuracy differentiation formulas, 653–656 important relationships/formulas, 694 integrals used in this Part, 597 integration of equations See Integration of equations Mathcad, 667–668 mathematical background, 595–597 MATLAB, 661–667 method of undetermined coefficients, 641–643 methods, compared, 692 Newton-Cotes See Newton-Cotes integration formulas noncomputer methods, 588–590 overview, 597–599 partial derivatives, 660–661 Richardson extrapolation, 656–658 root-mean-square current, 675–678 sailboat, 673–675 scope/preview, 597–599 simple strip method, 590 study objectives, 599, 600 trade-offs, 692–693 uncertain data, 660 work, calculation of, 678–681 Numerical double integral, 625–627 Numerical error, 98–103 Numerical integration, 590 Numerical library, 46 Numerical methods accuracy, 55 defined, engineering practice, 4–5 error, 52 hyperbolic equations, 930 iterative approach, 57 linear algebraic equations, 337–338 noncomputer methods, precision, 55–56 rapid growth, trade-offs, 107–109 why studied, Numerical Recipe, 46 Numerical round-off errors, 62–76 Numerically unstable, 97 O Objective function, 344, 346 Octal number system, 62 Odd function, 534 cha01064_Ind.qxd 964 3/26/09 10:55 AM Page 964 INDEX ODE See Ordinary differential equations Ohm’s law, 322, 576, 835 One-dimensional parabolic PDEs, 929 One-dimensional problems, 347 One-dimensional unconstrained optimization, 351–366 bracketing methods, 352 Brent’s method, 363, 364 global vs local extremum, 351 golden-section search, 352–359 Newton’s method, 361–362 open methods, 352 parabolic interpolation, 359–361 trade-offs, 436 One-point iteration, 143 One-sided interval, 445 One-step methods, 708 Open integration formulas, 625 Open methods, 142–173 bracketing methods, compared, 142 Brent’s method, 159–164 modified Newton-Raphson, 165–167, 223 modified secant method, 157–159 multiple roots, 164–167 Newton-Raphson method See NewtonRaphson method secant method, 154–159 simple fixed-point iteration, 143–148 systems of nonlinear equations, 167–171 Operation counting, 252–254 Optimal steepest ascent, 381–383 Optimization, 340–437 case studies, 413–435 computer objectives, 348 constrained See Constrained optimization dimensionality, 347 engineering practice, 342 equilibrium and minimum potential energy, 426–428 Excel, 399–405 fundamental elements, 346 historical overview, 342 least-cost design of tank, 413–417 least-cost treatment of wastewater, 418–422 Mathcad, 409–410 mathematical background, 346–347 MATLAB, 405–409 maximum power transfer for circuit, 422–426 multidimensional See Multidimensional unconstrained optimization noncomputer methods, 342 one-dimensional See One-dimensional unconstrained optimization overview, 348, 349 parachute, 343–346 references, 437 root location, contrasted, 341 scope/preview, 348, 349 study objectives, 348, 350 trade-offs, 436–437 Optimum, 341 Ordinary differential equations, 696–841 boundary-value problems See Boundaryvalue problems case studies, 808–837 chaos, 815–819 computer objectives, 705 eigenvalue problems See Eigenvalue problems engineering practice, 699–700 important relationships/formulas, 839–841 Mathcad, 803–805 mathematical background, 701–703 MATLAB, 799–803 methods, compared, 838 multistep methods See Multistep methods noncomputer methods, 698–699 overview, 703–705 predator-prey models, 815–819 reactor, 808–815 RK methods See Runge-Kutta methods scope/preview, 703–705 simulating transient current for electric circuit, 819–824 stiffness, 752–756 study objectives, 705, 706 swinging pendulum, 824–827 trade-offs, 838–839 transient response of reactor, 808–815 Orthogonal polynomials, 583 Overconstrained, 347 Overdamped case, 176 Overdetermined, 339 Overflow error, 67 Overrelaxation, 305 Overview of book, 5–8 P p norm, 289 Parabola, 491 Parabolic equations, 871–887 ADI scheme, 883–886 comparison of one-dimensional methods, 882–883 convergence, 875 Crank-Nicolson method, 880–881 derivative boundary conditions, 875 explicit methods, 872–876, 883 heat conduction equation, 871–872 higher-order temporal approximations, 875–876 MacCormack’s method, 876 method of lines, 875 simple implicit method, 876–880, 883 stability, 875 two spatial dimensions, 883–886 Parabolic interpolation, 359–361 Parachutist problem air resistance, 14 algorithm, 34 analytical solution, 14–15 error, 18, 52 evaluating integrals, 611–612 Excel, 37–40 Gauss quadrature, 647 gravity, 13 numerical/analytical solution, compared, 18 numerical solution, 17 schematic diagram, 13 Parameter estimation, 813 Parameters, 12 Parametric Technology Corporation (PTC), 45 Parthenon, 354 Partial derivative, 586 Partial derivatives, 660–661 Partial differential equations, 842–930 area of focus, 843 case studies, 913–928 classification, 844 computer objectives, 848–849 deflections of a plate, 917–919 elliptic equations See Elliptic equations engineering practice, 844–846 Excel, 906–908 finite difference methods, 850–887 finite-element method See Finite-element method important relationships/formulas, 930 linear equations, 843 mass balance of reactor, 913–917 Mathcad, 909–910 MATLAB, 908–909 overview, 847–848 parabolic equations See Parabolic equations precomputer methods, 846 reactor, 913–917 references, 930 cha01064_Ind.qxd 3/26/09 10:55 AM Page 965 INDEX scope/preview, 847–848 series of springs, 922–925 study objectives, 848, 849 trade-offs, 929 two-dimensional electrostatic field problems, 919–922 Partial pivoting, 260–262 Pattern directions, 371 Pattern searches, 371–372 PDE See Partial differential equations Penalty functions, 398 Pentadiagonal system, 314 Percent relative error, 57 Perfection, 52 Period, 526 Periodic function, 525 Phase line spectra, 536, 537 Phase-plane representation, 817 Phase shift, 527 Phases of engineering problem solving, Piecewise functions, 50 Pipe friction, 209–213 Pivot coefficient, 249 Pivot element, 249 Pivot equation, 249 Pivoting, 260–262 Place value, 62 Plane, 889 Point-slope method, 708 Poisson equation, 807, 835, 852 Polynomial See Roots of polynomials Polynomial deflation, 178–180 Polynomial evaluation and differentiation, 177–180 Polynomial method, 792–794 Polynomial regression, 470–474 Polynomials, 115 Population, 444 Population models, 561–565 Positional notation, 62 Positive definite matrix, 300n Posttest loop, 30 Potential energy, 426 Potentiometers, 422 Powell’s method, 371, 372 Power equation, 467–468 Power method, 794–796 Power spectrum, 549 Practical issues, 21 Practice applications See Case studies Precision, 55–56, 109 Predator-prey equations, 799–800 Predator-prey models, 815–819 965 Predictor, 720 Predictor-corrector approach, 721 Predictor equation, 720–721 Predictor modifier, 762 Prescriptive models, 342 Pretest loop, 30 Principal diagonal, 231 Problem solving See Engineering problem solving Problem-solving process, 12 Program development cost, 108 Programming See Computer programming Programming effort required, 109 Programming languages See Excel; Mathcad; MATLAB Propagated truncation error, 711 Propagation problems, 845, 846 Proportionality, 287 Pseudocode, 28, 46 Pseudoplastics, 578 PTC, 45 Q QD algorithm, 224 QR factorization, 480 QR method, 798 Quadratic convergence, 150 Quadratic interpolation, 491–493 Quadratic polynomial, 491 Quadratic programming, 346 Quadratic splines, 512–515 Quadrature, 590 Quantizing errors, 67–69 Quasi-Newton methods, 385 Quotient difference (QD) algorithm, 224 R r, 460 r2, 460 Rainwater, 205–207 Ralston’s method, 730 Random search, 368–370 Rate equation, 697 Rate of convergence, 108 Razdow, Allen, 45 Reaction kinetics, 813 Reactor, 808–815, 913–917 Reactors, 315–318 Redlich-Kwong equation of state, 214 References See Advanced methods/additional references References (bibliography), 952–954 Relative error, 97 Relaxation, 305 Repetition, 29–32 Residual, 455 Resistor circuits, 322–324 Reynolds number, 210, 215, 219 Richardson extrapolation, 656–658 Richardson’s extrapolation, 632–636 RK methods See Runge-Kutta methods Romberg integration, 632, 636–638 Root-mean-square current, 675–678 Root polishing, 180 Roots of equations, 112–225 advanced methods/additional references, 224–225 bracketing methods See Bracketing methods bracketing/open methods, compared, 142 case studies, 202–222 computer objectives, 119 electrical circuit design, 207–209 engineering practice, 114–115 Excel, 190–193 graphical methods, 120–124, 142 greenhouse gases, 205–207 ideal/nonideal gas laws, 202–205 important relationships/formulas, 225 Mathcad, 197–199 mathematical background, 115–116 MATLAB, 193–196 methods, compared, 223 noncomputer methods, 113 open methods See Open methods optimization, contrasted, 341 overview, 116–118 pipe friction, 209–213 problem areas, 116 QD algorithm, 224 rainwater, 205–207 root polishing, 180 roots of polynomials See Roots of polynomials scope/preview, 116–118 study objectives, 118 trade-offs, 223–224 Roots of polynomials, 174–201 Bairstow’s method, 185–189 characteristic equation, 175 conventional methods, 180–181 Excel, 190–193 factored form of polynomial, 178 general solution, 175 Jenkins-Traub method, 190 Laguerre’s method, 190 cha01064_Ind.qxd 966 3/26/09 10:55 AM Page 966 INDEX Roots of polynomials—Cont Mathcad, 197–199 MATLAB, 193–196 Müller’s method, 181–185 ODE, 174 overdamped/critically damped/underdamped, 176–177 polynomial deflation, 178–180 polynomial evaluation and differentiation, 177–180 root polishing, 180 Roots of quadratic algorithm, 189 Rosenbrock method, 803 Rosin-Rammler-Bennet (RRB) equation, 690 Round-off error, 62–76 adding large and small number, 72–73 addition, 70 arithmetic manipulations of computer numbers, 70–76 chopping, 67 computer representation of numbers, 62–70 division, 71 extended precision, 69–70 floating-point representation, 64–69 Gauss elimination, 255 inner products, 76 integer representation, 62–64 large computations, 71–72 multiplication, 71 normalization, 65 number systems, 62 overflow error, 67 quantizing errors, 67–69 rounding, 67, 68 smearing, 74 subtraction, 71 subtractive cancellation, 73–74 underflow “hole,” 67 Rounding, 67, 68 Row, 230 Row-sum norm, 289, 290 Row vectors, 230 RRB equation, 690 Run-time cost, 108 Runge-Kutta Fehlberg, 745–746 Runge-Kutta methods, 707–751 adaptive See Adaptive Runge-Kutta methods Euler’s method See Euler’s method fourth-order RK methods, 733–735 Heun’s method, 720–724, 726, 727, 756–764 higher-order RK methods, 735 methods, compared, 735–736 midpoint method, 724–726, 727, 730 pseudocode, 737 Ralston’s method, 730 Runge-Kutta Fehlberg, 745–746 second-order RK methods, 728–732 step-size control, 746–747 systems of equations, 737–742 third-order RK methods, 732–733 Runge’s function, 523 S Saddle, 376 Sailboat, 673–675 Sample, 444, 447 Sample mean, 445 Sande-Tukey algorithm, 544–548 Saturation-growth-rate equation, 467, 468 Saturation-growth-rate model, 561–565 Scaling, 258, 262–265 Secant formula, 215 Secant method, 154–159 additional features, 154, 157 algorithm, 157 false-position method, compared, 155–157 formula, 155 graphical depiction, 154 modified method, 157–159 multiple roots, 164, 165 Second Adams-Bashforth formula, 770 Second Adams-Moulton formula, 771 Second derivative, 586 Second finite divided difference, 493 Second forward finite divided difference, 94 Second-order closed Adams formula, 771 Second-order equation, 697 Second-order open Adams formula, 770 Second-order Ralston RK method, 730 Second-order RK methods, 728–732 Secondary variables, 857 Selection, 28–29 Sensitivity analysis, 21 Sequence, 28 Series of springs, 922–925 Shadow price, 421 Shape function, 890 Shooting method, 780–783 Signed magnitude method, 62 Significance level, 446 Significand, 64 Significant digits, 54 Significant figures, 53–54 Simple fixed-point iteration, 143–148 Simple implicit method, 876–880, 883 Simple strip method, 590 Simplex method, 393–398 Simplex procedure, 342 Simpson’s 1/3 rule, 613–618 Simpson’s 3/8 rule, 618–620 Simpson’s rules, 613–620 algorithms, 620, 621 multiple-application Simpson’s 1/3 rule, 616–618 pseudocode, 621 Simpson’s 1/3 rule, 613–618 Simpson’s 3/8 rule, 618–620 uneven data, 623 Simulated annealing, 370 Simulating transient current for electric current, 819–824 Simultaneous linear algebraic equations See Linear algebraic equations Simultaneous nonlinear equations, 267–269 Simultaneous overrelaxation, 305 Single-value decomposition, 583 Single-variable optimization, 352 Singular system, 243 Singular systems, 259 Sinusoid, 526 Sinusoidal functions, 525–531 Slide rule, Small vs large systems, 21 Smearing, 74 Software cost, 108 Software packages See Excel; Mathcad; MATLAB Solution technique, 852–858 Solver, 191–193 SOR, 305 Special matrix, 296 Specific growth rate, 561 Spectral norm, 289 Spline, 509 Spline functions, 509 Spline interpolation, 509–519 cubic splines, 509, 512, 515–519 linear splines, 509–512 quadratic splines, 512–515 superiority, 509, 510 Spreadsheet, 37 See also Excel Spring-mass system, 324–326 Square matrix, 231 Stability, 97, 108, 775–776, 875 Stage extraction process, 327 Standard deviation, 442, 445 Standard error of the estimate, 460 cha01064_Ind.qxd 3/26/09 10:55 AM Page 967 INDEX Standard normal estimate, 446 Starting point, 108 Static instability, 917 Statistical inference, 445 Statistically determinate truss, 318–322 Statistics, 441–450 Steady-state, 19 Steady-state analysis of system of reactors, 315–318 Steepest ascent method, 378–383 Stefan-Boltzmann constant, 105 Stefan-Boltzmann law, 105 Step halving, 744 Stiff system, 752 Stiffness, 752–756 Stiffness matrix, 892 Stimulus-response computations, 286–287 Stopping criterion, 57 Straightening, 90 Strange attractors, 819 Streeter-Phelps model, 431 Strip method, 590 Structured programming, 26–35 Student-t, 448 Subdomain method, 897 Subroutine, 35 Subtraction, 71 Subtractive cancellation, 73–74 Successive overrelaxation, 305 Successive substitution, 143 Superposition, 287 SVD method, 583 Swamee-Jain equation, 210 Swinging pendulum, 699, 824–827 Symmetric form, 502 Symmetric matrix, 231, 296 Synthetic division, 178 System condition, 287–288 Systems of nonlinear equations, 167–171 T t distribution, 448 Table look-up, 523 Tableau, 395 Tabu search, 370 Taylor series, 78–94 approximation of function with infinite number of derivatives, 83 approximation of polynomial, 80 backward difference approximation, 90 centered difference approximation, 90 finite difference approximations of higher derivatives, 93–94 967 finite-divided-difference approximations of derivatives, 92–93 first-order approximation, 79 first theorem of mean for integrals, 79 forward difference approximation, 90 Newton-Raphson method, 150 nonlinearity, 86–90 nth-order expansion, 82 numerical differentiation, 90–94 remainder, 79, 80, 82, 84–86 second-order approximation, 79 second theorem of mean for integrals, 79 step size, 86–90 Taylor series expansion, 80 theorem, 79 truncation errors, 86 what is it, 78 zero-order approximation, 78 Taylor series expansion, 80 Taylor’s formula, 79 Taylor’s theorem, 79 Temperature of heated plate, 855–856 Terminal velocity, 15 Termination criteria, 57 The MathWorks, Inc., 41 Thermocline, 565 Third-order RK methods, 732–733 Thomas algorithm, 297 Three-point Gauss-Legendre formula, 647 Time domain, 534–538 Time plane, 534 Time-variable, 18 Total numerical error, 98–103 Total sum of the squares, 460 Trace, 235 Trade-offs curve fitting, 580–581 linear algebraic equations, 337–338 numerical differentiation/integration, 692–693 numerical methods, 107–109 ODEs, 838–839 optimization, 436–437 PDEs, 929 roots of equations, 223–224 Transcendental function, 116 Transient, 18 Transient responses of reactor, 808–815 Transpose, 235 Trapezoidal rule, 603–613 algorithms, 611 area under straight line, 640–641 computer program, 611–612 conclusions, 612–613 derivation, 603, 606 error, 605 error corrections, 634–635 formula, 603 graphical depiction, 604 Heun’s method, 757 multiple-application rule, 607–610 pseudocode, 611 single application, 606–607 unequal segments, 622 Trend analysis, 441 Triangular matrix, 231 Tridiagonal matrix, 231 Tridiagonal system, 297–298 Triple root, 164 True error, 56 True fractional relative error, 56 True local truncation error, 711 True mean, 445 True solution, 710 True value, 56 Truncation, 67 Truncation error, 78, 711 Truncation errors and Taylor series, 79–106 blunders, 103–104 condition, 97 data uncertainty, 104 error propagation, 94–98 formulation errors, 104 stability, 97 Taylor series See Taylor series total numerical error, 98–103 truncation errors, 78 Truss, 318–322 Tukey, J W., 543 Twiddle factors, 545 Two-dimensional electrostatic field problems, 919–922 Two-dimensional interpolation, 520–521 Two-dimensional parabolic PDEs, 929 Two-equation Newton-Raphson approach, 170, 171 Two-point Gauss-Legendre formula, 643–646 Two-segment trapezoidal rule, 627 Two-sided interval, 445–446 U Uncertain data, 104 Uncertainty, 55 Unconditionally stable, 754 Unconstrained optimization, 347 Underdamped case, 177 cha01064_Ind.qxd 3/26/09 10:55 AM 968 Underdetermined, 338, 393 Underflow “hole,” 67 Underrelaxation, 305 Underspecified, 393 Unexplained sum of the squares, 460 Uniform-matrix norm, 289, 290 Uniform-vector norm, 289 Unimodal, 352 Univariate search method, 370 Unstable, 715 Upper triangular matrix, 231 V Van der Pol’s equation, 801 Van der Waals equation, 203 Vandermonde matrix, 294 Page 968 INDEX Variable metric methods, 385 Variance, 442 Variational approach, 891 VBA macro, 38 See also Excel Vector norms, 288–290 Vibrating string, 846 Videoangiography, 684 Voltage balance, 20 Volterra, Vito, 815 Volume integral, 594 Volume-integral approach, 866 Von Karman equation, 219 Wave equation, 844, 846 Well-conditioned systems, 255 WHILE, 42 Wolf, Johann Rudolph, 567 Wolf sunspot number, 567 Word, 62 Work, calculation of, 678–681 W Zero-order approximation, 78 Waste minimization, 429 Wastewater treatment, 418–422 Y Yield stress, 572 Young’s modulus, 432 Z cha01064_Ind.qxd 3/26/09 10:55 AM Page 969 cha01064_Ind.qxd 3/26/09 10:55 AM Page 970 cha01064_Ind.qxd 3/26/09 10:55 AM Page 971 cha01064_Ind.qxd 3/26/09 10:55 AM Page 972 cha01064_Ind.qxd 3/26/09 10:55 AM Page 973 cha01064_Ind.qxd 3/26/09 10:55 AM Page 974 Sixth Edition Features include: which are based on exciting new areas such as bioengineering and differential equations students using this text will be able to apply their new skills to their chosen field Electronic Textbook Options an online resource where students can purchase the complete text in a digital format at almost half the cost of the traditional textbook Students can access the text online for one year learning, which include full text search, notes and highlighting, and email tools for sharing contact your sales representative or visit www.CourseSmart.com Sixth Edition Numerical Methods for Engineers Chapra Canale Steven C Chapra Raymond P Canale MD DALIM #1009815 03/12/09 CYAN MAG YELO BLK For more information, please visit www.mhhe.com/chapra for Engineers adaptive quadrature Numerical Methods The sixth edition of Numerical Methods for Engineers offers an innovative and accessible presentation of numerical methods; the book has earned the Meriam-Wiley award, which is given by the American Society for Engineering Education for the best textbook Because software packages are now regularly used for numerical analysis, this eagerly anticipated revision maintains its strong focus on appropriate use of computational tools [...]... involve numerical methods The intelligent use of these programs is often predicated on knowledge of the basic theory underlying the methods 3 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 4 Numerical methods. .. 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 steven .chapra@ tufts.edu... 3/20/09 1:21 PM Page 1 Numerical Methods for Engineers cha01064_p01.qxd 3/23/09 4:32 PM Page 2 PART ONE cha01064_p01.qxd 3/20/09 1:22 PM Page 3 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... CHAPTER 25 Runge-Kutta Methods 707 25.1 Euler’s Method 708 25.2 Improvements of Euler’s Method 719 25.3 Runge-Kutta Methods 727 25.4 Systems of Equations 737 25.5 Adaptive Runge-Kutta Methods 742 Problems 750 CHAPTER 26 Stiffness and Multistep Methods 752 26.1 Stiffness 752 26.2 Multistep Methods 756 Problems 776 CHAPTER 27 Boundary-Value and Eigenvalue Problems 778 27.1 General Methods for Boundary-Value... 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... 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... same time, you will also learn to acknowledge and control the errors of approximation that are part and parcel of largescale numerical calculations 5 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... partial differential equations numerically In the present text, we will emphasize finitedifference 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 PT1.3 ORIENTATION Some orientation might be helpful before proceeding with our introduction to numerical methods The following is... 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. .. (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