Chemistry form ws5 3.1 a answer key

Numerical Methods for Engineers

SEVENTH EDITION

Steven C. Chapra Berger Chair in Computing and Engineering

Tufts University

Raymond P. Canale Professor Emeritus of Civil Engineering

University of Michigan

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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.

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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

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To

Margaret and Gabriel Chapra

Helen and Chester Canale

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iv

CONTENTS

PREFACE xiv

ABOUT THE AUTHORS xvi

PART ONE

MODELING, PT1.1 Motivation 3 COMPUTERS, AND PT1.2 Mathematical Background 5 ERROR ANALYSIS 3 PT1.3 Orientation 8

CHAPTER 1 Mathematical Modeling and Engineering Problem Solving 11

1.1 A Simple Mathematical Model 11 1.2 Conservation Laws and Engineering 18 Problems 21

CHAPTER 2 Programming and Software 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 55

3.1 Signifi cant Figures 56 3.2 Accuracy and Precision 58 3.3 Error Defi nitions 59 3.4 Round-Off Errors 65 Problems 79

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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 106 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 113

PART TWO

ROOTS OF PT2.1 Motivation 117 EQUATIONS 117 PT2.2 Mathematical Background 119

PT2.3 Orientation 120

CHAPTER 5 Bracketing Methods 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 141 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 176 7.2 Computing with Polynomials 179 7.3 Conventional Methods 182

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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 192 Problems 202

CHAPTER 8 Case Studies: Roots of Equations 204

8.1 Ideal and Nonideal Gas Laws (Chemical/Bio Engineering) 204 8.2 Greenhouse Gases and Rainwater (Civil/Environmental Engineering) 207 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 227

PART THREE

LINEAR ALGEBRAIC PT3.1 Motivation 231 EQUATIONS 231 PT3.2 Mathematical Background 233

PT3.3 Orientation 241

CHAPTER 9 Gauss Elimination 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 278 10.1 LU Decomposition 278 10.2 The Matrix Inverse 287 10.3 Error Analysis and System Condition 291 Problems 297

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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 311 Problems 316

CHAPTER 12 Case Studies: Linear Algebraic Equations 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 350 PT4.3 Orientation 351

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 370

14.1 Direct Methods 371 14.2 Gradient Methods 375 Problems 388

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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 443 PT5.3 Orientation 452

CHAPTER 17 Least-Squares Regression 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 491 18.2 Lagrange Interpolating Polynomials 502 18.3 Coeffi cients 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

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CONTENTS ix

CHAPTER 19 Fourier Approximation 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 584

PART SIX

NUMERICAL PT6.1 Motivation 587 DIFFERENTIATION PT6.2 Mathematical Background 597 AND PT6.3 Orientation 599 INTEGRATION 587

CHAPTER 21 Newton-Cotes Integration Formulas 603

21.1 The Trapezoidal Rule 605 21.2 Simpson’s Rules 615 21.3 Integration with Unequal Segments 624 21.4 Open Integration Formulas 627 21.5 Multiple Integrals 627 Problems 629

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x CONTENTS

CHAPTER 22 Integration of Equations 633

22.1 Newton-Cotes Algorithms for Equations 633 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 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 663 Problems 670

CHAPTER 24 Case Studies: Numerical Integration and Differentiation 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 695

PART SEVEN

ORDINARY PT7.1 Motivation 699 DIFFERENTIAL PT7.2 Mathematical Background 703 EQUATIONS 699 PT7.3 Orientation 705

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CONTENTS xi

CHAPTER 25 Runge-Kutta Methods 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 755

26.1 Stiffness 755 26.2 Multistep Methods 759 Problems 779

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 842

PART EIGHT

PARTIAL PT8.1 Motivation 845 DIFFERENTIAL PT8.2 Orientation 848 EQUATIONS 845

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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 885 Problems 888

CHAPTER 31 Finite-Element Method 890

31.1 The General Approach 891 31.2 Finite-Element Application in One Dimension 895 31.3 Two-Dimensional Problems 904 31.4 Solving PDEs with Software Packages 908 Problems 912

CHAPTER 32 Case Studies: Partial Differential Equations 915

32.1 One-Dimensional Mass Balance of a Reactor (Chemical/Bio Engineering) 915

32.2 Defl ections 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

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CONTENTS xiii

APPENDIX A: THE FOURIER SERIES 933

APPENDIX B: GETTING STARTED WITH MATLAB 935

APPENDIX C: GETTING STARTED WITH MATHCAD 943

BIBLIOGRAPHY 954

INDEX 957

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xiv

PREFACE

It has been over twenty years since we published the fi rst edition of this book. Over that period, our original contention that numerical methods and computers would fi gure more prominently in the engineering curriculum—particularly in the early parts—has been dra- matically 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 fi rst 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 en- deavored 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 environ- ments. This knowledge carries over to standard programming languages such as Visual Basic, Fortran 90, and C/C11. We believe that the current fl ight from computer programming represents something of a “dumbing down” of the engineering curricu- lum. 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-fi les.” This book is designed to empower them to do that.

Beyond these fi ve original principles, the seventh edition has new and expanded problem sets. Most of the problems have been modifi ed so that they yield different numerical solu- tions 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 effi ciently, and retain more knowledge for greater success. LearnSmart assesses

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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 numeri- cal 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 man- uscript and Arpana Kumari of Aptara also did an outstanding job in the book’s fi nal 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. Use- ful 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 benefi ted 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 edi- tion. 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

Raymond P. Canale Lake Leelanau, Michigan

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xvi

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 Pro- tection 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 environ- mental 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 comput- ers, 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 scientifi c papers and reports. He has also designed and developed personal computer software to facilitate en- gineering 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 fi rms and industry and governmental agencies as a consultant and expert witness.

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Numerical Methods for Engineers

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PART ONE

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3

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, effi cient 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 fi repower, the widespread availability of com- puters (especially personal computers) and their partnership with numerical methods has had a signifi cant infl uence on the actual engineering problem-solving process. In the precomputer era there were generally three different ways in which engineers approached problem solving:

1. 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.

2. 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.

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

MODELING, COMPUTERS, AND ERROR ANALYSIS

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4 MODELING, COMPUTERS, AND ERROR ANALYSIS

Today, computers and numerical methods provide an alternative for such compli- cated calculations. Using computer power to obtain solutions directly, you can approach these calculations without recourse to simplifying assumptions or time-intensive tech- niques. 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)