Numerical Methods for Engineers
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1 Steven C. Chapra Berger Chair in Computing and Engineering Tufts University RaymondP. Canale Professor Emeritus of Civil Engineering University of Michigan Numerical Methods for Engineers With Software and Programming Applications Fourth Edition / &J0 * Mc 1 Grawl Hill 1 Boston BurrRidge, IL Dubuque, IA Madison, Wl New York San Francisco St. Louis Bangkok Bogota Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal New Delhi Santiago Seoul Singapore Sydney Taipei Toronto
2 PREFACE xvi ABOUT THE AUTHORS xviii PART ONE MODELING, COMPUTERS, AND ERROR ANALYSIS 3 PT 1.1 Motivation 3 PT 1.2 Mathematical Backgrounc PT 1.3 Orientation 8 CHAPTER 1 Mathematical Modeling and Engineering Problem Solving A Simple Mathematical Model Conservation Laws and Engineering 1 8 Problems 21 CHAPTER 2 Programming and Software Packages and Programming Structured Programming Modular Programming Excel MATLAB Other Languages and Libraries 45 Problems 46 CHAPTER 3 Approximations and Round-Off Errors Significant Figures Accuracy and Precision Error Definitions 54 VII
3 viii CONTENTS 3.4 Round-Off Errors 57 Problems 72 CHAPTER 4 Truncation Errors and the Taylor Series The Taylor Series Error Propagation Total Numerical Error Blunders, Formulation Errors, and Data Uncertainty Problems EPILOGUE: PART ONE 99 PT 1.4 Trade-Offs 99 PT 1.5 Important Relationships and Formulas 102 PT 1.6 Advanced Methods and Addifional References 102 PART TWO ROOTS OF EQUATIONS 105 PT2.1 Motivation 105 PT 2.2 Mathematical Background PT2.3 Orientation CHAPTER 5 Bracketing Methods 1T Graphical Methods The Bisection Method The False-Position Method Incremental Searches and Determining Initial Guesses Problems CHAPTER 6 Open Methods Simple Fixed-Point Iteration The Newton-Raphson Method The Secant Method Multiple Roors Systems of Nonlinear Equations 153 Problems 157 CHAPTER 7 Roots of Polynomials Polynomials in Engineering and Science Computing with Polynomials Conventional Methods Müller's Method Bairstow's Method Other Methods 176
4 ix 7.7 Root Location with Libraries and Packages 176 Problems 185 CHAPTER 8 Engineering Applications: Roots of Equations Ideal and Nonideal Gas Laws (Chemical/Bio Engineering) Open-Channel Flow (Civil/Environmental Engineering) Design of an Electric Circuit (Electrical Engineering) Vibration Analysis (Mechanical/Aerospace Engineering) 196 Problems 203 EPILOGUE: PART TWO 212 PT2.4 Trade-Offs 212 PT 2.5 Important Relationships and Formulas 213 PT 2.6 Advanced Methods and Additional References 213 PART THREE LINEAR ALGEBRAIC PT 3.1 Motivation 217 EQUATIONS 217 PT 3.2 Mathematical Background 219 PT3.3 Orientation 227 CHAPTER 9 Gauss Elimination Solving Small Numbers of Equations NaTve Gauss Elimination Pitfalls of Elimination Methods Techniques for Improving Solutions Complex Systems Nonlinear Systems of Equations Gauss-Jordan Summary 261 Problems 261 CHAPTER 10 LU Decomposition and Matrix Inversion LU Decomposition The Matrix Inverse Error Analysis and System Condition 277 Problems 283 CHAPTER 11 Special Matrices and Gauss-Seidel Special Matrices Gauss-Seidel 289
5 1 1.3 Linear Algebraic Equafions with Libraries and Packages 296 Problems 303 CHARTER 12 Engineering Applications: Linear Algebraic Equations Steady-State Analysis of a System of Reactors (Chemical/Bio Engineering) Analysis of a Statically Determinate Truss (Civil/Environmental Engineering) Currents and Voltages in Resistor Circuits (Electrical Engineering) Spring-Mass Systems (Mechanical/Aerospace Engineering) 314 Problems 317 EPILOGUE: PART THREE 327 PT 3.4 Trade-Offs 327 PT3.5 Important Relationships and Formulas 328 PT 3.6 Advanced Methods and Additional References 328 PART FOUR OPTIMIZATION 331 PT4.1 Motivation 331 PT 4.2 Mathematical Background 336 PT4.3 Orientation 337 CHAPTER 13 One-Dimensional Unconstrained Optimization Golden-Section Search Quadratic Interpolation Newton's Method 351 Problems 353 CHAPTER 14 Multidimensional Unconstrained Optimization Direct Methods Gradient Methods 360 Problems 373 CHAPTER 15 Constrained Optimization Linear Programming Nonlinear Constrained Optimization Optimization with Packages 387 Problems 398 CHAPTER 16 Engineering Applications: Optimization Least-Cost Design of a Tank (Chemical/Bio Engineering) Least-Cost Treatment of Wastewater (Civil/Environmental Engineering) 405
6 xi 16.3 Maximum Power Transfer for a Circuit (Electrical Engineering) Mountain Bike Design (Mechanical/Aerospace Engineering) 413 Problems 415 EPILOGUE: PART FOUR 422 PT 4.4 Trade-Offs 422 PT 4.5 Additional References 423 PART FIVE CURVE FITTING 425 PT 5.1 Motivation 425 PT5.2 Mathematical Background 427 PT5.3 Orientation 436 CHAPTER 17 Least-Squares Regression Linear Regression Polynomial Regression Multiple Linear Regression General Linear Least Squares Nonlinear Regression 468 Problems 471 CHAPTER 18 Interpolation Newton's Divided-Difference Interpolating Polynomials Lagrange Interpolating Polynomials Coefficients of an Interpolating Polynomial Inverse Interpolation Additional Comments Spline Interpolation 495 Problems 505 CHAPTER 19 Fourier Approximation Curve Fitting with Sinusoidal Functions Continuous Fourier Series Frequency and Time Domains Fourier Integral and Transform Discrete Fourier Transform (DFT) Fast Fourier Transform (FFT) The Power Spectrum Curve Fitting with Libraries and Packages 533 Problems 542
7 xii CONTENTS CHAPTER 20 Engineering Applications: Curve Fitting Linear Regression and Population Models (Chemical/Bio Engineering) Use of Splines to Estimate Heat Transfer (Civil/Environmental Engineering) Fourier Analysis (Electrical Engineering) Analysis of Experimental Data (Mechanical/Aerospace Engineering) 551 Problems 553 EPILOGUE: PART FIVE 563 PT 5.4 Trade-Offs 563 PT5.5 Important Relationships and Formulas 564 PT 5.6 Advanced Methods and Additional References 566 PART SIX NUMERICAL DIFFERENTIATION AND INTEGRATION 569 PT6.1 Motivation 569 PT 6.2 Mathematical Background PT6.3 Orientation CHAPTER 21 Newton-Cotes Integration Formulas The Trapezoidal Rule Simpson's Rules Integration with Unequal Segments Open Integration Formulas Multiple Integrals 608 Problems 610 CHAPTER 22 Integration of Equations Newton-Cotes Algorithms for Equations Romberg Integration Gauss Quadrature Improper Integrals 627 Problems 631 CHAPTER 23 Numerical Differentiation High-Accuracy Differentiation Formulas Richardson Extrapolation Derivatives of Unequally Spaced Data Derivatives and Integrals for Data with Errors Numerical Integration/Differentiation with Libraries and Packages Problems
8 xiii CHAPTER 24 Engineering Applications: Numerical Integration and Differentiation Integration to Determine the Total Quantity of Heat (Chemical/Bio Engineering) Effective Force on the Mast of a Racing Sailboat (Civil/Environmental Engineering) Root-Mean-Square Current by Numerical Integration (Electrical Engineering) Numerical Integration to Compute Work (Mechanical/Aerospace Engineering) 653 Problems 657 EPILOGUE: PART SIX 667 PT 6.4 Trade-Offs 667 PT 6.5 Important Relationships and Formulas 668 PT 6.6 Advanced Methods and Additional References 668 PART SEVEN ORDINARY PT7.1 Motivation 671 DIFFERENTIAL PT 7.2 Mathematical Background 675 EQUATIONS 671 PT7.3 Orientation 677 CHAPTER 25 Runge-Kutta Methods Euler's Method Improvements of Euler's Method Runge-Kutta Methods Systems of Equations Adaptive Runge-Kutta Methods 716 Problems 724 CHAPTER 26 Stiffness and Multistep Methods Stiffness Multistep Methods 730 Problems 750 CHAPTER 27 Boundary-Value and Eigenvalue Problems General Methods for Boundary-Value Problems Eigenvalue Problems ODEs and Eigenvalues with Libraries and Packages 771 Problems 779
9 xiv CONTENTS CHAPTER 28 Engineering Applications: Ordinary Differential Equations Using ODEs to Analyze the Transient Response of a Reactor (Chemical/Bio Engineering) Predator-Prey Models and Chaos (Civil/Environmental Engineering) Simulating Transient Current for an Electric Circuit (Electrical Engineering) The Swinging Pendulum (Mechanical/Aerospace Engineering) 797 Problems 801 EPILOGUE: PART SEVEN 808 PT 7.4 Trade-Offs 808 PT 7.5 Important Relationships and Formulas 809 PT 7.6 Advanced Methods and Additional References 809 PART EIGHT PARTI AL PT8.1 Motivation 813 DIFFERENTIAL PT 8.2 Orientation 816 EQUATIONS 813 CHAPTER 29 Finite Difference: Elliptic Equations The Laplace Equation Solution Techniques Boundary Conditions The Control-Volume Approach Software to Solve Elliptic Equations 837 Problems 838 CHAPTER 30 Finite Difference: Parabolic Equations The Heat Conduction Equation Explicit Methods A Simple Implicit Method The Crank-Nicolson Method Parabolic Equations in Two Spatial Dimensions 852 Problems 855 CHAPTER 31 Finite-Element Method The General Approach Finite-Element Application in One Dimension Two-Dimensional Problems Solving PDEs with Libraries and Packages 875 Problems 881
10 xv CHAPTER 32 Engineering Applications: Partial Differential Equations One-Dimensional Mass Balance of a Reactor (Chemical/Bio Engineering) Deflections of a Plate (Civil/Environmental Engineering) Two-Dimensional Electrostatic Field Problems (Electrical Engineering) Finite-Element Solution of a Series of Springs (Mechanical/Aerospace Engineering) 893 Problems 797 EPILOGUE: PART EIGHT 899 PT 8.3 Trade-Offs 899 PT 8.4 Important Relationships and Formulas 899 PT 8.5 Advanced Methods and Additional References 900 APPENDIX A: THE FOURIER SERIES 901 APPENDIX B: GETTING STARTED WITH MATLAB 903 BIBLIOGRAPHY 911 INDEX 915
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