Numerical Methods for Engineers
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1 Numerical Methods for Engineers SEVENTH EDITION Steven C Chopra Berger Chair in Computing and Engineering Tufts University Raymond P. Canal Professor Emeritus of Civil Engineering of Michiaan University Mc Graw Hill Education : ; : /^:!K,';»33 M.^Ar;:/SS^m ^tammmm
2 PREFACE xiv ABOUT THE AUTHORS xvi PART ONE MODELING, PTl.l Motivation 3 COMPUTERS, AND PTl.2 Mathematical Background 5 ERROR ANALYSIS 3 PTl 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 Mathcad Other Languages and Libraries 48 Problems 49 CHAPTER 3 Approximations and Round-Off Errors Significant Figures Accuracy and Precision Error Definitions Round-Off Errors 65 Problems 79 iv
3 v CHAPTER 4 Truncation Errors and the Taylor Series The Taylor Series Error Propagation Total Numerical Error Blunders, Formulation Errors, and Data Uncertainty 106 Problems 108 EPILOGUE: PART ONE 110 PT1.4 Tradeoffs 110 PT1.5 Important Relationships and Formulas 113 PT1.6 Advanced Methods and Additional References 11 3 PART TWO ROOTS OF PT2.1 Motivation 117 EQUATIONS 117 PT2.2 Mathematical Background 119 PT2.3 Orientation 120 CHAPTER 5 Bracketing Methods Graphical Methods The Bisection Method The False-Position Method Incremental Searches and Determining Problems 142 Initial Guesses 141 CHAPTER 6 Open Methods Simple Fixed-Point Iteration The Newton-Raphson Method The Secant Method Brent's Method Multiple Roots Systems of Nonlinear Equations 169 Problems 173 CHAPTER 7 Roots of Polynomials Polynomials in Engineering and Science Computing with Polynomials Conventional Methods 1 82
4 vi CONTENTS 7.4 Mullen's Method Bairstow's Method Other Methods Root Location with Software Packages 192 Problems 202 CHAPTER 8 Case Studies: Roots of Equations Ideal and Nonideal Gas Laws (Chemical/Bio Engineering) Greenhouse Gases and Rainwater (Civil/Environmental Engineering) Design of an Electric Circuit (Electrical Engineering) 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 Solving Small Numbers of Equations Naive Gauss Elimination Pitfalls of Elimination Methods Techniques for Improving Solutions Complex Systems Nonlinear Systems of Equations Gauss-Jordan Summary 275 Problems 275 CHAPTER 10 LU Decomposition and Matrix Inversion LU Decomposition The Matrix Inverse Error Analysis and System Condition 291 Problems 297
5 vii CHAPTER 11 Special Matrices and Gauss-Seidel Special Matrices Gauss-Seidel Linear Algebraic Equations with Software Packages 31 1 Problems 316 CHAPTER 12 Case Studies: 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) 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 Golden-Section Search Parabolic Interpolation Newton's Method Brent's Method 366 Problems 368 CHAPTER 14 Multidimensional Unconstrained Optimization Direct Methods Gradient Methods 375 Problems 388
6 viii CONTENTS CHAPTER 15 Constrained Optimization Linear Programming Nonlinear Constrained Optimization Optimization with Software Packages 402 Problems 413 CHAPTER 16 Case Studies: Optimization Least-Cost Design of a Tank (Chemical/Bio Engineering) Least-Cost Treatment of Wastewater (Civil/Environmental Engineering) Maximum Power Transfer for a Circuit (Electrical Engineering) 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 Linear Regression Polynomial Regression Multiple Linear Regression General Linear Least Squares Nonlinear Regression 483 Problems 487 CHAPTER 18 Interpolation Newton's Divided-Difference Interpolating Polynomials Lagrange Interpolating Polynomials Coefficients of an Interpolating Polynomial Inverse Interpolation Additional Comments Spline Interpolation Multidimensional Interpolation 521 Problems 524
7 ix 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 Software Packages 552 Problems 561 CHAPTER 20 Case Studies: 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) 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 DIFFERENTIATION AND INTEGRATION 587 PT6.1 Motivation 587 PT6.2 Mathematical Background 597 PT6.3 Orientation 599 CHAPTER 21 Newton-Cotes Integration Formulas The Trapezoidal Rule Simpson's Rules Integration with Unequal Segments Open Integration Formulas Multiple Integrals 627 Problems 629
8 X CONTENTS CHAPTER 22 Integration of Equations Newton-Cotes Algorithms for Equations Romberg Integration Adaptive Quadrature Gauss Quadrature Improper Integrals 650 Problems 653 CHAPTER 23 Numerical Differentiation High-Accuracy Differentiation Formulas Richardson Extrapolation Derivatives of Unequally Spaced Data Derivatives and Integrals for Data with Errors Partial Derivatives Numerical Integration/Differentiation with Software Packages 663 Problems 670 CHAPTER 24 Case Studies: 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) 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 DIFFERENTIAL EQUATIONS 699 PT7.1 Motivation 699 PT7.2 Mathematical Background 703 PT7.3 Orientation 705
9 xi CHAPTER 25 Runge-Kutta Methods Euler's Method Improvements of Euler's Method Runge-Kutta Methods Systems of Equations Adaptive Runge-Kutta Methods 744 Problems 752 CHAPTER 26 Stiffness and Multistep Methods Stiffness Multistep Methods 759 Problems 779 CHAPTER 27 Boundary-Value and Eigenvalue Problems General Methods for Boundary-Value Problems Eigenvalue Problems Odes and Eigenvalues with Software Packages 801 Problems 808 CHAPTER 28 Case Studies: 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) 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 DIFFERENTIAL EQUATIONS 845 PT8.1 Motivation 845 PT8.2 Orientation 848
10 xii CONTENTS CHAPTER 29 Finite Difference: Elliptic Equations The Laplace Equation Solution Technique Boundary Conditions The Control-Volume Approach Software to Solve Elliptic Equations 869 Problems 870 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 885 Problems 888 CHAPTER 31 Finite-Element Method The General Approach Finite-Element Application in One Dimension Two-Dimensional Problems Solving PDEs with Software Packages 908 Problems 912 CHAPTER 32 Case Studies: 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) 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
11 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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