This book is designed to support a one-semester course in numerical methods. It has been written for students who want to learn and apply numerical methods in order to solve problems in engineering and science. As such, the methods are motivated by problems rather than by mathematics. That said, sufficient theory is provided so that students come away with insight into the techniques and their shortcomings.
MATLAB provides a great environment for such a course. Although other environments (e.g., Excel/VBA, Mathcad) or languages (e.g., Fortran 90, C++, Python) could have been chosen, MATLAB presently offers a nice combination of handy programming features with powerful built-in numerical capabilities. On the one hand, its M-file programming environment allows students to implement moderately complicated algorithms in a structured and coherent fashion. On the other hand, its built-in, numerical capabilities empower students to solve more difficult problems without trying to “reinvent the wheel.”
Algorithms Presented Using MATLAB M-files. Instead of using pseudocode, this book presents algorithms as well-structured MATLAB M-files. Aside from being useful computer programs, these provide students with models for their own M-files that they will develop as homework exercises.
Author(s): Steven C. Chapra
Publisher: McGraw Hill
Year: 2023
Language: English
Pages: 753
Cover
Title Page
Copyright
Dedication
About the Author
Contents
Preface viii
Acknowledgments xiv
Part One Modeling, Computers, and Error Analysis 1
1.1 Motivation 1
1.2 Part Organization 2
chapter 1: Mathematical Modeling, Numerical Methods, and Problem Solving 4
1.1 A Simple Mathematical Model 5
1.2 Conservation Laws in Engineering and Science 12
1.3 Numerical Methods Covered in This Book 13
1.4 Case Study: It’s a Real Drag 17
Problems 20
chapter 2: MATLAB Fundamentals 27
2.1 The Matlab Environment 28
2.2 Assignment 29
2.3 Mathematical Operations 36
2.4 Use of Built-In Functions 39
2.5 Graphics 42
2.6 Other Resources 46
2.7 Case Study: Exploratory Data Analysis 46
Problems 49
chapter 3: Programming with MATLAB 53
3.1 M-Files 54
3.2 Input-output 61
3.3 Structured Programming 65
3.4 Nesting and Indentation 79
3.5 Passing Functions to M-Files 82
3.6 Case Study: Bungee Jumper Velocity 88
Problems 92
chapter 4: Roundoff and Truncation Errors 100
4.1 Errors 101
4.2 Roundoff Errors 107
4.3 Truncation Errors 116
4.4 Total Numerical Error 127
4.5 Blunders, Model Errors, and Data Uncertainty 132
Problems 133
Part Two Roots and Optimization 137
2.1 Overview 137
2.2 Part Organization 138
chapter 5: Roots: Bracketing Methods 140
5.1 Roots in Engineering and Science 141
5.2 Graphical Methods 142
5.3 Bracketing Methods and Initial Guesses 143
5.4 Bisection 148
5.5 False Position 154
5.6 Case Study: Greenhouse Gases and Rainwater 158
Problems 161
chapter 6: Roots: Open Methods 166
6.1 Simple Fixed-Point Iteration 167
6.2 The Wegstein Method 173
6.3 Newton-Raphson 177
6.4 Secant Methods 182
6.5 Brent’s Method 184
6.6 Matlab Function: fzero 189
6.7 Polynomials 191
6.8 Case Study: Pipe Friction 194
Problems 199
chapter 7: Optimization 206
7.1 Introduction and Background 207
7.2 One-Dimensional Optimization 210
7.3 Multidimensional Optimization 219
7.4 Case Study: Equilibrium and Minimum Potential Energy 221
Problems 223
Part Three Linear Systems 231
3.1 Overview 231
3.2 Part Organization 233
chapter 8: Linear Algebraic Equations and Matrices 235
8.1 Matrix Algebra Overview 237
8.2 Solving Linear Algebraic Equations with Matlab 246
8.3 Case Study: Currents and Voltages in Circuits 248
Problems 252
chapter 9: Gauss Elimination 256
9.1 Solving Small Numbers of Equations 257
9.2 Naive Gauss Elimination 262
9.3 Pivoting 270
9.4 Tridiagonal Systems 273
9.5 Case Study: Model of a Heated Rod 276
Problems 279
chapter 10: LU Factorization 284
10.1 Overview of Lu Factorization 285
10.2 Gauss Elimination as Lu Factorization 286
10.3 Cholesky Factorization 293
10.4 Matlab Left Division 296
Problems 297
chapter 11: Matrix Inverse and Condition 298
11.1 The Matrix Inverse 298
11.2 Error Analysis and System Condition 302
11.3 Case Study: Indoor Air Pollution 307
Problems 310
chapter 12: Iterative Methods 315
12.1 Linear Systems: Gauss-Seidel 315
12.2 Nonlinear Systems 322
12.3 Case Study: Chemical Reactions 330
Problems 333
chapter 13: Eigenvalues 336
13.1 Eigenvalues and Eigenvectors—The Basics 338
13.2 Applications of Eigenvalues and Eigenvectors 341
13.3 Physical Settings—Mass-Spring Systems 347
13.4 The Power Method 350
13.5 MATLAB Function: eig 352
13.6 Case Study: Eigenvalues and Earthquakes 353
Problems 356
Part Four Curve Fitting 359
4.1 Overview 359
4.2 Part Organization 361
chapter 14: Linear Regression 362
14.1 Statistics Review 364
14.2 Random Numbers and Simulation 369
14.3 Linear Least-Squares Regression 374
14.4 Linearization of Nonlinear Relationships 382
14.5 Computer Applications 386
14.6 Case Study: Enzyme Kinetics 389
Problems 394
chapter 15: General Linear Least-Squares and Nonlinear Regression 401
15.1 Polynomial Regression 401
15.2 Multiple Linear Regression 405
15.3 General Linear Least Squares 407
15.4 Qr Factorization and the Backslash Operator 410
15.5 Nonlinear Regression 411
15.6 Case Study: Fitting Experimental Data 413
Problems 415
chapter 16: Fourier Analysis 420
16.1 Curve Fitting with Sinusoidal Functions 421
16.2 Continuous Fourier Series 427
16.3 Frequency and Time Domains 430
16.4 Fourier Integral and Transform 431
16.5 Discrete Fourier Transform (DFT) 434
16.6 The Power Spectrum 439
16.7 Case Study: Sunspots 441
Problems 442
chapter 17: Polynomial Interpolation 445
17.1 Introduction to Interpolation 446
17.2 Newton Interpolating Polynomial 449
17.3 Lagrange Interpolating Polynomial 457
17.4 Inverse Interpolation 460
17.5 Extrapolation and Oscillations 461
Problems 465
chapter 18: Splines and Piecewise Interpolation 469
18.1 Introduction to Splines 469
18.2 Linear Splines 471
18.3 Quadratic Splines 475
18.4 Cubic Splines 478
18.5 Piecewise Interpolation in MATLAB 484
18.6 Multidimensional Interpolation 489
18.7 Smoothing of Data Series 491
18.8 Case Study: Heat Transfer 501
Problems 505
Part Five Integration and Differentiation 511
5.1 Overview 511
5.2 Part Organization 512
chapter 19: Numerical Integration Formulas 514
19.1 Introduction and Background 515
19.2 Newton-Cotes Formulas 518
19.3 The Trapezoidal Rule 520
19.4 Simpson’s Rules 527
19.5 Higher-Order Newton-Cotes Formulas 533
19.6 Integration with Unequal Segments 534
19.7 Open Methods 538
19.8 Multiple Integrals 538
19.9 Case Study: Computing Work with Numerical Integration 541
Problems 544
chapter 20: Numerical Integration of Functions 550
20.1 Introduction 550
20.2 Romberg Integration 551
20.3 Gauss Quadrature 556
20.4 Adaptive Quadrature 563
20.5 Case Study: Root-Mean-Square Current 566
Problems 570
chapter 21: Numerical Differentiation 574
21.1 Introduction and Background 575
21.2 High-Accuracy Differentiation Formulas 579
21.3 Richardson Extrapolation 582
21.4 Tangent Line Differentiation of Functions 584
21.5 Derivatives of Unequally Spaced Data 587
21.6 Differentiation of Noisy Data 590
21.7 Partial Derivatives 596
21.8 Numerical Differentiation with Matlab 596
21.9 Case Study: Visualizing Fields 601
Problems 603
Part Six Ordinary Differential Equations 609
6.1 Overview 609
6.2 Part Organization 613
chapter 22: Initial-Value Problems 615
22.1 Overview 617
22.2 Euler’s Method 617
22.3 Improvements of Euler’s Method 623
22.4 Runge-Kutta Methods 629
22.5 Systems of Equations 634
22.6 Case Study: Predator-Prey Models and Chaos 640
Problems 645
chapter 23: Adaptive Methods and Stiff Systems 651
23.1 Adaptive Runge-Kutta Methods 651
23.2 Multistep Methods 660
23.3 Stiffness 664
23.4 Matlab Application: Bungee Jumper with Cord 670
23.5 Case Study: Pliny’s Intermittent Fountain 671
Problems 676
chapter 24: Boundary-Value Problems 682
24.1 Introduction and Background 683
24.2 The Shooting Method 687
24.3 Finite-Difference Methods 694
24.4 MATLAB Function: bvp4c 701
Problems 704
Appendix A: MATLAB Built-in Functions 710
Appendix B: MATLAB M-File functions 712
Appendix C: INTRODUCTION TO SIMULINK 713
Bibliography 721
Index 723
