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  Applied Numerical Methods with MATLAB for Engineers and Scientists

CONTENTS
About the Author iv
Preface 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
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vi CONTENTS
3.3 Structured Programming 65
3.4 Nesting and Indentation 79
3.5 Passing Functions to M-Files 81
3.6 Case Study: Bungee Jumper Velocity 87
Problems 91
CHAPTER 4
Roundoff and Truncation Errors 99
4.1 Errors 100
4.2 Roundoff Errors 106
4.3 Truncation Errors 114
4.4 Total Numerical Error 125
4.5 Blunders, Model Errors, and Data Uncertainty 130
Problems 131
Part Two Roots and Optimization 135
2.1 Overview 135
2.2 Part Organization 136
CHAPTER 5
Roots: Bracketing Methods 138
5.1 Roots in Engineering and Science 139
5.2 Graphical Methods 140
5.3 Bracketing Methods and Initial Guesses 141
5.4 Bisection 146
5.5 False Position 152
5.6 Case Study: Greenhouse Gases and Rainwater 156
Problems 159
CHAPTER 6
Roots: Open Methods 164
6.1 Simple Fixed-Point Iteration 165
6.2 Newton-Raphson 169
6.3 Secant Methods 174
6.4 Brent’s Method 176
6.5 MATLAB Function: fzero 181
6.6 Polynomials 183
6.7 Case Study: Pipe Friction 186
Problems 191
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CONTENTS vii
CHAPTER 7
Optimization 198
7.1 Introduction and Background 199
7.2 One-Dimensional Optimization 202
7.3 Multidimensional Optimization 211
7.4 Case Study: Equilibrium and Minimum Potential Energy 213
Problems 215
Part Three Linear Systems 223
3.1 Overview 223
3.2 Part Organization 225
CHAPTER 8
Linear Algebraic Equations and Matrices 227
8.1 Matrix Algebra Overview 229
8.2 Solving Linear Algebraic Equations with MATLAB 238
8.3 Case Study: Currents and Voltages in Circuits 240
Problems 244
CHAPTER 9
Gauss Elimination 248
9.1 Solving Small Numbers of Equations 249
9.2 Naive Gauss Elimination 254
9.3 Pivoting 261
9.4 Tridiagonal Systems 264
9.5 Case Study: Model of a Heated Rod 266
Problems 270
CHAPTER 10
LU Factorization 274
10.1 Overview of LU Factorization 275
10.2 Gauss Elimination as LU Factorization 276
10.3 Cholesky Factorization 283
10.4 MATLAB Left Division 286
Problems 287
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viii CONTENTS
CHAPTER 11
Matrix Inverse and Condition 288
11.1 The Matrix Inverse 288
11.2 Error Analysis and System Condition 292
11.3 Case Study: Indoor Air Pollution 297
Problems 300
CHAPTER 12
Iterative Methods 305
12.1 Linear Systems: Gauss-Seidel 305
12.2 Nonlinear Systems 312
12.3 Case Study: Chemical Reactions 320
Problems 323
CHAPTER 13
Eigenvalues 326
13.1 Mathematical Background 328
13.2 Physical Background 331
13.3 The Power Method 333
13.4 MATLAB Function: eig 336
13.5 Case Study: Eigenvalues and Earthquakes 337
Problems 340
Part Four Curve Fitting 343
4.1 Overview 343
4.2 Part Organization 345
CHAPTER 14
Linear Regression 346
14.1 Statistics Review 348
14.2 Random Numbers and Simulation 353
14.3 Linear Least-Squares Regression 358
14.4 Linearization of Nonlinear Relationships 366
14.5 Computer Applications 370
14.6 Case Study: Enzyme Kinetics 373
Problems 378
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CONTENTS ix
CHAPTER 15
General Linear Least-Squares and Nonlinear Regression 385
15.1 Polynomial Regression 385
15.2 Multiple Linear Regression 389
15.3 General Linear Least Squares 391
15.4 QR Factorization and the Backslash Operator 394
15.5 Nonlinear Regression 395
15.6 Case Study: Fitting Experimental Data 397
Problems 399
CHAPTER 16
Fourier Analysis 404
16.1 Curve Fitting with Sinusoidal Functions 405
16.2 Continuous Fourier Series 411
16.3 Frequency and Time Domains 414
16.4 Fourier Integral and Transform 415
16.5 Discrete Fourier Transform (DFT) 418
16.6 The Power Spectrum 423
16.7 Case Study: Sunspots 425
Problems 426
CHAPTER 17
Polynomial Interpolation 429
17.1 Introduction to Interpolation 430
17.2 Newton Interpolating Polynomial 433
17.3 Lagrange Interpolating Polynomial 441
17.4 Inverse Interpolation 444
17.5 Extrapolation and Oscillations 445
Problems 449
CHAPTER 18
Splines and Piecewise Interpolation 453
18.1 Introduction to Splines 453
18.2 Linear Splines 455
18.3 Quadratic Splines 459
18.4 Cubic Splines 462
18.5 Piecewise Interpolation in MATLAB 468
18.6 Multidimensional Interpolation 473
18.7 Case Study: Heat Transfer 476
Problems 480
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x CONTENTS
Part Five Integration and Differentiation 485
5.1 Overview 485
5.2 Part Organization 486
CHAPTER 19
Numerical Integration Formulas 488
19.1 Introduction and Background 489
19.2 Newton-Cotes Formulas 492
19.3 The Trapezoidal Rule 494
19.4 Simpson’s Rules 501
19.5 Higher-Order Newton-Cotes Formulas 507
19.6 Integration with Unequal Segments 508
19.7 Open Methods 512
19.8 Multiple Integrals 512
19.9 Case Study: Computing Work with Numerical Integration 515
Problems 518
CHAPTER 20
Numerical Integration of Functions 524
20.1 Introduction 524
20.2 Romberg Integration 525
20.3 Gauss Quadrature 530
20.4 Adaptive Quadrature 537
20.5 Case Study: Root-Mean-Square Current 540
Problems 544
CHAPTER 21
Numerical Differentiation 548
21.1 Introduction and Background 549
21.2 High-Accuracy Differentiation Formulas 552
21.3 Richardson Extrapolation 555
21.4 Derivatives of Unequally Spaced Data 557
21.5 Derivatives and Integrals for Data with Errors 558
21.6 Partial Derivatives 559
21.7 Numerical Differentiation with MATLAB 560
21.8 Case Study: Visualizing Fields 565
Problems 567
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CONTENTS xi
Part six Ordinary Differential Equations 573
6.1 Overview 573
6.2 Part Organization 577
CHAPTER 22
Initial-Value Problems 579
22.1 Overview 581
22.2 Euler’s Method 581
22.3 Improvements of Euler’s Method 587
22.4 Runge-Kutta Methods 593
22.5 Systems of Equations 598
22.6 Case Study: Predator-Prey Models and Chaos 604
Problems 609
CHAPTER 23
Adaptive Methods and Stiff Systems 615
23.1 Adaptive Runge-Kutta Methods 615
23.2 Multistep Methods 624
23.3 Stiffness 628
23.4 MATLAB Application: Bungee Jumper with Cord 634
23.5 Case Study: Pliny’s Intermittent Fountain 635
Problems 640
CHAPTER 24
Boundary-Value Problems 646
24.1 Introduction and Background 647
24.2 The Shooting Method 651
24.3 Finite-Difference Methods 658
24.4 MATLAB Function: bvp4c 665
Problems 668
APPENDIX A: MATLAB BUILT-IN FUNCTIONS 674
APPENDIX B: MATLAB M-FILE FUNCTIONS 676
APPENDIX C: INTRODUCTION TO SIMULINK 677
BIBLIOGRAPHY 685
INDEX 687