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Chapter 1 Introduction 1

1.1 Background 1

1.2 Representation of Numbers on a Computer 4

1.3 Errors in Numerical Solutions 10

1.3.1 Round-Off Errors 10

1.3.2 Truncation Errors 13

1.3.3 Total Error 14

1.4 Computers and Programming 15

1.5 Problems 18

Chapter 2 Solving Nonlinear Equations 23

2.1 Background 23

2.2 Estimation of Errors in Numerical Solutions 25

2.3 Bisection Method 27

2.4 Regula Falsi Method 30

2.5 Newton’s Method 32

2.6 Secant Method 37

2.7 Fixed-Point Iteration Method 40

2.8 Use of MATLAB Built-In Functions for Solving Nonlinear Equations 43

2.8.1 The fzero Command 44

2.8.2 The roots Command 45

2.9 Equations with Multiple Solutions 45

2.10 Systems of Nonlinear Equations 47

2.10.1 Newton s Method for Solving a System of Nonlinear Equations 48

2.10.2 Fixed-Point Iteration Method for Solving a System of Nonlinear Equations 52

2.11 Problems 54

Chapter 3 Solving a System of Linear Equations 65

3.1 Background 65

3.1.1 Overview of Numerical Methods for Solving a System of Linear Algebraic Equations 66

3.2 Gauss Elimination Method 68

3.2.1 Potential Difficulties When Applying the Gauss Elimination Method 76

3.3 Gauss Elimination with Pivoting 78

3.4 Gauss-Jordan Elimination Method 81

3.5 LU Decomposition Method 84

3.5.1 LU Decomposition Using the Gauss Elimination Procedure 86

3.5.2 LU Decomposition Using Crouts Method 87

3.5.3 LU Decomposition with Pivoting 94

3.6 Inverse of a Matrix 94

3.6.1 Calculating the Inverse with the LU Decomposition Method 95

3.6.2 Calculating the Inverse Using the Gauss-Jordan Method 97

3.7 Iterative Methods 98

3.7.1 Jacobi Iterative Method 99

3.7.2 Gauss-Seidel Iterative Method 99

3.8 Use of MATLAB Built-In Functions for Solving a System of Linear Equations 102

3.8.1 Solving a System of Equations Using MATLABs Left and Right Division 102

3.8.2 Solving a System of Equations Using MATLABs Inverse Operation 103

3.8.3 MATLABs Built-In Function for L U Decomposition 104

3.8.4 Additional MATLAB Built-In Functions 105

3.9 Tridiagonal Systems of Equations 107

3.10 Error，Residual，Norms，and Condition Number 112

3.10.1 Error and Residual 112

3.10.2 Norms and Condition Number 114

3.11 Ill-Conditioned Systems 117

3.12 Eigenvalues and Eigenvectors 121

3.12.1 The Basic Power Method 124

3.12.2 The Inverse Power Method 128

3.12.3 The Shifted Power Method 129

3.12.4 The QR Factorization and Iteration Method 129

3.12.5 Use of MA TLAB Built-In Functions for Determining Eigenvalues and Eigenvectors 139

3.13 Problems 141

Chapter 4 Curve Fitting and Interpolation 153

4.1 Background 153

4.2 Curve Fitting with a Linear Equation 155

4.2.1 Measuring How Good Is a Fit 155

4.2.2 Linear Least-Squares Regression 157

4.3 Curve Fitting with Nonlinear Equation by Writing the Equation in a Linear Form 161

4.4 Curve Fitting with Quadratic and Higher-Order Polynomials 165

4.5 Interpolation Using a Single Polynomial 170

4.5.1 Lagrange Interpolating Polynomials 172

4.5.2 Newton’s Interpolating Polynomials 176

4.6 Piecewise （Spline） Interpolation 183

4.6.1 Linear Splines 183

4.6.2 Quadratic Splines 185

4.6.3 Cubic Splines 189

4.7 Use of MATLAB Built-In Functions for Curve Fitting and Interpolation 196

4.8 Curve Fitting with a Linear Combination of Nonlinear Functions 198

4.9 Problems 201

Chapter 5 Numerical Differentiation 211

5.1 Background 211

5.2 Finite Difference Approximation of the Derivative 213

5.3 Finite Difference Formulas Using Taylor Series Expansion 218

5.3.1 Finite Difference Formulas of First Derivative 218

5.3.2 Finite Difference Formulas for the Second Derivative 223

5.4 Summary of Finite Difference Formulas for Numerical Differentiation 225

5.5 Differentiation Formulas Using Lagrange Polynomials 227

5.6 Differentiation Using Curve Fitting 228

5.7 Use of MATLAB Built-In Functions for Numerical Differentiation 228

5.8 Richardson’s Extrapolation 230

5.9 Error in Numerical Differentiation 233

5.10 Numerical Partial Differentiation 235

5.11 Problems 238

Chapter 6 Numerical Integration 249

6.1 Background 249

6.1.1 Overview ofApproaches in Numerical Integration 250

6.2 Rectangle and Midpoint Methods 252

6.3 Trapezoidal Method 254

6.3.1 Composite Trapezoidal Method 255

6.4 Simpson’s Methods 258

6.4.1 Simpson s 1/3 Method 258

6.4.2 Simpson s 3/8 Method 261

6.5 Gauss Quadrature 263

6.6 Evaluation of Multiple Integrals 269

6.7 Use of MATLAB Built-In Functions for Integration 270

6.8 Estimation of Error in Numerical Integration 272

6.9 Richardson’s Extrapolation 274

6.10 Romberg Integration 277

6.11 Improper Integrals 280

6.11.1 Integrals with Singularities 280

6.11.2 Integrals with Unbounded Limits 281

6.12 Problems 282

Chapter 7 Ordinary Differential Equations：Initial-Value Problems 293

7.1 Background 293

7.2 Euler’s Methods 298

7.2.1 Euler’s Explicit Method 298

7.2.2 Analysis of Truncation Error in Euler’s Explicit Method 302

7.2.3 Euler’s Implicit Method 306

7.3 Modified Euler’s Method 309

7.4 Midpoint Method 312

7.5 Runge-Kutta Methods 313

7.5.1 Second-Order Runge-Kutta Methods 314

7.5.2 Third-Order Runge Kutta Methods 318

7.5.3 Fourth-Order Runge-Kutta Methods 319

7.6 Multistep Methods 325

7.6.1 Adams-Bashforth Method 326

7.6.2 Adams-Moulton Method 327

7.7 Predictor-Corrector Methods 328

7.8 System of First-Order Ordinary Differential Equations 330

7.8.1 Solving a System of First-Order ODEs Using Euler’s Explicit Method 332

7.8.2 Solving a System ofFirst-Order ODEs Using Second-Order Runge-Kutta Method （Modified Euler Version） 332

7.8.3 Solving a System of First-Order ODEs Using the Classical Fourth-Order Runge-Kutta Method 339

7.9 Solving a Higher-Order Initial Value Problem 340

7.10 Use of MATLAB Built-In Functions for Solving Initial-Value Problems 345

7.10.1 Solving a Single First-Order ODE Using MA TLAB 346

7.10.2 Solving a System of First-Order ODEs Using MATLAB 352

7.11 Local Truncation Error in Second-Order Range-Kutta Method 355

7.12 Step Size For Desired Accuracy 356

7.13 Stability 360

7.14 Stiff Ordinary Differential Equations 362

7.15 Problems 365

Chapter 8 Ordinary Differential Equations：Boundary-Value Problems 377

8.1 Background 377

8.2 The Shooting Method 380

8.3 Finite Difference Method 388

8.4 Use of MATLAB Built-In Functions for Solving Boundary Value Problems 398

8.5 Error and Stability in Numerical Solution of Boundary Value Problems 403

8.6 Problems 405

Appendix A Introductory MATLAB 415

A.1 Background 415

A.2 Starting with MATLAB 415

A.3 Arrays 420

A.4 Mathematical Operations with Arrays 425

A.5 Script Files 430

A.6 Plotting 432

A.7 User-Defined Functions and Function Files 434

A.8 Anonymous Functions 436

A.9 Function functions 438

A.10 Subfunctions 441

A.11 Programming in MATLAB 443

A.11.1 Relational and Logical Operators 443

A.11.2 Conditional Statements，if-else Structures 444

A.11.3 Loops 447

A.12 Problems 448

Appendix B MATLAB Programs 453

Index 457