Table of Contents

Chapter 1 Introduction 1
1.1 Numerical Analysis: What and Why? 1
1.2 Review of Calculus 5
1.3 Computer Arithmetic and Error Analysis 8
1.4 Algorithms and Convergence 14
1.5 Numerical Software 18
1.6 Reading: Golub and Moler 19
1.7 Self Assessment Exercises 21
Chapter 2 Solution of Nonlinear Equations 22
2.1 Introduction 22
2.2 Bisection Method 24
2.3 Fixed Point Iteration 29
2.4 Newton's Method 33
2.5 Secant Method 38
2.6 Survey of Methods and Software 41
2.7 Reading: Newton 43
2.8 Self Assessment Exercises 44
Chapter 3 Solution of Linear Systems 46
3.1 Introduction 46
3.2 Gaussian Elimination Method 49
3.2.1 Gaussian Elimination 49
3.2.2 Gaussian Elimination Algorithm 52
3.2.3 Partial Pivoting 56
3.2.4 Scaled Partial Pivoting 59
3.3 LU Factorization 60
3.4 Vector and Matrix Norms 67
3.5 Error of Calculated Solution 70
3.5.1 Error of Calculated Solution 70
3.5.2 A Practical Error Bound 72
3.6 Residual Correction Method 74
3.7 Jacobi Iterative Method 75
3.7.1 Jacobi Iteration 75
3.7.2 Convergence of the Jacobi Method 77
3.8 Gauss-Seidel Iterative Method 80
3.8.1 Gauss-Seidel Iteration 80
3.8.2 Termination of the Iteration 82
3.9 Survey of Methods and Software 84
3.9.1 MATLAB Functions 84
3.9.2 Comparison of Direct and Iterative Methods 84
3.9.3 Some Modern Methods and Software 84
3.10 Reading: Gauss， Jacobi and Seidel 85
3.11 Self Assessment Exercises 88
Chapter 4 Polynomial Interpolation and Curve Fitting 91
4.1 Introduction 91
4.2 Lagrange Form 92
4.3 Newton's Divided Difference Formula 95
4.4 Error of the Interpolating Polynomial 99
4.5 Runge's Phenomenon 101
4.6 Curve Fitting 103
4.6.1 Overview 103
4.6.2 Fitting a Straight Line 105
4.6.3 Fitting Linear Forms 106
4.6.4 Polynomial Fitting 107
4.6.5 Fitting Exponential Functions 109
4.7 Survey of Methods and Software 110
4.8 Reading: Lagrange and Runge 112
4.9 Self Assessment Exercises 114
Chapter 5 Numerical Differentiation and Numerical Integration 115
5.1 Introduction 115
5.2 Numerical Differentiation 117
5.3 Numerical Integration Based on Interpolation 127
5.3.1 Integration via Polynomial Interpolation 128
5.3.2 Trapezoid Rule 129
5.3.3 Simpson's Rule 130
5.4 Composite Trapezoid Rule 131
5.5 Romberg Integration 132
5.6 Reading: Cotes， Simpson and Romberg 140
5.7 Self Assessment Exercises 143
Chapter 6 Solution of Ordinary Differential Equations 145
6.1 Introduction 145
6.2 Euler's Method 148
6.3 Local and Global Truncation Errors 152
6.4 Taylor's Methods 154
6.5 Runge-Kutta Methods 156
6.6 Stability 163
6.7 First Order Systems 166
6.8 Reading: Euler， Kutta and Taylor 168
6.9 Self Assessment Exercises 171
Reference 173
Appendix A Introduction to MATLAB 175
A.1 A Glance of MATLAB 175
A.2 Starting Up 177
A.3 MATLAB as a Calculator 178
A.4 Plotting 187
A.5 Programming 191
A.6 Commonly Used Commands Summary 198
Appendix B Answers for Selected Exercises 202
Appendix C Common Mathematical Knowledge 205

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