數值分析

數值分析Numerical Analysis（第2版）本書採用中、英兩種語言編寫，各章都配有大量的習題及上機實驗題目，並附有部分習題的參考答案及數學專業軟件Mathematica和Matlab的簡介。 ^[1] 

本書介紹了科學計算中常用數值分析的基礎理論及計算機實現方法。主要內容包括：誤差分析、插值、函數逼近、數值積分和數值微分、非線性方程的數值解法、線性方程組的直接解法、線性方程組的迭代解法、常微分方程的數值解法及相應的上機實驗內容等。各章都配有大量的習題及上機實驗題目，並附有部分習題的參考答案及數學專業軟件Mathematica和Matlab的簡介。本書採用中、英兩種語言編寫，適合作為數學、計算機和其他理工類各專業本科“數值分析（計算方法）”雙語課程的教材或參考用書，也可供從事科學計算的相關技術人員參考。

蘇岐芳，副教授，台州學院數學與信息工程學院副院長 ^[1] 

1 Error Analysis ......1

1.1 Introduction ............ 1

1.2 Sources of Errors .... 2

1.3 Errors and Significant Digits.......... 4

1.4 Error Propagation ... 8

1.5 Qualitative Analysis and Control ofErrors ............ 9

1.5.1 Ill-condition Problem and ConditionNumber....................... 9

1.5.2 The Stability of Algorithm .. 10

1.5.3 The Control of Errors .......... 11

1.6 Computer Experiments.................14

1.6.1 Functions Needed in the Experimentsby Mathematica ...... 14

1.6.2 Experiments byMathematica...................... 14

1.6.3 Functions Needed in the Experimentsby Matlab................ 16

1.6.4 Experiments by Matlab ....... 16

Exercises 1..................... 17

2 Interpolating.......19

2.1 Introduction .......... 20

2.2 Basic Concepts ..... 21

2.3 Lagrange Interpolation................. 22

2.3.1 Linear and Parabolic Interpolation.............. 22

2.3.2 Lagrange InterpolationPolynomial............. 24

2.3.3 Interpolation Remainder and ErrorEstimate....................... 25

2.4 Divided-differences and NewtonInterpolation .... 29

2.5 Differences and Newton DifferenceFormulae..... 33

2.5.1 Differences .. 33

2.5.2 Newton Difference Formulae ......................35

2.6 Hermite Interpolation................... 38

2.7 Piecewise Low DegreeInterpolation.................... 42

2.7.1 Ill-posed Properties of High DegreeInterpolation .............. 42

2.7.2 Piecewise Linear Interpolation.................... 43

2.7.3 Piecewise Cubic HermiteInterpolation....... 44

2.8 Cubic Spline Interpolation............45

2.8.1 Definition of Cubic Spline... 45

2.8.2 The Construction of Cubic Spline............... 46

2.9 Computer Experiments.................49

2.9.1 Functions Needed in the Experimentsby Mathematica ...... 49

2.9.2 Experiments byMathematica...................... 50

2.9.3 Experiments by Matlab ....... 56

Exercises 2................... 64

3 Best Approximation ...................68

3.1 Introduction .......... 68

3.2 Norms ................... 69

3.2.1 Vector Norms ......................69

3.2.2 Matrix Norms ......................74

3.3 Spectral Radius..... 76

3.4 Best Linear Approximation .......... 79

3.4.1 Basic Concepts andTheories....................... 79

3.4.2 Best Linear Approximation . 81

3.5 Discrete Least Squares Approximation................ 82

3.6 Least Squares Approximation andOrthogonal Polynomials........ 87

3.7 Rational Function Approximation 94

3.7.1 Continued Fractions ............ 94

3.7.2 Padé Approximation............ 97

3.8 Computer Experiments.................99

3.8.1 Functions Needed in The Experimentsby Mathematica..... 99

3.8.2 Experiments byMathematica.................... 100

3.8.3 Functions Needed in The Experimentsby Matlab ............ 106

3.8.4 Experiments by Matlab ..... 106

Exercises 3................. 111

4 Numerical Integration and Differentiation........114

4.1 Introduction ........ 115

4.2 Interpolatory Quadratures...........116

4.2.1 Interpolatory Quadratures.. 116

4.2.2 Degree of Accuracy........... 117

4.3 Newton-Cotes QuadratureFormula.................... 118

4.4 Composite Quadrature Formula . 123

4.4.1 Composite Trapezoidal Rule .....................123

4.4.2 Composite Simpson’s Rule....................... 124

4.5 Romberg Integration...................125

4.5.1 Recursive Trapezoidal Rule...................... 125

4.5.2 Romberg Algorithm .......... 126

4.5.3 Richardson’s Extrapolation....................... 128

4.6 Gaussian Quadrature Formula .... 129

4.7 Multiple Integrals....................... 134

4.8 Numerical Differentiation...........135

4.8.1 Numerical Differentiation . 135

4.8.2 Differentiation PolynomialInterpolation .. 137

4.8.3 Richardson’s Extrapolation....................... 141

4.9 Computer Experiments............... 144

4.9.1 Functions Needed in the Experimentsby Mathematica .... 144

4.9.2 Experiments byMathematica.................... 144

4.9.3 Experiments by Matlab ..... 149

Exercises 4................... 153

5 Solution of Nonlinear Equations......................156

5.1 Introduction ........ 156

5.2 Basic Theories .... 158

5.3 Bisection Method 159

5.4 Iterative Method and ItsConvergence................ 162

5.4.1 Fixed Point and Iteration ... 162

5.4.2 Global Convergence.......... 163

5.4.3 Local Convergence............ 165

5.4.4 Order of Convergence ....... 167

5.5 Accelerating Convergence.......... 168

5.6 Newton’s Method .......................170

5.6.1 Newton’s Method and Its Convergence.... 170

5.6.2 Reduced Newton Method and Newton’sDescent Method ....................... 172

5.6.3 The Case of MultipleRoots....................... 173

5.7 Secant Method and Muller Method.................... 174

5.7.1 Secant Method................... 174

5.7.2 Muller Method................... 175

5.8 Systems of Nonlinear Equations. 176

5.9 Computer Experiments............... 179

5.9.1 Functions Needed in the Experimentsby Mathematica .... 179

5.9.2 Experiments byMathematica.................... 180

5.9.3 Experiments by Matlab ..... 185

Exercises 5................. 188

6 Direct Methods for Solving Linear Systems....191

6.1 Introduction ........ 192

6.2 Gaussian Elimination..................193

6.2.1 Basic GaussianElimination....................... 193

6.2.2 Triangular Decomposition. 197

6.3 Gaussian Elimination with ColumnPivoting ..... 200

6.4 Methods of the TriangularDecomposition......... 202

6.4.1 The Direct Methods of The TriangularDecomposition .... 202

6.4.2 The Square Root Method .. 203

6.4.3 The Speedup Method......... 206

6.5 Analysis of Round-off Errors ..... 210

6.5.1 Condition Number............. 210

6.5.2 Iterative Refinement .......... 214

6.6 Computer Experiments............... 215

6.6.1 Functions Needed in the Experimentsby Mathematica .... 215

6.6.2 Experiments byMathematica.................... 215

6.6.3 Functions Needed in the Experimentsby Matlab.............. 222

6.6.4 Experiments by Matlab ..... 222

Exercises 6................... 227

7 Iterative Techniques for Solving LinearSystems ....................230

7.1 Introduction ........ 231

7.2 Basic Iterative Methods ..............233

7.2.1 Jacobi Method ................... 234

7.2.2 Gauss-Seidel Method ........ 236

7.2.3 SOR Method...................... 237

7.3 Iterative Method Convergence ... 238

7.3.1 Basic Theorems ................. 238

7.3.2 Some Special Systems ofEquations.......... 243

7.4 Computer Experiments............... 247

7.4.1 Functions Needed in The Experiments byMathematica... 247

7.4.2 Experiments byMathematica.................... 247

7.4.3 Experiments by Matlab ..... 251

Exercises 7................... 255

8 Numerical Solution of OrdinaryDifferential Equations ............258

8.1 Introduction ........ 258

8.2 The Existence and Uniqueness ofSolutions....... 260

8.3 Taylor-Series Method.................262

8.4 Euler’s Method ... 263

8.5 Single-step Methods ...................267

8.5.1 Single-step Methods.......... 267

8.5.2 Local Truncation Error ...... 267

8.6 Runge-Kutta Methods ................268

8.6.1 Second-Order Runge-KuttaMethod.......... 268

8.6.2 Fourth-Order Runge-KuttaMethod........... 270

8.7 Multistep Methods......................271

8.7.1 General Formulas of MultistepMethods... 272

8.7.2 Adams Explicit and ImplicitFormulas...... 273

8.8 Systems and Higher-Order DifferentialEquations..................... 275

8.8.1 Vector Notation ................. 276

8.8.2 Taylor-Series Method forSystems............ 278

8.8.3 Fourth-Order Runge-Kutta Formula forSystems.............. 279

8.9 Computer Experiments............... 281

8.9.1 Functions Needed in the Experimentsby Mathematica .... 281

8.9.2 Experiments byMathematica.................... 281

8.9.3 Experiments by Matlab ..... 286

Exercises 8................... 290

Appendix ...............293

Appendix A Mathematica Basic Operations............ 293

Appendix B Matlab Basic Operations...................... 309

Appendix C Answers to SelectedQuestion.............. 327

Reference..............332 ^[1]