诺森数值分析双语教程檀结庆主编9787030763310科学出版社

目录

前言

Chapter 1 Mathematical Preliminaries (数学基础知识) 1

1.1 Mathematics English (数学英语) 1

1.2 Review of calculus (微积分回顾) 4

1.2.1 Limisndcntinuity (极限和连续 ) 4

1.2.2 Differentiability (可微) 6

1.. Integration (积分) 6

1.2.4 Taylor polynomials and series (泰勒多项式和级数) 7

1.2.5 Examples (例题) 8

1.3 Errors and significant digits (误差和有效数字 ) 9

1.3.1 Source of errors (误差的来源) 9

1.3.2 Absolute error and relative error (误差和相对误差) 11

1.3.3 Significant digit (or figure) (有效数字) 11

1.3.4 How to avoid the loss of accuracy (如何避免精度的丢失) 12

1.3.5 Examples (例题) 12

1.4本章要点 (Highlights) 14

1.5问题讨论 (estions for discussion) 14

1.6关键术语 (Key terms) 14

1.7延伸阅读 (Extending reading) 15

1.7.1 背景知识 15

1.7.2 数学家传记：泰勒 (Taylor) 16

1.7.3 数学家传记：黎曼 (Riemann) 16

1.8习题 (Exercises) 18 Chapter 2 Direct Methods for Solving Linear Systems (解线方程组的直接法) 21

2.1 Gauss elimination method (Gauss消元法 ) 21

2.1.1 Some preliminaries (预备知识) 21

2.1.2 Gauss elimination with backward-substitution process(可回代的 Gauss 消元法)

2.2 Pivoting strategies (选主元策略) 27

2.2.1 Partial pivoting (maximal column pivoting) (优选列主元) 28

2.2.2 Scaled partial pivoting (scaled-column pivoting) (按比例列主元) 29

. Matrix factorization (矩阵分解法) 31

..1 Doolittle factorization (Doolittle分解) 32

..2 Crout factorization (Crout分解) 38

.. Permutation matrix (置换矩阵) 38

2.4 Special types of matrices (特殊形式矩阵的三角分解) 39

2.4.1 Strictly diagonally dominant matrix (严格对角占优矩阵 ) 39

2.4.2 Positive definite matrix (正定矩阵) 41

2.4.3 Strictly diagonally dominant tridiagonal matrix (严格对角占优三对角矩阵) 42

2.5本章算法程序及实例 (Algorithms and examples) 45

2.5.1 Gauss消元法 (Gauss elimination method) 45

2.5.2 选主元策略 (Pivoting strategies) 46

2.5.3 LU分解法 (LU decoition) 48

2.6本章要点 (Hightlights) 49

2.7问题讨论 (estions for discussion) 49

2.8 关键术语 (Key terms) 50

2.9 延伸阅读 (Extending reading) 51

2.10习题 (Exercises) 54 Chapter 3 Iterative Techniques in Matrix Algebra (矩阵代数迭代技术) 57

3.1 Norms of vectors and matrices (向量范数与矩阵范数) 58

3.1.1 Vector norm (向量范数 ) 58

3.1.2 Distance between vectors (向量之间的距离) 59

3.1.3 Matrix norm and distance (矩阵范数和距离) 60

3.1.4 Examples (例题) 61

3.2 Eigenvalues and eigenvectors (特征值和特征向量 ) 62

3.2.1 Eigenvalues and eigenvectors (特征值和特征向量) 63

3.2.2 Spectral radius (谱半径) 63

3.. Convergent matrices (收敛矩阵 ) 64

3.2.4 Examples (例题) 64

3.3 Iterative techniques for solving linear systems (解线方程组的迭代法 ) 66

3.3.1 Jacobi iterative method (Jacobi迭代法) 67

3.3.2 Gauss-Seidel iterative method (Gauss-Seidel迭代法) 68

3.3.3 General iteration method (一般迭代法) 69

3.3.4 Examples (例题) 70

3.4 Convergence analysis and SOR iterative method (收敛分析与 SOR迭代法) 72

3.4.1 Convergence analysis (收敛分析) 72

3.4.2 SOR iterative method (SOR迭代法) 73

3.4.3 SOR iterative method in matrix form (矩阵形式的 SOR迭代法) 74

3.4.4 Examples (例题) 75

3.5 Condition number and iterative refinement (条件数和迭代优化 ) 77

3.5.1 Condition number (条件数) 77

3.5.2 Iterative refinement (迭代优化) 79

3.5.3 Examples (例题) 80

3.6本章算法程序及实例 (Algorithms and examples) 82

3.6.1 雅可比迭代法 (Jacobi iterative method) 82

3.6.2 高斯-赛德尔迭代法 (Gauss-Seidel iterative method) 83

3.6.3 SOR迭代法 (SOR iterative method) 84

3.7本章要点 (Highlights) 85

3.8问题讨论 (estions for discussion) 86

3.9关键术语 (Key terms) 87

3.10延伸阅读 (Extending reading) 87

3.10.1 背景知识 87

3.10.2 数学家传记：高斯 (Gauss) 88

3.10.3 数学家传记：雅可比 (Jacobi) 89

3.11习题 (Exercises) 90 Chapter 4 Solutions of Equations in One Variable (一元方程求根) 98

4.1 Bisection method (二分法 ) 99

4.2 Fixed-point iteration and error analysis (不动点迭代及误差分析 ) 101

4.2.1 Fixed-point iteration (不动点迭代法) 101

4.2.2 Convergence analysis and error estimation (收敛分析和误差估计) 101

4.. The order of convergence (收敛阶) 104

4.3 Newton’s method (牛顿法) 105

4.3.1 Newton’s method and convergence analysis (牛顿法及其收敛分析) 105

4.3.2 How to handle multiple roots using Newton’s method (如何采用牛顿法处理重根问题) 107

4.4 The secant method (弦截法 ) 111

4.5本章算法程序及实例 (Algorithms and examples) 113

4.5.1 二分法求方程的根 (Root finding by bisection method) 113

4.5.2 不动点迭代法求方程的根 (Root finding by fix point iteration) 113

4.5.3 牛顿法求方程的根 (Root finding by Newton’s method) 114

4.5.4 牛顿法求一元方程重根(未知重数) (Multiple root finding by Newton’s method) 115

4.5.5 割线法求方程的根 (Root finding by secant method) 116

4.6本章要点 (Highlights) 117

4.7问题讨论 (estions for discussion) 118

4.8关键术语 (Key terms) 118

4.9延伸阅读 (Extending reading) 119

4.10习题 (Exercises) 127 Chapter 5 Interpolation by Polynomials (多项式插值) 129

5.1 Lagrange interpolation (Lagrange插值) 130

5.1.1 Linear interpolation (线插值) 130

5.1.2 dratic interpolation(二次插值) 131

5.1.3 nth-order polynomial interpolation ( n次多项式插值) 132

5.1.4 Uniqueness of interpolation (插值的专享) 133

5.1.5 Lagrange error formula (Lagrange误差公式) 134

5.1.6 Examples (例题) 135

5.2 Neville interpolation (Neville 插值) 139

5.3 Newton interpolation (Newton插值) 143

5.3.1 Definition of divided differences (差商的定义) 144

5.3.2 Newton’s expansion of a function (函数的 Newton展开) 145

5.3.3 Properties of divided differences (差商的质) 146

5.3.4 Computation of Newton’s interpolant (Newton插值的计算) 151

5.3.5 The relationship between divided differences and derivatives (差商与导数的关系) 153

5.3.6 Relations between Newton’s expansion and Taylor’s expansion (Newton展开与 Taylor展开之间的关系) 154

5.3.7 Comparisons among Lagrange, Neville and Newton interpolations (Lagrange插值、Neville插值与 Newton插值之间的比较) 154

5.3.8 Newton forward divided-difference formula (Newton向前差商公式) 156

5.3.9 Newton forward difference formula (Newton向前差分公式) 157

5.3.10 Newton backward divided-difference formula (Newton向后差商公式) 159

5.3.11 Newton backward-difference formula (Newton向后差分公式) 160

5.4 Hermite interpolation (Hermite插值) 164

5.4.1 Two-point Hermite interpolation (两点 Hermite插值) 164

5.4.2 General Hermit interpolation (一般 Hermite插值) 166

5.4.3 Examples (例题) 169

5.5 Cubic spline interpolation (三次样条插值) 172

5.5.1 Runge phenomenon (Runge现象) 172

5.5.2 Piecewise linear interpolation (分段线插值) 172

5.5.3 Piecewise cubic interpolation (分段三次插值) 174

5.5.4 Definition of cubic splines (三次样条的定义) 176

5.5.5 Derivation of cubic splines (三次样条的推导) 177

5.5.6 Examples (例题) 179

5.6 本章算法程序及实例 (Algorithms and examples) 187

5.6.1 拉格朗日插值 (Lagrange interpolation) 187

5.6.2 Neville 插值 (Neville interpolation) 189

5.6.3 牛顿插值 (Newton interpolation) 190

5.6.4 牛顿向前差商插值 (Interpolation by Newton forward divided differences) 192

5.6.5 牛顿向后差商插值 (Interpolation by Newton backward divided differences) 194

5.6.6 埃尔米特插值 (Hermite interpolation) 196

5.6.7 分段线插值 (Piecewise linear interpolation) 197

5.6.8 分段三次插值 (Piecewise cubic interpolation) 198

5.6.9 三次样条插值 1 (边界条件：固支边界) (Cubic spline interpolation with clamped boundary) 200

5.6.10 三次样条插值 2 (边界条件为自然边界) (Cubic spline interpolation with natural boundary ) 202

5.7 本章要点 (Hightlights) 204

5.8 问题讨论 (estions for discussion) 205

5.9 关键术语 (Key terms) 205

5.10 延伸阅读 (Extending reading) 207

5.10.1 有理函数插值 (Interpolation by rational functions) 207

5.10.2 Thiele型连分式插值 (Interpolation by Thiele type continued fractions) 209

5.10.3 Padé逼 (adé approximation) 212

5.10.4 数学家简介: 牛顿 (Newton) 212

5.10.5 数学家简介: 拉格朗日 (Lagrange) 213

5.10.6 数学家简介: 埃尔米特 (Hermite) 214

5.11习题 (Exercises) 215 Chapter 6 Approximation Theory (逼近论) 2

6.1 Discrete least squares approximation (离散昀二乘近) 2

6.1.1 Linear regression (线回归) 224

6.1.2 Criteria for the “best” fit (很好拟合准则) 224

6.1.3 Least squares fit of a straight line (二乘直线拟合) 225

6.1.4 Polynomial fitting (polynomial regression) (多项式拟合(多项式回归)) 226

6.1.5 Exponential fitting (指数拟合) 227

6.2 Orthogonal polynomials and least squares approximation (正交多项式和昀小平方逼近 ) 228

6.2.1 Basic ideas (基本思想) 228

6.2.2 Linearly independent functions (线无关函数) 0

6.. Orthogonal functions (正交函数)

6.2.4 Gram-Schmidt process (Gram-Schmidt 正交化)

6.3 Chebyshev polynomials and economization of power series (Chebyshev多项式与幂级数约化 ) 5

6.3.1 Definition of Chebyshev polynomial Tn(x)(Chebyshev多项式Tn(x)的定义)

6.3.2 Orthogonality of the Chebyshev polynomials (Chebyshev多项式的正交)

6.3.3 The zeros and extreme points of Tn(x)(Tn(x)的零点与极值点)

6.3.4 Minimization property (极小质)

6.3.5 Application of minimization property in polynomial interpolation (极小质在多项式插值中的应用)

6.3.6 Economization of power series (幂级数的约化) 240

6.4本章算法程序及实例 (Algorithms and examples) 241

6.4.1 二乘法 (Discrete least squares approximation) 241

6.4.2 指数拟合 (Exponential fitting) 243

6.4.3 Gram-Schmidt正交化 (Gram-Schmidt process) 245

6.4.4 勒让德正交多项式 (Legendre orthogonal polynomials) 245

6.4.5 切比雪夫正交多项式 (Chebyshev orthogonal polynomials) 246

6.4.6 很好平方逼近 (Least squares approximation) 247

6.5本章要点 (Hightlights) 248

章为数学英语和数学基础知识，主要内容包括：常见数学公式和数学表达式的读法、微积分基本概念和主要定理回顾、误差与相对误差的概念等；第二章为求解线方程组的直接方法，主要内容包括：Gauss消去法、主元法、矩阵分解法、特殊矩阵等；第三章为矩阵代数迭代技术，主要内容包括：Jacobi迭代法、Gauss-Seidel迭代法、SOR迭代法、收敛分析、条件数、迭代优化等；第四章为一元方程求根，主要内容包括：二分法、不动点迭代、牛顿法、割线法、收敛阶分析等；第五章为多项式插值，主要内容包括：Lagrange插值、Neville插值、Newton插值、Hermite插值、三次样条插值等；第六章为逼近论，主要内容包括：离散乘近、正交多项式、平方逼近、Chebyshev多项式与幂级数约化等；第七章为数值微分与数值积分，主要内容包括：数值微分、Richardson外推、数值积分、复化数值积分、Romberg积分、Gauss求积等；第八章为常微分方程初值问题，主要内容包括：初值问题基本理论、Euler方法、Runge-Kutta方法、多步方法等。