目錄
前言
Chapter 1 Mathematical Preliminaries (數(shù)學(xué)基礎(chǔ)知識(shí)) 1
1.1 Mathematics English (數(shù)學(xué)英語(yǔ)) 1
1.2 Review of calculus (微積分回顧) 4
1.2.1 Limits and continuity (極限和連續(xù)性 ) 4
1.2.2 Differentiability (可微性) 6
1.2.3 Integration (積分) 6
1.2.4 Taylor polynomials and series (泰勒多項(xiàng)式和級(jí)數(shù)) 7
1.2.5 Examples (例題) 8
1.3 Errors and significant digits (誤差和有效數(shù)字 ) 9
1.3.1 Source of errors (誤差的來(lái)源) 9
1.3.2 Absolute error and relative error (絕對(duì)誤差和相對(duì)誤差) 11
1.3.3 Significant digit (or figure) (有效數(shù)字) 11
1.3.4 How to avoid the loss of accuracy (如何避免精度的丟失) 12
1.3.5 Examples (例題) 12
1.4本章要點(diǎn) (Highlights) 14
1.5問(wèn)題討論 (Questions for discussion) 14
1.6關(guān)鍵術(shù)語(yǔ) (Key terms) 14
1.7延伸閱讀 (Extending reading) 15
1.7.1 背景知識(shí) 15
1.7.2 數(shù)學(xué)家傳記：泰勒 (Taylor) 16
1.7.3 數(shù)學(xué)家傳記：黎曼 (Riemann) 16
1.8習(xí)題 (Exercises) 18 Chapter 2 Direct Methods for Solving Linear Systems (解線性方程組的直接法) 21
2.1 Gauss elimination method (Gauss消元法 ) 21
2.1.1 Some preliminaries (預(yù)備知識(shí)) 21
2.1.2 Gauss elimination with backward-substitution process(可回代的 Gauss 消元法) 23
2.2 Pivoting strategies (選主元策略) 27
2.2.1 Partial pivoting (maximal column pivoting) (最大列主元) 28
2.2.2 Scaled partial pivoting (scaled-column pivoting) (按比例列主元) 29
2.3 Matrix factorization (矩陣分解法) 31
2.3.1 Doolittle factorization (Doolittle分解) 32
2.3.2 Crout factorization (Crout分解) 38
2.3.3 Permutation matrix (置換矩陣) 38
2.4 Special types of matrices (特殊形式矩陣的三角分解) 39
2.4.1 Strictly diagonally dominant matrix (嚴(yán)格對(duì)角占優(yōu)矩陣 ) 39
2.4.2 Positive definite matrix (正定矩陣) 41
2.4.3 Strictly diagonally dominant tridiagonal matrix (嚴(yán)格對(duì)角占優(yōu)三對(duì)角矩陣) 42
2.5本章算法程序及實(shí)例 (Algorithms and examples) 45
2.5.1 Gauss消元法 (Gauss elimination method) 45
2.5.2 選主元策略 (Pivoting strategies) 46
2.5.3 LU分解法 (LU decomposition) 48
2.6本章要點(diǎn) (Hightlights) 49
2.7問(wèn)題討論 (Questions for discussion) 49
2.8 關(guān)鍵術(shù)語(yǔ) (Key terms) 50
2.9 延伸閱讀 (Extending reading) 51
2.10習(xí)題 (Exercises) 54 Chapter 3 Iterative Techniques in Matrix Algebra (矩陣代數(shù)迭代技術(shù)) 57
3.1 Norms of vectors and matrices (向量范數(shù)與矩陣范數(shù)) 58
3.1.1 Vector norm (向量范數(shù) ) 58
3.1.2 Distance between vectors (向量之間的距離) 59
3.1.3 Matrix norm and distance (矩陣范數(shù)和距離) 60
3.1.4 Examples (例題) 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 (譜半徑) 63
3.2.3 Convergent matrices (收斂矩陣 ) 64
3.2.4 Examples (例題) 64
3.3 Iterative techniques for solving linear systems (解線性方程組的迭代法 ) 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 (例題) 70
3.4 Convergence analysis and SOR iterative method (收斂性分析與 SOR迭代法) 72
3.4.1 Convergence analysis (收斂性分析) 72
3.4.2 SOR iterative method (SOR迭代法) 73
3.4.3 SOR iterative method in matrix form (矩陣形式的 SOR迭代法) 74
3.4.4 Examples (例題) 75
3.5 Condition number and iterative refinement (條件數(shù)和迭代優(yōu)化 ) 77
3.5.1 Condition number (條件數(shù)) 77
3.5.2 Iterative refinement (迭代優(yōu)化) 79
3.5.3 Examples (例題) 80
3.6本章算法程序及實(shí)例 (Algorithms and examples) 82
3.6.1 雅可比迭代法 (Jacobi iterative method) 82
3.6.2 高斯-賽德?tīng)柕�代�?(Gauss-Seidel iterative method) 83
3.6.3 SOR迭代法 (SOR iterative method) 84
3.7本章要點(diǎn) (Highlights) 85
3.8問(wèn)題討論 (Questions for discussion) 86
3.9關(guān)鍵術(shù)語(yǔ) (Key terms) 87
3.10延伸閱讀 (Extending reading) 87
3.10.1 背景知識(shí) 87
3.10.2 數(shù)學(xué)家傳記：高斯 (Gauss) 88
3.10.3 數(shù)學(xué)家傳記：雅可比 (Jacobi) 89
3.11習(xí)題 (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 (不動(dòng)點(diǎn)迭代及誤差分析 ) 101
4.2.1 Fixed-point iteration (不動(dòng)點(diǎn)迭代法) 101
4.2.2 Convergence analysis and error estimation (收斂性分析和誤差估計(jì)) 101
4.2.3 The order of convergence (收斂階) 104
4.3 Newton’s method (牛頓法) 105
4.3.1 Newton’s method and convergence analysis (牛頓法及其收斂性分析) 105
4.3.2 How to handle multiple roots using Newton’s method (如何采用牛頓法處理重根問(wèn)題) 107
4.4 The secant method (弦截法 ) 111
4.5本章算法程序及實(shí)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 不動(dòng)點(diǎn)迭代法求方程的根 (Root finding by fix point iteration) 113
4.5.3 牛頓法求方程的根 (Root finding by Newton’s method) 114
4.5.4 牛頓法求一元方程重根(未知重?cái)?shù)) (Multiple root finding by Newton’s method) 115
4.5.5 割線法求方程的根 (Root finding by secant method) 116
4.6本章要點(diǎn) (Highlights) 117
4.7問(wèn)題討論 (Questions for discussion) 118
4.8關(guān)鍵術(shù)語(yǔ) (Key terms) 118
4.9延伸閱讀 (Extending reading) 119
4.10習(xí)題 (Exercises) 127 Chapter 5 Interpolation by Polynomials (多項(xiàng)式插值) 129
5.1 Lagrange interpolation (Lagrange插值) 130
5.1.1 Linear interpolation (線性插值) 130
5.1.2 Quadratic interpolation(二次插值) 131
5.1.3 nth-order polynomial interpolation ( n次多項(xiàng)式插值) 132
5.1.4 Uniqueness of interpolation (插值的唯一性) 133
5.1.5 Lagrange error formula (Lagrange誤差公式) 134
5.1.6 Examples (例題) 135
5.2 Neville interpolation (Neville 插值) 139
5.3 Newton interpolation (Newton插值) 143
5.3.1 Definition of divided differences (差商的定義) 144
5.3.2 Newton’s expansion of a function (函數(shù)的 Newton展開(kāi)) 145
5.3.3 Properties of divided differences (差商的性質(zhì)) 146
5.3.4 Computation of Newton’s interpolant (Newton插值的計(jì)算) 151
5.3.5 The relationship between divided differences and derivatives (差商與導(dǎo)數(shù)的關(guān)系) 153
5.3.6 Relations between Newton’s expansion and Taylor’s expansion (Newton展開(kāi)與 Taylor展開(kāi)之間的關(guān)系) 154
5.3.7 Comparisons among Lagrange, Neville and Newton interpolations (Lagrange插值、Neville插值與 Newton插值之間的比較) 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 (兩點(diǎn) Hermite插值) 164
5.4.2 General Hermit interpolation (一般 Hermite插值) 166
5.4.3 Examples (例題) 169
5.5 Cubic spline interpolation (三次樣條插值) 172
5.5.1 Runge phenomenon (Runge現(xiàn)象) 172
5.5.2 Piecewise linear interpolation (分段線性插值) 172
5.5.3 Piecewise cubic interpolation (分段三次插值) 174
5.5.4 Definition of cubic splines (三次樣條的定義) 176
5.5.5 Derivation of cubic splines (三次樣條的推導(dǎo)) 177
5.5.6 Examples (例題) 179
5.6 本章算法程序及實(shí)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 牛頓插值 (Newton interpolation) 190
5.6.4 牛頓向前差商插值 (Interpolation by Newton forward divided differences) 192
5.6.5 牛頓向后差商插值 (Interpolation by Newton backward divided differences) 194
5.6.6 埃爾米特插值 (Hermite interpolation) 196
5.6.7 分段線性插值 (Piecewise linear interpolation) 197
5.6.8 分段三次插值 (Piecewise cubic interpolation) 198
5.6.9 三次樣條插值 1 (邊界條件：固支邊界) (Cubic spline interpolation with clamped boundary) 200
5.6.10 三次樣條插值 2 (邊界條件為自然邊界) (Cubic spline interpolation with natural boundary ) 202
5.7 本章要點(diǎn) (Hightlights) 204
5.8 問(wèn)題討論 (Questions for discussion) 205
5.9 關(guān)鍵術(shù)語(yǔ) (Key terms) 205
5.10 延伸閱讀 (Extending reading) 207
5.10.1 有理函數(shù)插值 (Interpolation by rational functions) 207
5.10.2 Thiele型連分式插值 (Interpolation by Thiele type continued fractions) 209
5.10.3 Padé逼近 (Padé approximation) 212
5.10.4 數(shù)學(xué)家簡(jiǎn)介: 牛頓 (Newton) 212
5.10.5 數(shù)學(xué)家簡(jiǎn)介: 拉格朗日 (Lagrange) 213
5.10.6 數(shù)學(xué)家簡(jiǎn)介: 埃爾米特 (Hermite) 214
5.11習(xí)題 (Exercises) 215 Chapter 6 Approximation Theory (逼近論) 223
6.1 Discrete least squares approximation (離散昀小二乘逼近) 223
6.1.1 Linear regression (線性回歸) 224
6.1.2 Criteria for the “best” fit (最佳擬合準(zhǔn)則) 224
6.1.3 Least squares fit of a straight line (最小二乘直線擬合) 225
6.1.4 Polynomial fitting (polynomial regression) (多項(xiàng)式擬合(多項(xiàng)式回歸)) 226
6.1.5 Exponential fitting (指數(shù)擬合) 227
6.2 Orthogonal polynomials and least squares approximation (正交多項(xiàng)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 (線性無(wú)關(guān)函數(shù)) 230
6.2.3 Orthogonal functions (正交函數(shù)) 233
6.2.4 Gram-Schmidt process (Gram-Schmidt 正交化) 233
6.3 Chebyshev polynomials and economization of power series (Chebyshev多項(xiàng)式與冪級(jí)數(shù)約化 ) 235
6.3.1 Definition of Chebyshev polynomial Tn(x)(Chebyshev多項(xiàng)式Tn(x)的定義) 236
6.3.2 Orthogonality of the Chebyshev polynomials (Chebyshev多項(xiàng)式的正交性) 236
6.3.3 The zeros and extreme points of Tn(x)(Tn(x)的零點(diǎn)與極值點(diǎn)) 237
6.3.4 Minimization property (極小性質(zhì)) 238
6.3.5 Application of minimization property in polynomial interpolation (極小性質(zhì)在多項(xiàng)式插值中的應(yīng)用) 239
6.3.6 Economization of power series (冪級(jí)數(shù)的約化) 240
6.4本章算法程序及實(shí)例 (Algorithms and examples) 241
6.4.1 最小二乘法 (Discrete least squares approximation) 241
6.4.2 指數(shù)擬合 (Exponential fitting) 243
6.4.3 Gram-Schmidt正交化 (Gram-Schmidt process) 245
6.4.4 勒讓德正交多項(xiàng)式 (Legendre orthogonal polynomials) 245
6.4.5 切比雪夫正交多項(xiàng)式 (Chebyshev orthogonal polynomials) 246
6.4.6 最佳平方逼近 (Least squares approximation) 247
6.5本章要點(diǎn) (Hightlights) 248
6.6問(wèn)題討論 (Questions for discussion) 249
6.7 關(guān)鍵術(shù)語(yǔ) (Key terms) 250
6.8 延伸閱讀 (Extending reading) 251
6.9習(xí)題 (Exercises) 259 Chapter 7 Numerical Differentiation and Integration (數(shù)值微分與數(shù)值積分) 263
7.1 Numerical differentiation (數(shù)值微分) 264
7.1.1 Forward-difference formula (向前差分公式) 264
7.1.2 Backward-difference formula (向后差分公式) 265
7.1.3 Three-point formula (三點(diǎn)公式) 265
7.1.4 Five-point formula (五點(diǎn)公式) 266
7.1.5 Approximation to higher derivatives (高階導(dǎo)數(shù)逼近) 269
7.1.6 Effect of round-off error (舍入誤差影響) 271
7.1.7 Examples (例題) 272
7.2 Richardson’s extrapolation (Richardson外推) 273
7.2.1 Basic idea (基本思想) 273
7.2.2 Examples (例題) 276
7.3 Elements of numerical integration (數(shù)值積分) 281
7.3.1 Basic idea (基本思想) 282
7.3.2 Midpoint rule (中點(diǎn)公式) 283
7.3.3 Trapezoidal rule (梯形公式) 283
7.3.4 Simpson’s rule (Simpson公式) 284
7.3.5 Newton-Cotes formulas (Newton-Cotes公式) 286
7.3.6 Degree of accuracy (代數(shù)精度) 287
7.3.7 Examples (例題) 287
7.4 Composite numerical integration (復(fù)化數(shù)值積分 ) 289
7.4.1 Composite midpoint rule (復(fù)化中點(diǎn)公式) 290
7.4.2 Composite trapezoidal rule (復(fù)化梯形公式) 290
7.4.3 Composite Simpson’s rule (復(fù)化 Simpson公式) 291
7.4.4 Stability (穩(wěn)定性) 292
7.5 Romberg integration (Romberg積分) 293
7.6 Gaussian quadrature (Gauss求積) 296
7.6.1 Basic idea (基本思想) 297
7.6.2 Two-point Gaussian quadrature (兩點(diǎn) Gauss求積公式) 298
7.6.3 Gaussian nodes (Gauss結(jié)點(diǎn)) 299
7.6.4 The error estimation for the Gaussian quadrature (Gauss求積的誤差估計(jì)) 300
7.6.5 Method to get Gaussian quadrature (Gauss求積的方法 ) 301
7.6.6 Examples (例題) 302
7.7 本章算法程序及實(shí)例 (Algorithms and examples) 304
7.7.1 數(shù)值微分 (Numerical differentiation) 304
7.7.2 數(shù)值積分 (Numerical integration) 308
7.8 本章要點(diǎn) (Highlights) 312
7.9 問(wèn)題討論 (Questions for discussion) 313
7.10 關(guān)鍵術(shù)語(yǔ) (Key terms) 316
7.11 延伸閱讀 (Extending reading) 317
7.11.1 Gauss求積公式的收斂性 (The Convergence of Gauss quadrature) 317
7.11.2 Radau求積公式與 Lobatto求積公式 (Radau quadrature and Lobatto quadrature) 320
7.11.3 Euler-Maclaurin 展開(kāi) (Euler-Maclaurin expansion) 330
7.12 習(xí)題 (Exercises) 338 Chapter 8 Initial-Value Problems for Ordinary Differential Equations (常微分方程初值問(wèn)題) 345
8.1 The elementary theory of initial- value problem (初值問(wèn)題基本理論 ) 345
8.1.1 Convex set (凸集) 346
8.1.2 Lipschitz condition (李普希茨條件) 346
8.1.3 Picard’s theorem (Picard定理) 347
8.1.4 Well-posed problem (適定問(wèn)題) 351
8.2 Euler’s method (Euler方法) 354
8.2.1 Basic ideas of Euler’s method (Euler方法的基本思想) 354
8.2.2 Order of a method (方法的階) 355
8.2.3 Implicit Euler’s scheme (隱式 Euler格式) 356
8.2.4 Two-step Euler’s scheme (兩步 Euler格式) 356
8.2.5 Trapezoidal scheme (梯形格式) 357
8.2.6 Modified Euler scheme (改進(jìn)的 Euler格式) 359
8.2.7 Error bounds for the Euler’s scheme (Euler格式的誤差界) 360
8.3 Runge-Kutta method (Runge-Kutta方法) 363
8.3.1 Taylor polynomials in two variables (二元 Taylor多項(xiàng)式) 363
8.3.2 Basic ideas of Runge-Kutta method (Runge-Kutta方法的基本思想) 364
8.3.3 Runge-Kutta method of order two (二階 Runge-Kutta方法) 365
8.3.4 Runge-Kutta method of order three (三階 Runge-Kutta方法) 366
8.3.5 Runge-Kutta method of order four (四階 Runge-Kutta方法) 366
8.4 Multistep methods (多步方法 ) 367
8.4.1 Adams explicit schemes (Adams顯式格式) 367
8.4.2 Adams implicit schemes (Adams隱式格式) 370
8.5本章算法程序及實(shí)例 (Algorithms and examples) 371
8.5.1 Euler方法 (Euler’s scheme) 371
8.5.2 隱式 Euler方法 (Euler implicit scheme) 373
8.5.3 梯形方法 (Trapezoidal scheme) 374
8.5.4 改進(jìn)的 Euler公式 (Modified Euler’s scheme) 375
8.5.5 二階龍格-庫(kù)塔方法 (Runge-Kutta order two) 377
8.5.6 三階龍格-庫(kù)塔方法 (Runge-Kutta order three) 379
8.5.7 四階龍格-庫(kù)塔方法 (Runge-Kutta order four) 380
8.6本章要點(diǎn) (Hightlights) 382
8.7問(wèn)題討論 (Questions for discussion) 383
8.8關(guān)鍵術(shù)語(yǔ) (Key terms) 384
8.9延伸閱讀 (Extending reading) 385
8.10習(xí)題 (Exercises) 391
References (參考文獻(xiàn)) 396