第1章 插值與逼近.......................................................1
1.1 問題介紹..........................................................1
1.2 多項式插值.......................................................2
1.2.1 概述.......................................................2
1.2.2 Lagrange插值..............................................4
1.2.3 Newton插值...............................................6
1.2.4 分片線性插值..............................................8
1.2.5 Hermite插值..............................................10
1.3 徑向基函數(shù)插值..................................................13
1.3.1 概述......................................................13
1.3.2 再生核空間...............................................16
1.3.3 誤差估計..................................................18
1.4 最佳逼近.........................................................20
1.4.1 最小二乘擬合.............................................20
1.4.2 最佳一致逼近.............................................22
1.4.3 最佳平方逼近.............................................23
1.4.4 正交多項式...............................................24
1.5 注記.............................................................26
習(xí)題1................................................................27
第2章 數(shù)值微分與數(shù)值積分.............................................31
2.1 問題介紹.........................................................31
2.2 數(shù)值微分.........................................................31
2.2.1 Taylor展開求導(dǎo)...........................................31
2.2.2 插值型求導(dǎo)...............................................33
2.3 數(shù)值積分.........................................................35
2.3.1 中點(diǎn)、梯形和Simpson求積公式..........................35
2.3.2 Newton-Cotes求積公式...................................37
2.3.3 復(fù)合求積公式.............................................39
2.3.4 Romberg求積公式........................................40
2.3.5 Gauss求積公式...........................................41
2.4 注記.............................................................45
習(xí)題2................................................................46
第3章 求解線性方程組..................................................49
3.1 問題介紹.........................................................49
3.2 直接法...........................................................50
3.2.1 LU分解..................................................50
3.2.2 Cholesky分解.............................................52
3.2.3 QR分解..................................................53
3.3 基本迭代法......................................................56
3.3.1 三種基本迭代法...........................................56
3.3.2 收斂性準(zhǔn)則...............................................61
3.4 共軛梯度法......................................................62
3.5 注記.............................................................66
習(xí)題3................................................................66
第4章 求解非線性方程組...............................................70
4.1 問題介紹.........................................................70
4.2 非線性方程的迭代法.............................................70
4.2.1 二分法....................................................71
4.2.2 不動點(diǎn)迭代...............................................72
4.2.3 Newton迭代..............................................74
4.2.4 割線法....................................................75
4.3 非線性方程組的迭代法...........................................78
4.3.1 基本非線性迭代法.........................................78
4.3.2 Newton迭代法............................................80
4.3.3 Broyden算法.............................................81
4.4 注記.............................................................83
習(xí)題4................................................................84
第5章 矩陣特征值計算..................................................86
5.1 問題介紹.........................................................86
5.2 冪方法...........................................................87
5.2.1 乘冪法....................................................87
5.2.2 反冪法....................................................88
5.3 QR迭代.........................................................90
5.4 Rayleigh商迭代..................................................92
5.5 注記.............................................................94
習(xí)題5................................................................94
第6章 常微分方程數(shù)值方法.............................................96
6.1 歐拉方法.........................................................97
6.2 Runge-Kutta方法...............................................100
6.2.1 方法介紹.................................................100
6.2.2 常用的Runge-Kutta方法................................101
6.3 線性多步法.....................................................105
6.4 注記............................................................106
習(xí)題6...............................................................107
第7章 有限差分方法...................................................109
7.1 偏微分方程及其分類............................................110
7.2 拋物型方程有限差分方法........................................112
7.2.1 1-D拋物型方程離散.....................................112
7.2.2 穩(wěn)定性、相容性和收斂性.................................114
7.2.3 2-D拋物型方程離散.....................................117
7.2.4 ADI格式................................................118
7.3 雙曲型方程有限差分方法........................................120
7.3.1 基本差分方法............................................120
7.3.2 守恒律...................................................122
7.3.3 二階雙曲型方程..........................................123
7.4 橢圓型方程有限差分方法........................................126
7.4.1 基本差分方法............................................126
7.4.2 其他應(yīng)用.................................................127
7.5 注記............................................................129
習(xí)題7...............................................................129
第8章 有限元方法.....................................................132
8.1 一維橢圓型方程離散............................................132
8.2 二維橢圓型方程離散............................................135
8.3 有限元收斂理論.................................................137
8.3.1 變分問題解的存在性.....................................137
8.3.2 Sobolev空間.............................................138
8.3.3 有限元插值理論..........................................140
8.3.4 誤差估計.................................................142
8.4 一些常見有限元.................................................143
8.4.1 P1,P2有限元............................................143
8.4.2 Q1,Q2有限元............................................145
8.4.3 其他有限元..............................................147
8.5 注記............................................................149
習(xí)題8...............................................................149
第9章 無網(wǎng)格方法.....................................................152
9.1 Kansa方法.....................................................152
9.2 對稱配點(diǎn)方法...................................................153
9.3 Galerkin配點(diǎn)方法..............................................154
9.4 多尺度配點(diǎn)方法.................................................155
9.5 基本解方法.....................................................157
9.5.1 PDEs的基本解..........................................157
9.5.2 齊次方程的求解..........................................158
9.5.3 非齊次方程的求解.......................................160
9.6 注記............................................................161
習(xí)題9...............................................................161
參考文獻(xiàn).................................................................165