第 1 章 MATLAB基本运算 .............................................................................................................................. 1
1.1 生成矩阵.................................................................................................................................................................... 1
1.2 加减运算.................................................................................................................................................................... 4
1.3 乘法运算.................................................................................................................................................................... 5
1.4 除法运算.................................................................................................................................................................... 6
1.5 矩阵分解运算 ............................................................................................................................................................ 7
1.6 矩阵求秩.................................................................................................................................................................... 7
1.7 复数矩阵.................................................................................................................................................................... 8
1.8 三角函数运算 .......................................................................................................................................................... 10
1.9 指数和对数运算....................................................................................................................................................... 11
1.10 常见分布随机数..................................................................................................................................................... 13
第 2 章 常用运算............................................................................................................................................ 18
2.1 多项式运算 .............................................................................................................................................................. 18
2.2 常规矩阵求逆 .......................................................................................................................................................... 23
2.3 对称正定矩阵求逆 ................................................................................................................................................... 27
2.4 托普利兹矩阵求逆 ................................................................................................................................................... 29
2.5 求一般行列式的值 ................................................................................................................................................... 33
2.6 产生随机数 .............................................................................................................................................................. 35
第 3 章 矩阵分解............................................................................................................................................ 39
3.1 对称正定矩阵的乔利斯基分解................................................................................................................................. 39
3.2 矩阵的三角分解....................................................................................................................................................... 41
3.3 一般实矩阵的 QR 分解 ............................................................................................................................................ 43
3.4 一般实矩阵的奇异值分解 ........................................................................................................................................ 47
3.5 奇异值分解法求广义逆............................................................................................................................................ 61
第 4 章 特征值与特征向量 ............................................................................................................................. 64
4.1 约化实矩阵为赫申伯格矩阵..................................................................................................................................... 64
4.2 双重步 QR 法 ........................................................................................................................................................... 66
4.3 约化对称矩阵为对称三对角阵................................................................................................................................. 73
4.4 变形 QR 法............................................................................................................................................................... 77
4.5 雅可比法.................................................................................................................................................................. 81
4.6 雅可比过关法 .......................................................................................................................................................... 87
4.7 乘幂法 ..................................................................................................................................................................... 90
10 MATLAB 数值算法手册
第 5 章 线性方程组 ........................................................................................................................................ 95
5.1 全选主元高斯消去法 ............................................................................................................................................... 95
5.2 全选主元高斯-约当消去法 ...................................................................................................................................... 99
5.3 追赶法 ....................................................................................................................................................................102
5.4 列选主元高斯消去法 ..............................................................................................................................................105
5.5 分解法 ....................................................................................................................................................................110
5.6 平方根法.................................................................................................................................................................114
5.7 列文逊法.................................................................................................................................................................117
5.8 高斯-赛德尔迭代法 ................................................................................................................................................122
5.9 共轭梯度法 .............................................................................................................................................................124
5.10 豪斯荷尔德变换法 ................................................................................................................................................126
5.11 广义逆法...............................................................................................................................................................129
5.12 病态方程组求解....................................................................................................................................................132
第 6 章 非线性方程 ...................................................................................................................................... 135
6.1 对分法 ....................................................................................................................................................................135
6.2 牛顿迭代法 .............................................................................................................................................................137
6.3 埃特金迭代法 .........................................................................................................................................................139
6.4 试位法 ....................................................................................................................................................................141
6.5 连分式法.................................................................................................................................................................142
6.6 QR 法......................................................................................................................................................................146
6.7 牛顿下山法 .............................................................................................................................................................147
第 7 章 非线性方程组................................................................................................................................... 156
7.1 梯度法 ....................................................................................................................................................................156
7.2 拟牛顿法.................................................................................................................................................................159
7.3 广义逆法.................................................................................................................................................................163
7.4 蒙特卡罗法 .............................................................................................................................................................167
第 8 章 数据插值.......................................................................................................................................... 175
8.1 拉格朗日插值 .........................................................................................................................................................175
8.2 连分式插值 .............................................................................................................................................................177
8.3 埃尔米特插值 .........................................................................................................................................................181
8.4 埃特金逐步插值......................................................................................................................................................182
8.5 光滑插值.................................................................................................................................................................185
8.6 三次样条插值 .........................................................................................................................................................191
8.7 二元插值.................................................................................................................................................................200
第 9 章 曲线拟合.......................................................................................................................................... 204
9.1 最小二乘曲线拟合 ..................................................................................................................................................204
9.2 切比雪夫曲线拟合 ..................................................................................................................................................208
9.3 里米兹法求最佳一致逼近多项式 ............................................................................................................................213
9.4 矩形域的最小二乘曲面拟合....................................................................................................................................217
目 录 11
第 10 章 数值积分........................................................................................................................................ 226
10.1 变步长梯形求积法 ................................................................................................................................................226
10.2 变步长辛卜生求积法.............................................................................................................................................228
10.3 自适应梯形求积法 ................................................................................................................................................229
10.4 龙贝格求积法........................................................................................................................................................231
10.5 连分式求积法........................................................................................................................................................233
10.6 分部求积法 ...........................................................................................................................................................237
10.7 勒让德-高斯求积法 ..............................................................................................................................................240
10.8 拉盖尔-高斯求积法 ..............................................................................................................................................242
10.9 埃尔米特-高斯求积法...........................................................................................................................................243
10.10 切比雪夫求积法 ..................................................................................................................................................244
10.11 蒙特卡罗求积法 ..................................................................................................................................................246
10.12 计算二重积分......................................................................................................................................................247
10.13 计算多重积分......................................................................................................................................................254
第 11 章 常微分方程组................................................................................................................................. 258
11.1 变步长欧拉法........................................................................................................................................................258
11.2 变步长龙格-库塔法 ..............................................................................................................................................261
11.3 变步长基尔法........................................................................................................................................................264
11.4 变步长默森法........................................................................................................................................................268
11.5 连分式法...............................................................................................................................................................272
11.6 变步长特雷纳法....................................................................................................................................................279
11.7 变步长维梯法........................................................................................................................................................284
11.8 双边法全区间积分 ................................................................................................................................................287
11.9 阿当姆斯预报校正法全区间积分 ..........................................................................................................................291
11.10 哈明法全区间积分 ..............................................................................................................................................295
11.11 吉尔法积分刚性方程组 .......................................................................................................................................298
11.12 二阶初值问题......................................................................................................................................................314
11.13 二阶边值问题......................................................................................................................................................322
第 12 章 数据分析........................................................................................................................................ 335
12.1 随机样本分析........................................................................................................................................................335
12.2 一元线性回归分析 ................................................................................................................................................340
12.3 多元线性回归分析 ................................................................................................................................................342
12.4 逐步回归分析........................................................................................................................................................346
12.5 半对数数据拟合....................................................................................................................................................357
12.6 对数数据拟合........................................................................................................................................................359
第 13 章 极值问题........................................................................................................................................ 362
13.1 一维极值连分式法 ................................................................................................................................................362
13.2 n 维极值连分式法 .................................................................................................................................................365
13.3 不等式约束线性规划问题求解 ..............................................................................................................................369
12 MATLAB 数值算法手册
13.4 单形调优法求 n 维极值 .........................................................................................................................................374
13.5 复形调优法求约束条件下的 n 维极值 ...................................................................................................................379
第 14 章 数学变换与滤波............................................................................................................................. 387
14.1 傅里叶级数逼近....................................................................................................................................................387
14.2 快速傅里叶变换....................................................................................................................................................389
14.3 快速沃尔什变换....................................................................................................................................................396
14.4 五点三次平滑........................................................................................................................................................398
14.5 卡尔曼滤波 ...........................................................................................................................................................400
14.6 α-β-γ 滤波 ............................................................................................................................................................406
第 15 章 序列排序........................................................................................................................................ 409
15.1 冒泡排序...............................................................................................................................................................409
15.2 快速排序...............................................................................................................................................................411
15.3 希尔排序...............................................................................................................................................................413
15.4 堆排序 ..................................................................................................................................................................415
第 16 章 特殊函数求值................................................................................................................................. 418
16.1 伽马函数...............................................................................................................................................................418
16.2 误差函数...............................................................................................................................................................423
16.3 贝塞尔函数 ...........................................................................................................................................................425
16.4 不完全贝塔函数....................................................................................................................................................441
16.5 概率分布函数........................................................................................................................................................444
16.6 积分函数...............................................................................................................................................................448
参考文献 ........................................................................................................................................................ 459
附录 A 内部运算符及函数一览..................................................................................................................... 460