数值分析_线性方程组迭代解法Hilbert矩阵(2)

1.005405 1.00207 1.001696 1.001447 1.0013497

1.003674 1.000894 1.000797 1.000762 1.0007839

0.999295 0.999034 0.999305 0.999535 0.9996135

0.992495 0.996642 0.997335 0.997852 0.9980307

相对误差 0.003614 0.001837 0.001438 0.001195 0.0015527

迭代步数 13292 2640 5455 7486 8372

1.00003 1.000076 1.000054 1.000024 1.0000038

1.000042 0.999079 0.99939 0.99987 1.0002181

0.997591 1.001761 1.000907 0.999243 0.9977531

15 x 1.004931 1.000961 1.001251 1.00287 1.0052607

1.002346 0.998896 0.999369 0.999369 0.9976825

0.997049 0.997963 0.998249 0.998244 0.9991342

0.9947 0.998424 0.998491 0.998406 0.9978538

0.996027 0.999638 0.999519 0.999396 0.999637

0.999425 1.000916 1.000665 1.000488 1.0002834

1.002972 1.001797 1.001488 1.001291 1.0013136

1.005237 1.002061 1.001775 1.001605 1.0015163

1.005425 1.001653 1.00147 1.001373 1.0013724

1.003242 1.000616 1.000601 1.000615 1.0006151

0.998732 0.999041 0.999238 0.999389 0.9994484

0.992121 0.997031 0.997467 0.99777 0.9978658

相对误差 0.003902 0.001506 0.001267 0.001349 0.0019123

迭代步数 12570 4206 5922 7032 7522

2. 共轭梯度法分析

令初始向量x0=u(1,1,……)T，给定不同的u值，计算不同情况下解的情况。计算结果如下。

n 项目 u=0 u=0.5 u=0.9 u=1.2 u=1.5

1.000000 1.000000 1.000000 1.000000 1.000000

x

1.000000 1.000000 1.000000 1.000000 1.000000

2

相对误差 2.32E-15

迭代步数 1 1.20E-15 2.83E-16 1.84E-15 1 1 1 1.10E-15 1

1.000000 1.000000 1.000000 1.000000 1.000000

x

3

1.000000 1.000000 1.000000 1.000000 1.000000

相对误差 9.50E-15 4.68E-15 2.67E-14 8.79E-14 4.31E-13 1.000000 1.000000 1.000000 1.000000 1.000000

迭代步数 2 2 2 3 3

1.000000 1.000000 1.000000 1.000000 1.000000

1.000000 1.000000 1.000000 1.000000 1.000000

x

1.000000 1.000000 1.000000 1.000000 1.000000

4

1.000000 1.000000 1.000000 1.000000 1.000000

相对误差 3.45E-09

迭代步数 4 1.73E-09 6.42E-10 5.39E-10 4 3 3 9.88E-10 4

0.999971 0.999986 0.999997 1.000006 1.000014

1.000542 1.000271 1.000054 0.999892 0.999729

x

5 0.997648 0.998824 0.999765 1.000468 1.001176 1.003565 1.001783 1.000357 0.999291 0.998218

0.998252 0.999126 0.999825 1.000348 1.000874

相对误差 0.0020781 0.001039 0.000208 0.000413 0.001039

迭代步数 5 5 4 5 5

1.000001 1.000001 0.999994 1.000000 0.999999

0.999967 0.999983 1.000118 1.000007 1.000017

1.000224 1.000112 0.999618 0.999955 0.999888

x

0.999418 0.999709 1.000239 1.000116 1.000291

6

1.000641 1.000321 1.000361 0.999872 0.999679

0.999747 0.999874 0.999667 1.000051 1.000126

相对误差 0.0003796 0.00019 0.000276 7.59E-05 0.0001898

迭代步数

7 x 6 6 4 6 6 1.000007 1.000004 1.000000 1.000006 1.000004

0.999916 0.999958 0.999988 0.999969 1.000018

1.000591 1.000296 1.000069 1.000022 0.999784

0.998793 0.999396 0.999876 1.000199 1.000582

1.000311 1.000156 1.000023 0.999829 0.999786

1.001243 1.000621 1.000121 0.999711 0.999357

0.999147 0.999574 0.999922 1.000277 1.000486

相对误差 0.000773 0.000386 7.70E-05 0.000181 0.0003929

迭代步数 4 4 7 4 4

1.000016 1.000008 1.000000 1.000000 1.000000

0.999728 0.999864 1.000001 0.999998 0.999996

1.001338 1.000669 0.999993 1.000013 1.000033

0.998233 0.999116 1.000020 0.999960 0.999899

x

0.999372 0.999686 0.999978 1.000044 1.000111

8

1.001385 1.000693 0.999994 1.000011 1.000028

1.001463 1.000731 1.000026 0.999949 0.999871

0.998465 0.999232 0.999988 1.000025 1.000062

相对误差 0.0012143 0.000607 1.50E-05 3.00E-05 7.49E-05

迭代步数 4 4 8 8 8

1.000028 1.000014 1.000003 0.999995 1.000000

0.999485 0.999743 0.999949 1.000104 0.999990

9 x 1.002178 1.001089 1.000218 0.999565 1.000076

0.997908 0.998954 0.999791 1.000419 0.999807

0.998337 0.999169 0.999834 1.000333 1.000121

1.000794 1.000397 1.000079 0.999842 1.000146

1.002301 1.001150 1.000230 0.999540 0.999919

1.001371 1.000686 1.000137 0.999726 0.999806

0.997581 0.998790 0.999758 1.000484 1.000135

相对误差 0.0016933 0.000847 0.000169 0.000339 0.0001256

迭代步数 4 4 4 4 9

1.000045 1.000023 1.000000 0.999991 1.000000

0.999172 0.999586 1.000000 1.000166 0.999998

1.003113 1.001556 1.000001 0.999377 1.000014

0.997807 0.998904 0.999995 1.000439 0.999982

0.997335 0.998667 1.000008 1.000533 0.999976

x

0.999815 0.999907 0.999999 1.000037 1.000040

10

1.002236 1.001118 0.999993 0.999553 1.000040

1.002857 1.001428 1.000000 0.999429 0.999969

1.000995 1.000497 1.000009 0.999801 0.999930

0.996579 0.998290 0.999995 1.000684 1.000052

相对误差 0.0021949 0.001097 5.07E-06 0.000439

迭代步数 4 4 10 4 3.59E-05 10

1.000070 1.000035 1.000007 0.999986 1.000000

0.998792 0.999396 0.999879 1.000242 1.000001

11 x 1.004111 1.002056 1.000411 0.999178 0.999987

0.997916 0.998958 0.999792 1.000417 1.000047

0.996446 0.998223 0.999645 1.000711 0.999941

0.998659 0.999329 0.999866 1.000268 0.999986

1.001608 1.000804 1.000161 0.999679 1.000050

1.003369 1.001685 1.000337 0.999326 1.000037

1.003049 1.001524 1.000305 0.999390 0.999969

1.000396 1.000198 1.000040 0.999921 0.999937

0.995503 0.997752 0.999550 1.000899 1.000045

相对误差 0.0027094 0.001355 0.000271 0.000542

迭代步数 4 4 4 4 3.91E-05 9

1.000000 1.000000 1.000000 1.000000 0.999950

0.999994 0.999997 0.999999 1.000001 1.000827

1.000051 1.000025 1.000005 0.999990 0.997426

0.999842 0.999921 0.999984 1.000032 1.000894

1.000146 1.000073 1.000015 0.999971 1.002146

1.000094 1.000047 1.000009 0.999981 1.001274

x

0.999910 0.999955 0.999991 1.000018 0.999682