Question

我有以下11x11线性方程组Ax = b with：

A = {
{1.0000000000000000, 8.0000000000000000, 6.0000000000000000, 12.0000000000000000, 24.0000000000000000, 24.0000000000000000, 8.0000000000000000, 6.0000000000000000, 24.0000000000000000, 24.0000000000000000, 24.0000000000000000},
{4.5999999999999996, 41.8531411531233601, 33.0479488942856037, 87.8349057232554173, 149.3783917109033439, 195.3689938163366833, 121.0451669808013690, 48.8422484540841708, 223.6406089026404516, 851.8470736603384239, 269.3015780207464900},
{21.1599999999999966, 218.9606780479085160, 182.0278210198854936, 642.9142219510971472, 929.7459962556697519, 1590.3768227003254196, 1831.4915561762611560, 397.5942056750813549, 2083.9634145976574473, 30235.1432043200838962, 3021.8058301860087340},
{97.3359999999999701, 1145.5240206653393216, 1002.6076877338904296, 4705.8591727678940515, 5786.8317341801457587, 12946.2633183243797248, 27711.6501551604087581, 3236.5658295810949312, 19419.1186238102454809, 1073154.9275125553831458, 33907.3782725576675148},
{447.7455999999998539, 5992.9723163999815370, 5522.3546042079124163, 34444.8913989153879811, 36017.8173980603314703, 105387.4349242659372976, 419295.1650431178859435, 26346.8587310664843244, 180954.3130575636751018, 38090161.8577392920851707, 380471.2698060897528194},
{0.0000000000000000, 34.2801357124991952, 168.4702728821191613, 2101.6181209908259007, 1236.1435394200643714, 6631.0420254749351443, 38374.2674650820554234, 4069.0485156323466072, 28291.8793721561523853, 7044717.1197200166061521, 60211.4334496619121637},
{2059.6297599999993508, 31353.0895356311411888, 30417.0821226643129194, 252121.9823892920394428, 224178.4848274685500655, 857893.2134182706940919, 6344206.6583608603104949, 214473.3033545676735230, 1686197.1981563565786928, 1351958038.0734937191009521, 4269229.7229307144880295},
{0.0000000000000000, 179.3414198404317403, 927.9328280691040618, 15382.9524602928686363, 7693.8805767663707229, 53979.1670196200575447, 580627.4516345988959074, 33123.5797620395824197, 263633.8804078772664070, 250042569.2999326586723328, 675626.4184535464737564},
{0.0000000000000000, 938.2502198978935439, 5111.0461132262771571, 112596.6815912620077142, 47887.4794405465727323, 439410.6478194649680518, 8785268.3545934017747641, 269638.3520710353623144, 2456635.0642409822903574, 8874917956.1941699981689453, 7581135.8600852200761437},
{0.0000000000000000, 938.2502198978935439, 0.0000000000000000, 56298.3407956310038571, 23943.7397202732863661, 319571.3802323381532915, 8785268.3545934017747641, 0.0000000000000000, 269630.6777825467870571, 3293783983.7421655654907227, 1735440.7390556528698653},
{0.0000000000000000, 70.9608494071368625, 1546.2151390406352220, 34063.2210755480555235, 13279.8613116998949408, 129911.1650312914862297, 2657756.2850107550621033, 183537.2854802548536099, 1654054.3836708476301283, 5487391301.6329326629638672, 5049794.3807012736797333}
};


b = {1, 6.167551546217714, 39.66265463865314, 267.9960092725794, 1918.2310370808632, 137.49061855461255, 14662.396462231256, 1216.4598834815756, 11424.520672986631, 3808.17355766221, 6082.299417407878};

矩阵显然是病态的，尽管可以用mathematica找到正确的解决方案：

x = {0.0775277, 0.00771443, 0.087553, 0.0208838, 8.47931*1e-7, 0.00197285, 0.0000611365, 0.00187375, 0.000283606, 3.82771*1e-9, 0.000788588};

我现在想要在C程序中使用这个和许多其他类似的矩阵来解决系统问题。我已经尝试了几乎所有的lapack函数来解决线性方程组，特别是：

dgesv
dsgesv
dgels
dgelss
dgelsy

但他们都给出了严重错误的结果。

此时我不希望从编程的角度出现任何错误/错误，因为尝试使用条件良好的矩阵我得到了正确的结果。我想它是概念上的东西，或者我必须使用其他工具..有什么我可以做的找到从数学库中获得正确的解决方案吗？

Answer 1

解决错误编码的线性方程通常很难。至少你不能使用那些一步式LAPACK API来获得满意的数字错误的答案。

作为一个良好的开端，您可以使用截断的SVD方法来获得更加数值稳定的结果。

https://en.m.wikipedia.org/wiki/Linear_least_squares_(mathematics)

这种方法计算量最大，但如果正规方程矩阵XTX非常恶劣（即，如果其条件数乘以机器的相对舍入误差相当大），则该方法特别有用。在这种情况下，包括反演中的最小奇异值仅仅会给解决方案增加数值噪声。这可以通过截断的SVD方法来解决，通过明确地将低于某个阈值的所有奇异值设置为零并因此忽略它们来提供更稳定和精确的答案，这一过程与因子分析密切相关。

更有效的方法可能涉及通过寻找预处理基质在解决之前使基质充分调节。您需要对原始矩阵的结构有一些了解。您可以在以下讨论中找到更多想法。

https://www.researchgate.net/post/How_can_I_solve_an_ill-conditioned_linear_system_of_equations

用Lapack＆amp; co解决病态线性方程组

1 个答案: