改善条件恶劣的矩阵 [英] Improving a badly conditioned matrix

问题描述

我有一个条件恶劣的矩阵，其 rcond() 是接近于零，因此，该矩阵的逆并不正确.我尝试使用 pinv() ，但这不能解决问题.这就是我采取的相反方法:

I have a badly conditioned matrix, whose rcond() is close to zero, and therefore, the inverse of that matrix does not come out to be correct. I have tried using pinv() but that does not solve the problem. This is how I am taking the inverse:

X = (A)\(b);

我正在寻找针对此问题的解决方案，并找到了此链接(最后一种解决方案)以改善矩阵.那里的解决方案建议使用此方法:

I looked up for a solution to this problem and found this link (last solution) for improving the matrix. The solution there suggests to use this:

A_new = A_old + c*eye(size(A_old));

c > 0所在的位置.到目前为止，采用该技术可以使矩阵A的条件更好，并且所得的结果看起来更好.但是，我使用了不同的c值进行了调查，得出的结果取决于所选c的值.

Where c > 0. So far employing this technique works in making the matrix A better conditioned and the resultant solution looks better. However, I investigated using different values of c and the resultant solution depends on the value of chosen c.

除了手动调查c的值外，还有没有一种自动的方法可以找到最佳解决方案的c值?

Other than manually investigating for the value of c, is there an automatic way through which I can find the value of c for which I get the best solution?

推荐答案

在离散逆理论中，将一个较小的值c添加到要反转的矩阵A的对角线上，这称为阻尼反转，而要添加的小值c被称为Marquardt-Levenberg系数.有时矩阵A的特征值为零或接近零，因此矩阵变得奇异.向对角线元素添加较小的阻尼系数使其稳定. c的值越大，阻尼的值越大，您的矩阵求反更加稳定，但您离真正的解决方案还很远. c的值较小，阻尼的值较小，您的倒排矩阵与真正的倒排矩阵更接近，但可能会变得不稳定. 有时使用的一种自适应阻尼"技术是-从测试值c开始，对矩阵A求逆，然后减小c的值，再次进行求反，依此类推.当A再次变为奇数时，当您在倒数矩阵中获得怪异的值时停止，例如真正的大数字. 我认为这不能完全回答您的问题，但是将其放入评论中太久了.

In discrete inverse theory, adding a small value c to the diagonal of the matrix A about to be inverted, is called damping the inversion and the small value to be added c is called Marquardt-Levenberg coefficient. Sometimes matrix A has zero or close to zero eigenvalues, due to which the matrix becomes singular; adding a small damping coefficient to the diagonal elements makes it stable. Bigger is the value of c, bigger is the damping, your matrix inversion is more stable, but you are further away from true solution. Smaller is the value of c, smaller is the damping, closer is your inverted matrix to true inverted matrix but it might become unstable. One 'adaptive damping' technique sometimes used is - start with a test value of c, invert the matrix A, then decrease the value of c, do the inversion again and so on. stop when you get weird values in inverted matrix due to A becoming singular again, like really large numbers. I think this doesn't exactly answer your question, but it was too long to put it in a comment.

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