Very bad conditional linear system
I have a linear system for the solution, written as Ax = b A - a symmetric 175 by 175 square with a diagonal on it (i.e. Aii = 1) and the other entries range from 0 to 1 (i.e. 0
A is very ill-conditioned, not positive definite, its rank is 162 and its condition number is 3.5869e + 16
I spent several days to solve this problem in MATLAB, I tried almost every method I found, including \, pcg, bicg, bicgstab, bicgstabl, cgs, gmres, lsqr, minres, qmr, symmlq, tfqmr These methods gave me some solutions ... But I don't know how to trust them, or what solution to trust. Are there criteria for determining?
I will appreciate someone who can give me a solution that I can trust. Thank!
A and b are stored in .mat files and can be downloaded from Dropbox links:
https://www.dropbox.com/s/s6xlbq68juqs6xi/A.mat?dl=0
https://www.dropbox.com/s/pxl0hdup20hf2lr/b.mat?dl=0
use the following:
loads ('A.mat');
loads ('b.mat');
x = A \ b;
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Not sure if this helps, but let it go:
Basically, when, due to poor condition, it is difficult to calculate the following:
Instead, you will reduce
Being \ Gamma is usually the identity matrix.
You end up with the following equation for x:
To add to that, you usually want to add a "hyperparameter" to control how tightly you adjust the problem. So, \ Gamma instead of being just the identity matrix, it will be the number (i.e. 0.001) multiplied by the identity matrix size(A)
.
The last equation should be simple in Matlab. Let him go.
NOTE: this is not an answer. In fact, there is probably no clear answer to a poorly posed problem. It's just the way to go.
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