Is there any way to reduce the error in the calculation of eigenvectors when using eig function?

In the equation AV = BVD (cond(A) = 10^4, cond(B) = 10^8), my A and B matrices are highly ill-conditioned. The eigenvectors I am getting have an error around 100, whereas the actual value of eigenvector is in the order of 10^-2. I tried [V, D] = eig(A, B, 'qz'), but this didn't make much difference. So is there any way I can specify the error tolerance while using [V, D] = eig(A, B) or is there any other way of computing eigenvectors with the required tolerance in MATLAB?

Thanks.

[Adam]

Surely the fact that your matrices are highly ill-conditioned is the source of the problem? Pre-condition them and eig should be more accurate I would have thought.

Answer 1

[Christine Tobler]

I'm just catching up on MATLAB Answers: Can you tell us some more about your problem? How are you measuring the error in the eigenvectors?

The error that EIG (and most eigensolvers) is trying to minimize is that norm(A*V - B*V*D) should be small. Is this the error you are measuring?

[Yaswanth Sai]

Yes. (A*V-B*V*D) is the error I am calculating.

[Christine Tobler]

Okay, that seems bad. Would you be able to attach these matrices here as a .mat file? It would make it easier to look into this.

[Yaswanth Sai]

I was just thinking that it makes more sense if we calculate the error as (A*V-B*V*D)/(max(abs(B*V*D), abs(A*V))). For my matrices, this expression is coming out to be 10^-12. But I still think that (A*V-B*V*D) should not be as high as 100. Please find the attachment of the .mat file "matans.mat". This mat file contains the matrices A and B.

[Christine Tobler]

It seems that the problem is not the condition of A and B so much as the norms of A and B, which are both 3.1378e+15.

For any generalized eigenvalue problem, we can multiply both A and B by 1e15, and this will not affect the eigenvalues and eigenvectors, but the residual will be a factor 1e15 larger.

A colleague suggested that a good way to go would be to scale the residual by the norms of A and B:

 >> norm(A*V - B*V*D) / max(norm(A), norm(B))
 ans =
    8.0423e-10

That this is not 1e-15, I would now blame on the condition numbers of A and B.

[Yaswanth Sai]

So is there any other way we can improve the condition of matrices and then get the eigenvectors? I tried using 'balance'([V,D]=eig(balance(A),balance(B)), but that doesn't seem to work.

Thanks.