How accurate are eigenvalues/vectors returned by eig and eigs?

[Nick]

Does EIG or EIGS provide any guarantees as to the accuracy of the eigenvalues / eigenvectors returned? This is, how accurate is the numerical estimates of the eigenvalues/eigenvectors (I am especially concerned about the eigenvalues) to the "true" eigenvalues/vectors. I know that it will be quite accurate but I imagine it's also a function of how well conditioned the matrix is.

[Bruno Luong]

"Nick" wrote in message <kdbtrd$kfo$1...@newscl01ah.mathworks.com>...

> Does EIG or EIGS provide any guarantees as to the accuracy of the eigenvalues / eigenvectors returned?

No.

>This is, how accurate is the numerical estimates of the eigenvalues/eigenvectors (I am especially concerned about the eigenvalues) to the "true" eigenvalues/vectors. I know that it will be quite accurate but I imagine it's also a function of how well conditioned the matrix is.

Conditioning of eigen values calculation is NOT conditioning of the matrix. To make the long story short, it's some what related to the conditioning of V, the output of EIG. This is an intrinsic characteristic of eigen-value problem. On top of that, there is some error related to the algorithm itself. Direct method, such as EIG tends to be good. Iterative methods such as EIGS are notorious to be unstable, the first (corresponds to the largest) eigen vector estimated is usually OK, but things get worse for other vectors.

Bruno

[Nick]

If the matrix is (real) positive definite, then V should be orthogonal, so its condition number will be 1. Does that mean that the accuracy of all eigenvalue calculations on such matrices are (roughly) the same?

"Bruno Luong" <b.l...@fogale.findmycountry> wrote in message <kdcbi4$fl8$1...@newscl01ah.mathworks.com>...

[Bruno Luong]

"Nick" wrote in message <kds8k4$380$1...@newscl01ah.mathworks.com>...

> If the matrix is (real) positive definite, then V should be orthogonal, so its condition number will be 1. Does that mean that the accuracy of all eigenvalue calculations on such matrices are (roughly) the same?

I would said mostly yes, if the algorithm and precision of the input matrix are not an obstacle.

Bruno