Short synopsis: for a specific family of sparse matrices, the eigensolver seems to be unstable (kernel quitting) for certain examples, and when it works it seems to consistently return vectors which are not eigenvectors.
I would like to know if there are any known options or pre-conditioning methods which fix this, and which are feasible for large sparse matrices.
I have a family of sparse Hermitian cyclic-banded matrices $M$, and I want to calculate the smallest (absolute value) eigenvalue for each of them.
However, the kernel seems to unexpectedly quit (a problem which I have not diagnosed) during execution of Eigensystem.
In a possibly related problem, when Eigensystem executes without quitting the Kernel, it consistently returns vectors which are not eigenvectors.
An example $1220\times1220$ matrix which can be obtained using Import (not Get) can be found here. The matrix is hermitian (see In in image below) converting to a dense matrix and solving yields an eigenvector $\phi$ (In) which satisfies the eigenvector equation (Out).
Working with sparse matrices is much faster (compare $\tau$ from In and In) but is not even close to satisfying the eigenvector equation (Out).
Using Method->banded resolves the issue (Out), but the performance is horrible ($\tau$ from In is worse than the dense solve).
I believe the default method is Arnoldi, and I have been unable to fix this by playing around with the options. Arnoldi always returns a vector which does not satisfy the eigenvector equation, even if it is handed the correct solution as the sarting vector.
Is there anything I can do about this problem?