When reading the following paper:
- Ed S. Coakley, Vladimir Rokhlin, A fast divide-and-conquer algorithm for computing the spectra of real symmetric tridiagonal matrices, Applied and Computational Harmonic Analysis, Volume 34, Issue 3, May 2013.
I got the impression that somehow obtaining knowledge of the eigenvalues of a tridiagonal matrix might be combined with some sort of standard numerical routines to find eigenvectors even more quickly than without knowing the eigenvalues. The purpose of the paper itself is to propose a faster algorithm for calculating eigenvalues and even the abstract says
Such an algorithm may be combined with any of the existing methods for the determination of eigenvectors of a symmetric tridiagonal matrix with known eigenvalues.
However, I have had a difficult time finding what methods are preferred or fast for determining the eigenvector for a collection of known eigenvalues. I have tested shifted Lanczos-type algorithms where I am shifting toward each known eigenvalue, but their performance seems significantly slower than simply calculating all the eigenvalues and eigenvectors of the matrix simultaneously without any knowledge of the eigenvalues.
Can anyone suggest what algorithms might best take advantage of knowledge of the eigenvalues in order to compute eigenvectors, and in particular be competitive with more standard routines? Even better, can one point to relevant calls in LAPACK or other standard libraries?
