A thick-restart lanczos algorithm with polynomial filtering for hermitian eigenvalue problems

Polynomial filtering can provide a highly effective means of computing all eigenvalues of a real symmetric (or complex Hermitian) matrix that are located in a given interval, anywhere in the spectrum. This paper describes a technique for tackling this problem by combining a thick- restart version of the Lanczos algorithm with deation (\locking") and a new type of polynomial filter obtained from a least-squares technique. The resulting algorithm can be utilized in a \spectrum- slicing" approach whereby a very large number of eigenvalues and associated eigenvectors of the matrix are computed by extracting eigenpairs located in different subintervals independently from one another.