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 deflation (locking') and a new type of polynomial filters 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 sub-intervals independently from one another.