Using a low-storage contour-integral eigensolver for nonsymmetric matrices with collocation to compute vibrational spectra

Abstract

Collocation is advantageous for computing vibrational spectra because it obviates the need for quadrature. However, to use collocation, one must solve a non-Hermitian matrix eigenvalue problem. Simple, accurate Lanczos algorithms that require storing only a few vectors can be used only for symmetric matrices. Established eigensolvers for nonsymmetric problems require storing (and orthogonalizing) many vectors. For large matrices, they require a lot of memory. We propose a low-storage eigensolver combining the block Sakurai–Sugiura method and the multi-shift quasi-minimal residual linear solver. It requires storing only a few vectors. The method has many of the advantages of the well-known Cullum–Willoughby Lanczos approach for symmetric matrices.