Parallel Krylov Solvers for the Polynomial Eigenvalue Problem in SLEPc

Abstract

Polynomial eigenvalue problems are often found in scientific computing applications. When the coefficient matrices of the polynomial are large and sparse, usually only a few eigenpairs are required and projection methods are the best choice. We focus on Krylov methods that operate on the companion linearization of the polynomial but exploit the block structure with the aim of being memory-efficient in the representation of the Krylov subspace basis. The problem may appear in the form of a low-degree polynomial (quartic or quintic, say) expressed in the monomial basis, or a high-degree polynomial (coming from interpolation of a nonlinear eigenproblem) expressed in a nonmonomial basis. We have implemented a parallel solver in SLEPc covering both cases that is able to compute exterior as well as interior eigenvalues via spectral transformation. We discuss important issues such as scaling and restart and illustrate the robustness and performance of the solver with some numerical experiments.