A restarted Induced Dimension Reduction method to approximate eigenpairs of large unsymmetric matrices

Abstract

This work presents a new algorithm to compute eigenpairs of large unsymmetric matrices. Using the Induced Dimension Reduction method (IDR(ss)), which was originally proposed for solving systems of linear equations, we obtain a Hessenberg decomposition, from which we approximate the eigenvalues and eigenvectors of a matrix. This decomposition has two main advantages. First, IDR(ss) is a short-recurrence method, which is attractive for large scale computations. Second, the IDR(ss) polynomial used to create this Hessenberg decomposition is also used as a filter to discard the unwanted eigenvalues. Additionally, we incorporate the implicitly restarting technique proposed by D.C. Sorensen, in order to approximate specific portions of the spectrum and improve the convergence.