A Contour-Integral Based Method with Schur–Rayleigh–Ritz Procedure for Generalized Eigenvalue Problems

Abstract

Recently, a class of eigensolvers based on contour integrals has been developed for computing the eigenvalues inside a given region in the complex plane. The CIRR method (a Rayleigh–Ritz type method with contour integrals) is a classic example among this kind of methods. It first constructs a subspace to contain the eigenspace of interest via a set of contour integrals, and then uses the standard Rayleigh–Ritz procedure to extract desired eigenpairs. However, it was shown that the CIRR method may fail to find the desired eigenpairs when the considered eigenproblem is non-Hermitian. This fact motivates us to develop a non-Hermitian scheme for the CIRR method. To this end, we formulate a Schur–Rayleigh–Ritz procedure to extract the desired eigenpairs. The theoretical analysis shows that our new extraction scheme can make the CIRR method also applicable for the non-Hermitian problems. Some implementation issues arising in practical applications are also studied. Numerical experiments are reported to illustrate the numerical performance of our new method.