Acceleration of inverse subspace iteration with Newton’s method

Abstract

This work is focused on the computation of the invariant subspace associated with a separated group of eigenvalues near a specified shift of a large sparse matrix. First, we consider the inverse subspace iteration with the preconditioned GMRES method. It guarantees a convergence to the desired invariant subspace but the rate of convergence is at best linear. We propose to use it as a preprocessing for a Newton scheme which necessitates, at each iteration, the solution of a Sylvester type equation for which an iterative algorithm based on the preconditioned GMRES method is specially devised. This combination results in a fast and reliable method. We discuss the implementation aspects and propose a theory of convergence. Numerical tests are given to illustrate our approach.