Title: An improved Krylov eigenvalue strategy using the FEAST algorithm with inexact system solves,
Abstract: The FEAST eigenvalue algorithm is a subspace iteration algorithm that uses
contour integration in the complex plane to obtain the eigenvectors of a matrix
for the eigenvalues that are located in any user-defined search interval. By
computing small numbers of eigenvalues in specific regions of the complex
plane, FEAST is able to naturally parallelize the solution of eigenvalue
problems by solving for multiple eigenpairs simultaneously. The traditional
FEAST algorithm is implemented by directly solving collections of shifted
linear systems of equations; in this paper, we describe a variation of the
FEAST algorithm that uses iterative Krylov subspace algorithms for solving the
shifted linear systems inexactly. We show that this iterative FEAST algorithm
(which we call IFEAST) is mathematically equivalent to a block Krylov subspace
method for solving eigenvalue problems. By using Krylov subspaces indirectly
through solving shifted linear systems, rather than directly for projecting the
eigenvalue problem, IFEAST is able to solve eigenvalue problems using very
large dimension Krylov subspaces, without ever having to store a basis for
those subspaces. IFEAST thus combines the flexibility and power of Krylov
methods, requiring only matrix-vector multiplication for solving eigenvalue
problems, with the natural parallelism of the traditional FEAST algorithm. We
discuss the relationship between IFEAST and more traditional Krylov methods,
and provide numerical examples illustrating its behavior.