New Krylov Methods: Theory and Applications

University Of Maryland, College Park, College Park MD

Investigators

Abstract

ABSTRACT 0204084 Gilbert Stewart University of Maryland College Park Krylov sequence methods are widely used in the solution of large eigenproblems (e.g., the Lanczos and Arnoldi methods) and the solution of large linear systems (e.g., the conjugate gradient method and GMRES). In their ordinary applications these methods are well understood and have a firm theoretical basis. In more exotic applications, however, the theory and practice are not as advanced. This proposal in concerned with three of these applications. Residual Krylov methods. The property of being a Krylov sequence is very sensitive to error. A single error toward the beginning of the sequence causes permanent loss of accuracy in the approximations to eigenvectors in the Krylov subspace. This has the consequence that when shift-and-invert techniques are used to enhance the spectrum of a matrix, the resulting linear systems must be solved to full accuracy at each step of the process. It turns out, however, that if the sequence is extended using the residual for a targeted eigenvector, the sequence converges to that vector even when the shift-and- invert equations are solved inaccurately. The practical consequences of this observation, which needs new theory to support it, promises to be great.