Affiliation:
1. School of Mathematical and Statistical Sciences, Clemson University, O-110 Martin Hall, Box 340975, Clemson, SC 29634, USA
Abstract
A flexible extended Krylov subspace method (F-EKSM) is considered for numerical approximation of the action of a matrix function f(A) to a vector b, where the function f is of Markov type. F-EKSM has the same framework as the extended Krylov subspace method (EKSM), replacing the zero pole in EKSM with a properly chosen fixed nonzero pole. For symmetric positive definite matrices, the optimal fixed pole is derived for F-EKSM to achieve the lowest possible upper bound on the asymptotic convergence factor, which is lower than that of EKSM. The analysis is based on properties of Faber polynomials of A and (I−A/s)−1. For large and sparse matrices that can be handled efficiently by LU factorizations, numerical experiments show that F-EKSM and a variant of RKSM based on a small number of fixed poles outperform EKSM in both storage and runtime, and usually have advantages over adaptive RKSM in runtime.
Funder
U.S. National Science Foundation
Subject
General Mathematics,Engineering (miscellaneous),Computer Science (miscellaneous)