Recursion relations for the extended Krylov subspace method

The evaluation of matrix functions of the form f (A)v, where A is a large sparse or structured symmetric matrix, f is a nonlinear function, and v is a vector, is frequently subdivided into two steps: first an orthonormal basis of an extended Krylov subspace of fairly small dimension is determined, and then a projection onto this subspace is evaluated by a method designed for small problems. This paper derives short recursion relations for orthonormal bases of extended Krylov subspaces of the type K-m,K-mi+1 (A) = span {A(-m+1) v,...,A(-1) v, v, A v, ... ,A(mi) v}, m = 1,2,3,.., with i a positive integer, and describes applications to the evaluation of matrix functions and the computation of rational Gauss quadrature rules. (C) 2010 Elsevier Inc. All rights reserved.