Approximation of the matrix exponential operator by a structure-preserving block Arnoldi-type method

Abstract

The approximation of exp(A)V, where A is a real matrix and V a rectangular matrix, is the key ingredient of many exponential integrators for solving systems of ordinary differential equations. The use of Krylov subspace techniques in this context has been actively investigated; see Calledoni and Moret (1997) [10], Hochbruck and Lubich (1997) [17], Saad (1992) [20]. An appropriate structure preserving block method for approximating exp(A)V, where A is a large square real matrix and V a rectangular matrix, is given in Lopez and Simoncini (2006) [18]. A symplectic Krylov method to approximate exp(A)V was also proposed in Agoujil et al. (2012) [2] with V∈R2n×2. The purpose of this work is to describe a structure preserving block Krylov method for approximating exp(A)V when A is a Hamiltonian or skew-Hamiltonian 2n-by-2n real matrix and V is a 2n-by-2s matrix (s≪n). Our approach is based on block Krylov subspace methods that preserve Hamiltonian and skew-Hamiltonian structures.