Preserving geometric properties of the exponential matrix by block Krylov subspaces methods

Given a large square real matrix A and a rectangular tall matrix Q, many application problems require the approximation of the operation exp(A)Q. Under certain hypotheses on A, the matrix exp(A)Q preserves the orthogonality characteristics of Q; this property is particularly attractive when the associated application problem requires some geometric constraints to be satis¯ed. For small size problems numerical methods have been devised to approximate exp(A)Q while maintaining the structure properties. On the other hand, no algorithm for large A has been derived with similar preservation properties. In this paper we show that an appropriate use of the block Lanczos method allows one to obtain a structure preserving approximation to exp(A)Q when A is skew-symmetric or skew-symmetric and Hamiltonian. Moreover, for A Hamiltonian we derive a new variant of the block Lanczos method that again preserves the geometric properties of the exact scheme. Numerical results are reported to support our theoretical ¯ndings, with particular attention to the numerical solution of linear dynamical systems by means of structure preserving integrators.