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

Given a large square 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 satisfied. 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-Hermitian or skew-Hermitian 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 findings, with particular attention to the numerical solution of linear dynamical systems by means of structure preserving integrators.