The objective of this paper is to provide an algorithm for efficient estimation of the exp(A)V operator applied to particular types of matrices, namely a large, square, Hamiltonian matrix A and a rectangular matrix V. The numerical approximation of exp(A)V is met in many engineering applications. It requires differential calculus and needs efficient time and resource-saving techniques. The approximation is developed on the block Krylov subspace by applying a modified block symplectic Arnoldi algorithm for square matrices, which generates a symplectic basis for the block Krylov subspace; during this projection process, it preserves certain structural features of the input matrix. Structure-preserving techniques are able to simplify and make more efficient the computation process.

The paper starts by introducing the main concepts used throughout the paper, among them a matrix normalization technique implemented by the Arnoldi algorithm, presented in section 4. Sections 5 and 6 derive an exponential approximation for certain classes of Hamiltonian matrices, which preserves orthogonality or symplecticity properties. The numerical examples considered in section 7 investigate the error obtained by the proposed approximation algorithm and are meant to demonstrate the efficacy of the new approach. The involved matrices are generated in a similar manner as those used in previous research. The work addresses a quite particular field of research and is very thoroughly and unambiguously exposed.