Structure-preserving Krylov Subspace Approximations for the Matrix Exponential of Hamiltonian Matrices: A Comparative Study

Abstract: We study structure-preserving Krylov subspace methods for approximating the matrix-vector products f(H)b, where H is a large Hamiltonian matrix and f denotes either the matrix exponential or the related phi-function. Such computations are central to exponential integrators for Hamiltonian systems. Standard Krylov methods generally destroy the Hamiltonian structure under projection, motivating the use of Krylov bases with J-orthogonal columns that yield Hamiltonian projected matrices and symplectic reduced exponentials. We compare several such structure-preserving Krylov methods on representative Hamiltonian test problems, focusing on accuracy, efficiency, and structure preservation, and briefly discuss adaptive strategies for selecting the Krylov subspace dimension.