The Hamiltonian Extended Krylov Subspace Method

TL;DR

This paper introduces the Hamiltonian Extended Krylov Subspace (HEKS) method, an efficient algorithm for approximating functions of large Hamiltonian matrices while preserving their structure, with promising numerical results.

Contribution

The paper develops a novel HEKS algorithm that constructs a structured basis with short recurrences for Hamiltonian matrices, enabling efficient approximation of matrix functions.

Findings

HEKS achieves short recurrences involving at most five basis vectors.

The method effectively approximates functions of Hamiltonian matrices.

Numerical experiments show competitive performance with existing structure-preserving methods.

Abstract

An algorithm for constructing a $J$-orthogonal basis of the extended Krylov subspace $\mathcal{K}_{r,s}=\operatorname{range}\{u,Hu, H^2u,$ $ \ldots, $ $H^{2r-1}u, H^{-1}u, H^{-2}u, \ldots, H^{-2s}u\},$ where $H \in \mathbb{R}^{2n \times 2n}$ is a large (and sparse) Hamiltonian matrix is derived (for $r = s+1$ or $r=s$). Surprisingly, this allows for short recurrences involving at most five previously generated basis vectors. Projecting $H$ onto the subspace $\mathcal{K}_{r,s}$ yields a small Hamiltonian matrix. The resulting HEKS algorithm may be used in order to approximate $f(H)u$ where $f$ is a function which maps the Hamiltonian matrix $H$ to, e.g., a (skew-)Hamiltonian or symplectic matrix. Numerical experiments illustrate that approximating $f(H)u$ with the HEKS algorithm is competitive for some functions compared to the use of other (structure-preserving) Krylov subspace methods.