The Hamiltonian Extended Krylov Subspace Method

An algorithm for constructing a $J$-orthogonal basis of the extended Krylov subspace ${\mathcal{K}}_{r,s}=\mathrm{range}\{u,Hu,H^{2}u,$ $\dots ,$ $H^{2r-1}u,H^{-1}u,H^{-2}u,\dots ,H^{-2s}u\},$ where $H \in $\mathbb{R}$^{2n \times 2n}$isalarge(andsparse)Hamiltonianmatrixisderived(for$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.