Extension to nonsymmetric matrices

Extend the telescopic decomposition–based rational Krylov subspace algorithm for computing matrix functions f(A) from symmetric hierarchically semiseparable (HSS) matrices to nonsymmetric HSS matrices, and develop the corresponding methodology so that functions f(A) can be approximated for nonsymmetric inputs within the same hierarchical framework.

Background

The paper introduces a new algorithm that computes telescopic decompositions for approximations of matrix functions f(A) specifically under the assumption that A is symmetric and has HSS structure. The symmetry assumption is used to simplify the telescopic factors (U_tau = V_tau) and to enable a convergence analysis (Theorem thm:error_bound) for analytic functions on intervals covering the spectrum.

The authors explicitly note that extending their framework beyond symmetry is not addressed and identify it as an open direction. Achieving such an extension would broaden the applicability of the method to a wider class of hierarchical low-rank matrices and would require adapting both the telescopic decomposition framework and the associated rational Krylov updates to the nonsymmetric case.