Theoretical explanation for the algorithm’s performance on the matrix sign function

Establish a rigorous theoretical explanation—such as convergence guarantees or error bounds—for the observed good numerical performance of the telescopic decomposition–based rational Krylov subspace method when computing the matrix sign function of HSS matrices, despite the function’s discontinuity and the inapplicability of the paper’s analytic-function error analysis.

Background

The paper’s error analysis relies on rational approximation bounds for analytic functions on intervals containing the spectrum of A. The sign function is discontinuous and thus not covered by this analysis; nevertheless, numerical experiments show favorable performance of the algorithm for sign(A) in practical settings.

The authors explicitly point out the lack of theoretical justification for these results and identify providing such an explanation as an open problem. Addressing this would involve characterizing the behavior of the telescopic decomposition and rational Krylov updates under non-analytic functions and potentially leveraging structure in the spectrum (e.g., gaps around zero).