An efficient block rational Krylov solver for Sylvester equations with adaptive pole selection

TL;DR

This paper introduces an efficient block rational Krylov method for solving Sylvester equations, featuring adaptive pole selection and improved residual evaluation, extending convergence analysis to the block case.

Contribution

The paper's novelty lies in maintaining the last pole at infinity via pole reordering and extending convergence analysis to the block case, enabling adaptive pole strategies.

Findings

Efficient residual evaluation at each iteration.

Extended convergence analysis for block rational Krylov methods.

Effective adaptive pole selection strategies demonstrated.

Abstract

We present an algorithm for the solution of Sylvester equations with right-hand side of low rank. The method is based on projection onto a block rational Krylov subspace, with two key contributions with respect to the state-of-the-art. First, we show how to maintain the last pole equal to infinity throughout the iteration, by means of pole reodering. This allows for a cheap evaluation of the true residual at every step. Second, we extend the convergence analysis in [Beckermann B., An error analysis for rational Galerkin projection applied to the Sylvester equation, SINUM, 2011] to the block case. This extension allows to link the convergence with the problem of minimizing the norm of a small rational matrix over the spectra or field-of-values of the involved matrices. This is in contrast with the non-block case, where the minimum problem is scalar, instead of matrix-valued. Replacing the norm of the objective function with an easier to evaluate function yields several adaptive pole selection strategies, providing a theoretical analysis for known heuristics, as well as effective novel techniques.