Minimal residual Hermitian and skew-Hermitian splitting iteration method for the continuous Sylvester equation

TL;DR

This paper introduces the MRHSS iteration method, combining minimal residual techniques with HSS iteration, to efficiently solve the continuous Sylvester equation, demonstrating improved robustness and effectiveness over existing methods.

Contribution

The paper proposes a novel MRHSS iteration method that enhances solving the continuous Sylvester equation by integrating minimal residual techniques with HSS iteration.

Findings

MRHSS method outperforms HSS in numerical tests

MRHSS serves as an effective preconditioner for Krylov methods

Numerical results confirm robustness and efficiency

Abstract

By applying the minimal residual technique to the Hermitian and skew-Hermitian (HSS) iteration scheme, we introduce a non-stationary iteration method named minimal residual Hermitian and skew-Hermitian (MRHSS) iteration method to solve the continuous Sylvester equation. Numerical results verify the effectiveness and robustness of the MRHSS iteration method versus the HSS method for the continuous Sylvester equation. Moreover, by numerical computation, we show that the MRHSS splitting can be used as a splitting preconditioner and induce accurate, robust and effective preconditioned Krylov subspace iteration methods for solving the continuous Sylvester equation.