Preconditioned MHSS iterative algorithm and its accelerated method for solving complex Sylvester matrix equations

TL;DR

This paper develops a preconditioned iterative algorithm for solving large sparse complex Sylvester matrix equations, introducing an acceleration technique that improves convergence speed and robustness, validated through numerical experiments.

Contribution

It proposes a novel preconditioned PMHSS iterative algorithm and an accelerated version with theoretical convergence analysis and numerical validation.

Findings

Accelerated PMHSS converges faster than standard PMHSS.

The algorithms are robust and efficient for large sparse complex Sylvester equations.

Numerical experiments confirm improved performance and convergence speed.

Abstract

This paper introduces and analyzes a preconditioned modified of the Hermitian and skew-Hermitian splitting (PMHSS). The large sparse continuous Sylvester equations are solved by PMHSS iterative algorithm based on nonHermitian, complex, positive definite/semidefinite, and symmetric matrices. We prove that the PMHSS is converged under suitable conditions. In addition, we propose an accelerated PMHSS method consisting of two preconditioned matrices and two iteration parameters {\alpha}, \b{eta}. Theoretical analysis showed that the convergence speed of the accelerated PMHSS is faster compared to the PMHSS. Also, the robustness and efficiency of the proposed two iterative algorithms were demonstrated in numerical experiments.