A class of multiplicative splitting iterations for solving the continuous Sylvester equation

TL;DR

This paper introduces a new class of multiplicative splitting iterative methods for efficiently solving the continuous Sylvester equation, with convergence analysis and numerical validation.

Contribution

It proposes a novel multiplicative splitting iteration framework using positive definite and Hermitian-splitting techniques for the Sylvester equation.

Findings

Convergence conditions are established for the proposed method.

Numerical experiments demonstrate the method's efficiency.

The approach is effective for matrices with positive semi-definite properties.

Abstract

For solving the continuous Sylvester equation, a class of the multiplicative splitting iteration method is presented. We consider two symmetric positive definite splittings for each coefficient matrix of the continuous Sylvester equations and it can be equivalently written as two multiplicative splitting matrix equations. When both coefficient matrices in the continuous Sylvester equation are (non-symmetric) positive semi-definite, and at least one of them is positive definite; we can choose Hermitian and skew-Hermitian (HS) splittings of matrices $A$ and $B$, in the first equation, and the splitting of the Jacobi iterations for matrices $A$ and $B$, in the second equation in the multiplicative splitting iteration method. Convergence conditions of this method are studied and numerical experiments show the efficiency of this method.