On circulant and skew-circulant splitting algorithms for (continuous) Sylvester equations

TL;DR

This paper introduces a new iterative method called CSCS for efficiently solving large sparse Sylvester equations with Toeplitz matrices, providing convergence guarantees and computational advantages.

Contribution

The paper proposes the CSCS iterative method for Sylvester equations with Toeplitz matrices, including convergence analysis and efficiency comparison.

Findings

Convergence of CSCS depends on positive semi-definiteness of splitting factors.

Upper bound for the convergence factor is derived.

CSCS outperforms alternative methods in computational tests.

Abstract

We present a circulant and skew-circulant splitting (CSCS) iterative method for solving large sparse continuous Sylvester equations $AX + XB = C$, where the coefficient matrices $A$ and $B$ are Toeplitz matrices. A theoretical study shows that if the circulant and skew-circulant splitting factors of $A$ and $B$ are positive semi-definite and at least one is positive definite (not necessarily Hermitian), then the CSCS method converges to the unique solution of the Sylvester equation. In addition, we obtain an upper bound for the convergence factor of the CSCS iteration. This convergence factor depends only on the eigenvalues of the circulant and skew-circulant splitting matrices. A computational comparison with alternative methods reveals the efficiency and reliability of the proposed method.