The MGHSS for Solving Continuous Sylvester Equation  A X + X B = C

Yu-Ye Feng,Qing-Biao Wu,Xue-Na Jing

doi:10.1155/2021/6615728

Abstract

This paper proposes the modified generalization of the HSS (MGHSS) to solve a large and sparse continuous Sylvester equation, improving the efficiency and robustness. The analysis shows that the MGHSS converges to the unique solution of AX + XB = C unconditionally. We also propose an inexact variant of the MGHSS (IMGHSS) and prove its convergence under certain conditions. Numerical experiments verify the efficiency of the proposed methods.

Highlights

Under the above assumptions, it is sufficient to prove that equation (1) has a unique solution [1]
For large and sparse continuous Sylvester equations, iteration methods were used, such as the gradient-based algorithm [13,14,15,16,17,18]. Such an iteration method has been studied in recent years, taking advantage of the low-rank and sparsity of right-hand C in equation (1)
There are numerical research studies which focus on solving complex Sylvester matrix equation with large size, based on the HSS method for solving (1) which is proposed in [11]

Summary

Introduction

It is sufficient to prove that equation (1) has a unique solution [1]. Many authors considered such a linear matrix equation problem and concentrated on accelerating the HSS iteration on the continuous Sylvester equation (1) [7,8,9,10], which was first proposed in [11]. In the same direction of the research, this paper presents a modified GHSS method to solve the Complexity continuous Sylvester equations.

Results

Conclusion