On Hermitian and Skew hermitian Splitting Iteration Methods for Continuous Sylvester Equations

Abstract

In this paper, we propose the generalized modified Hermitian and skew-Hermitian splitting (GMHSS) approach for computing the generalized Lyapunov equation. The GMHSS iteration is convergent to the unique solution of the generalized Lyapunov equation. Moreover, we discuss the convergence analysis of the GMHSS algorithm. Further, the inexact version of the GMHSS (IGMHSS) method is formulated to improve the GMHSS method. Finally, some numerical experiments are carried out to demonstrate the effectiveness and competitiveness of the derived methods

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Authors and Affiliations

1. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Department of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan, 411105, P. R. China

   Juan Zhang & Huihui Kang

Corresponding author

Correspondence to Juan Zhang.

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Recommended by Associate Editor Le Van Hien under the direction of Editor Jessie (Ju H.) Park.

This work was supported in part by National Natural Science Foundation of China (11771370, 11771368), Natural Science Foundation of Hunan Province (2018JJ2376), and Project of Education Department of Hunan Province (18B057, 19A500).

Juan Zhang received her B.S., M.S. and Ph.D. degrees in the Department of Mathematics and Computational Science from Xiangtan University, China, in 2006, 2009, and 2013, respectively. Since 2015, she has been an associate professor in the Department of Mathematics and Computational Science at Xiangtan University. Her research interests include control theory and its application, matrix analysis and its application.

Huihui Kang received her B.S. degree in the School of Mathematics and Statistics from Xinyang University, China, in 2016. Now, she is studying for a master degree in the School of Mathematics and Computational Science at Xiangtan University. Her research interests include matrix equation and numerical computation.

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Zhang, J., Kang, H. The Generalized Modified Hermitian and Skew-Hermitian Splitting Method for the Generalized Lyapunov Equation. Int. J. Control Autom. Syst. 19, 339–349 (2021). https://doi.org/10.1007/s12555-020-0053-1

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• Received: 18 January 2020

• Revised: 23 March 2020

• Accepted: 15 April 2020

• Published: 15 September 2020

• Issue Date: January 2021

• DOI : https://doi.org/10.1007/s12555-020-0053-1

Keywords

• Bilinear systems
• controllability
• convergence
• GMHSS method
• IGMHSS method
• Hermitian and skew-Hermitian splitting
• the generalized Lyapunov equation

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