# SSHI methods for solving general linear matrix equations

SSHI methods for solving general linear matrix equations Purpose – The purpose of this paper is to find the efficient iterative methods for solving the general matrix equation A1X + XA2 + A3XH + XHA4 = B (including Lyapunov and Sylvester matrix equations as special cases) with the unknown complex (reflexive) matrix X . Design/methodology/approach – By applying the principle of hierarchical identification and the Hermitian/skew‐Hermitian splitting of the coefficient matrix quadruplet A1; A2; A3; A4 the authors propose a shift‐splitting hierarchical identification (SSHI) method to solve the general linear matrix equation A1X + XA2 + A3XH + XHA4 = B . Also, the proposed algorithm is extended for finding the reflexive solution to this matrix equation. Findings – The authors propose two iterative methods for finding the solution and reflexive solution of the general linear matrix equation, respectively. The proposed algorithms have a simple, neat and elegant structure. The convergence analysis of the methods is also discussed. Some numerical results are given which illustrate the power and effectiveness of the proposed algorithms. Originality/value – So far, several methods have been presented and used for solving the matrix equations by using vec operator and Kronecker product, generalized inverse, generalized singular value decomposition (GSVD) and canonical correlation decomposition (CCD) of matrices. In several cases, it is difficult to find the solutions by using matrix decomposition and generalized inverse. Also vec operator and Kronecker product enlarge the size of the matrix greatly therefore the computations are very expensive in the process of finding solutions. To overcome these complications and drawbacks, by using the hierarchical identification principle and the Hermitian=skew‐Hermitian splitting of the coefficient matrix quadruplet (A1; A2; A3; A4), the authors propose SSHI methods for solving the general matrix equation. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Engineering Computations Emerald Publishing

# SSHI methods for solving general linear matrix equations

, Volume 28 (8): 16 – Nov 15, 2011
16 pages

/lp/emerald-publishing/sshi-methods-for-solving-general-linear-matrix-equations-IXeXE0mjnU
Publisher
Emerald Publishing
ISSN
0264-4401
DOI
10.1108/02644401111179018
Publisher site
See Article on Publisher Site

### Abstract

Purpose – The purpose of this paper is to find the efficient iterative methods for solving the general matrix equation A1X + XA2 + A3XH + XHA4 = B (including Lyapunov and Sylvester matrix equations as special cases) with the unknown complex (reflexive) matrix X . Design/methodology/approach – By applying the principle of hierarchical identification and the Hermitian/skew‐Hermitian splitting of the coefficient matrix quadruplet A1; A2; A3; A4 the authors propose a shift‐splitting hierarchical identification (SSHI) method to solve the general linear matrix equation A1X + XA2 + A3XH + XHA4 = B . Also, the proposed algorithm is extended for finding the reflexive solution to this matrix equation. Findings – The authors propose two iterative methods for finding the solution and reflexive solution of the general linear matrix equation, respectively. The proposed algorithms have a simple, neat and elegant structure. The convergence analysis of the methods is also discussed. Some numerical results are given which illustrate the power and effectiveness of the proposed algorithms. Originality/value – So far, several methods have been presented and used for solving the matrix equations by using vec operator and Kronecker product, generalized inverse, generalized singular value decomposition (GSVD) and canonical correlation decomposition (CCD) of matrices. In several cases, it is difficult to find the solutions by using matrix decomposition and generalized inverse. Also vec operator and Kronecker product enlarge the size of the matrix greatly therefore the computations are very expensive in the process of finding solutions. To overcome these complications and drawbacks, by using the hierarchical identification principle and the Hermitian=skew‐Hermitian splitting of the coefficient matrix quadruplet (A1; A2; A3; A4), the authors propose SSHI methods for solving the general matrix equation.

### Journal

Engineering ComputationsEmerald Publishing

Published: Nov 15, 2011

Keywords: Control theory; Programming and algorithm theory; Iterative methods; Shift‐splitting hierarchical identification; Matrix equation; Reflexive matrix; Iterative algorithm