Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations

A complex matrix P ∈ C n × n is said to be a generalized reflection if P = P H = P −1. Let P ∈ C n × n and Q ∈ C n × n be two generalized reflection matrices. A complex matrix A ∈ C n × n is called a generalized centro-symmetric with respect to ( P; Q), if A = PAQ. It is obvious that any n × n compl...

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Veröffentlicht in:	Applied mathematical modelling Jg. 35; H. 7; S. 3285 - 3300
Hauptverfasser:	Dehghan, Mehdi, Hajarian, Masoud
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Kidlington Elsevier Inc 01.07.2011 Elsevier
Schlagworte:	Algebra Exact sciences and technology Generalized coupled Sylvester matrix equations Iterative algorithms Iterative method Least Frobenius norm solution pair Linear and multilinear algebra, matrix theory Lyapunov matrix equation Mathematical analysis Mathematical models Mathematics Matrices Matrix methods Norms Optimal approximation solution pair Reflection Sciences and techniques of general use Sylvester matrix equation Least Frobenius norm solution pair Optimal approximation solution pair Iterative method Sylvester matrix equation Generalized coupled Sylvester matrix equations Lyapunov matrix equation Conjugate gradient method Operator equation Quadratic matrix Square matrix Frobenius norm Rounding error Reflection Lyapunov equation Quadratic equation Sylvester equation Symmetric matrix Optimal solution Optimal approximation Complex matrix Matrix equation Solvability
ISSN:	0307-904X
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Zusammenfassung:	A complex matrix P ∈ C n × n is said to be a generalized reflection if P = P H = P −1. Let P ∈ C n × n and Q ∈ C n × n be two generalized reflection matrices. A complex matrix A ∈ C n × n is called a generalized centro-symmetric with respect to ( P; Q), if A = PAQ. It is obvious that any n × n complex matrix is also a generalized centro-symmetric matrix with respect to ( I; I). In this work, we consider the problem of finding a simple way to compute a generalized centro-symmetric solution pair of the generalized coupled Sylvester matrix equations (GCSY) ∑ i = 1 l A i XB i + ∑ i = 1 l C i YD i = M , ∑ i = 1 l E i XF i + ∑ i = 1 l G i YH i = N , (including Sylvester and Lyapunov matrix equations as special cases) and to determine solvability of these matrix equations over generalized centro-symmetric matrices. By extending the idea of conjugate gradient (CG) method, we propose an iterative algorithm for solving the generalized coupled Sylvester matrix equations over generalized centro-symmetric matrices. With the iterative algorithm, the solvability of these matrix equations over generalized centro-symmetric matrices can be determined automatically. When the matrix equations are consistent over generalized centro-symmetric matrices, for any (special) initial generalized centro-symmetric matrix pair [ X(1), Y(1)], a generalized centro-symmetric solution pair (the least Frobenius norm generalized centro-symmetric solution pair) can be obtained within finite number of iterations in the absence of roundoff errors. Also, the optimal approximation generalized centro-symmetric solution pair to a given generalized centro-symmetric matrix pair [ X ∼ , Y ∼ ] can be derived by finding the least Frobenius norm generalized centro-symmetric solution pair of new matrix equations. Moreover, the application of the proposed method to find a generalized centro-symmetric solution to the quadratic matrix equation Q( X) = AX 2 + BX + C = 0 is highlighted. Finally, two numerical examples are presented to support the theoretical results of this paper.
Bibliographie:	ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 23
ISSN:	0307-904X
DOI:	10.1016/j.apm.2011.01.022