Matrix GPBiCG algorithms for solving the general coupled matrix equations

Masoud	Hajarian

Matrix GPBiCG algorithms for solving the general coupled matrix equations

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Author(s): Masoud Hajarian ¹
- Affiliations: 1: Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, General Campus, Evin, Tehran 19839, Iran
Source: Volume 9, Issue 1, 02 January 2015, p. 74 – 81
DOI: 10.1049/iet-cta.2014.0669 , Print ISSN 1751-8644, Online ISSN 1751-8652

Received 11/10/2013, Accepted 16/08/2014, Published 09/10/2014

Linear matrix equations have important applications in control and system theory. In the study, we apply Kronecker product and vectorisation operator to extend the generalised product bi-conjugate gradient (GPBiCG) algorithms for solving the general coupled matrix equations ∑ ^l _j=1(A)_i,1, _jX ₁ B_i ,_1,j+A_i ,_2,j X ₂ B_i, _2,j+…+A_i,l,jX_i,l,j ) = D_i for i = 1,2,…,l (including the (coupled) Sylvester, the second-order Sylvester and coupled Markovian jump Lyapunov matrix equations). We propose four effective matrix algorithms for finding solutions of the matrix equations. Numerical examples and comparison with other well-known algorithms demonstrate the effectiveness of the proposed matrix algorithms.

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