Matrix GPBiCG algorithms for solving the general coupled matrix equations

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 ∑lj=1(A)i,1,jX1Bi,1,j+Ai,...

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Published in:	IET control theory & applications Vol. 9; no. 1; pp. 74 - 81
Main Author:	Hajarian, Masoud
Format:	Journal Article
Language:	English
Published:	The Institution of Engineering and Technology 02.01.2015
Subjects:	Algorithms conjugate gradient methods Control systems control theory coupled Markovian jump Lyapunov matrix equation general coupled matrix equations generalised product bi‐conjugate gradient generalised Sylvester matrix equation GPBiCG algorithm Joining Kronecker product linear matrix equations Markov processes Mathematical analysis Mathematical models matrix algebra Operators system theory Systems theory vectorisation operator conjugate gradient methods generalised Sylvester matrix equation control theory coupled Markovian jump Lyapunov matrix equation Kronecker product generalised product bi-conjugate gradient general coupled matrix equations linear matrix equations vectorisation operator matrix algebra GPBiCG algorithm system theory
ISSN:	1751-8644, 1751-8652
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Abstract	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 ∑lj=1(A)i,1,jX1Bi,1,j+Ai,2,jX2Bi,2,j+…+Ai,l,jXi,l,j) = Di 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.
AbstractList	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 ∑lj=1(A)i,1,jX1Bi,1,j+Ai,2,jX2Bi,2,j+…+Ai,l,jXi,l,j) = Di 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. 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 capital sigma super()l sub(@)j= sub(1)(A) sub()i1, sub()jX sub(1) B sub()i sub(1), sub()j+ A sub()i sub(2,)jXd2B sub()i sub(2,)j+...+A sub()i,l,jXdi,l,j = D sub()ifor 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. 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 , j X 1 B i , 1, j + A i , 2, j X 2 B i, 2, j +…+ A i,l,j X 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. 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, j X1 Bi, 1,j +Ai, 2,j X2 Bi,2,j +…+Ai,l,j Xi,l,j) = Di 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.
Author	Hajarian, Masoud
Author_xml	– sequence: 1 givenname: Masoud surname: Hajarian fullname: Hajarian, Masoud email: m_hajarian@sbu.ac.ir organization: Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, General Campus, Evin, Tehran 19839, Iran
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Copyright	The Institution of Engineering and Technology 2021 The Authors. IET Control Theory & Applications published by John Wiley & Sons, Ltd. on behalf of The Institution of Engineering and Technology
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Keywords	conjugate gradient methods generalised Sylvester matrix equation control theory coupled Markovian jump Lyapunov matrix equation Kronecker product generalised product bi-conjugate gradient general coupled matrix equations linear matrix equations vectorisation operator matrix algebra GPBiCG algorithm system theory
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Snippet	Linear matrix equations have important applications in control and system theory. In the study, we apply Kronecker product and vectorisation operator to extend...
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SubjectTerms	Algorithms conjugate gradient methods Control systems control theory coupled Markovian jump Lyapunov matrix equation general coupled matrix equations generalised product bi‐conjugate gradient generalised Sylvester matrix equation GPBiCG algorithm Joining Kronecker product linear matrix equations Markov processes Mathematical analysis Mathematical models matrix algebra Operators system theory Systems theory vectorisation operator
Title	Matrix GPBiCG algorithms for solving the general coupled matrix equations
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