The GPBiCG($m,l$) Method for Solving General Matrix Equations
Received:January 20, 2019  Revised:March 05, 2019
Key Word: GPBiCG($m,l$) method   Krylov Subspace method   matrix equations   Kronecker product   vectorization operator  
Fund ProjectL:Supported by the National Natural Sciences Foundation of China (Grant Nos.11501079; 11571061) and in Part by the Higher Education Commission of Egypt.
Author NameAffiliation
Basemi I. Selim School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China
Mathematics and Computer Science Department, Faculty of Science, Menoufia University, Shebin El-Kom 32511, Egypt 
Lei DU School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China 
Bo YU School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China 
Xuanru ZHU School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China 
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Abstract:
      The generalized product bi-conjugate gradient (GPBiCG($m,l$)) method has been recently proposed as a hybrid variant of the GPBiCG and the BiCGSTAB methods to solve the linear system $Ax = b$ with non-symmetric coefficient matrix, and its attractive convergence behavior has been authenticated in many numerical experiments. By means of the Kronecker product and the vectorization operator, this paper aims to develop the GPBiCG($m,l$) method to solve the general matrix equation $$\sum^{p}_{i=1}{\sum^{s_{i}}_{j=1} A_{ij}X_{i}B_{ij}} = C,$$ and the general discrete-time periodic matrix equations $$\sum^{p}_{i=1}{\sum^{s_{i}}_{j=1} (A_{i,j,k}X_{i,k}B_{i,j,k}+C_{i,j, k}X_{i,k+1}D_{i,j,k})} = M_{k},~~k = 1, 2, \ldots,t,$$ which include the well-known Lyapunov, Stein, and Sylvester matrix equations that arise in a wide variety of applications in engineering, communications and scientific computations. The accuracy and efficiency of the extended GPBiCG($m,l$) method assessed against some existing iterative methods are illustrated by several numerical experiments.
Citation:
DOI:10.3770/j.issn:2095-2651.2019.04.008
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