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 Name  Affiliation  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 ElKom 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 biconjugate 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 nonsymmetric 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 discretetime 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 wellknown 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:20952651.2019.04.008 
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