The GPBiCOR Method for Solving the General Matrix Equation and the General Discrete-Time Periodic Matrix Equations

This paper is concerned with the numerical solutions of the general matrix equation <inline-formula> <tex-math notation="LaTeX">$\sum ^{p}_{i=1}{\sum ^{s_{i}}_{j=1} }\,\,{A_{ij}X_{i}}{B_{ij}} = C$ </tex-math></inline-formula>, and the general discrete-time periodic matrix equations <inline-formula> <tex-math notation="LaTeX">$\sum ^{p}_{i=1}\sum ^{s_{i}}_{j=1} (A_{i,j,k}X_{i,k}B_{i,j,k}+ C_{i,j, k}X_{i,k+1}D_{i,j,k}) = M_{k}$ </tex-math></inline-formula>, for <inline-formula> <tex-math notation="LaTeX">$k = 1, 2, \ldots ,t$ </tex-math></inline-formula>, which include the well-known Lyapunov, Stein, and Sylvester matrix equations that appear in a wide range of applications in engineering and mechanical problems. Recently the generalized product-type BiCOR method, denoted as GPBiCOR, has been originally proposed to solve the nonsymmetric linear systems <inline-formula> <tex-math notation="LaTeX">$Ax = b$ </tex-math></inline-formula>, and its significant convergence performance has been confirmed in many numerical results. By applying the Kronecker product and the vectorization operator, we develop a matrix form of the GPBiCOR method to approximate the solutions of the above-mentioned general matrix equation and general discrete-time periodic matrix equations. We present the theoretical background of the extended GPBiCOR method and its main computational aspects. Furthermore, several numerical examples of matrix equations arising in different applications are considered to exhibit the accuracy and the efficiency of the proposed method as compared with other popular methods in the literature.

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