Krylov Subspace Methods For Solving General Matrix Equations

The main goal of the present thesis is to develop some Krylov subspace methods that were originally proposed for solving the linear systems to deal with linear matrix equations.We con-sider the general matrix equations,the general discrete-time periodic matrix equations,the gen-eral coupled matrix equations,and the general coupled discrete-time periodic matrix equations which include the well-known Lyapunov,Stein,Sylvester,coupled Markovian jump Lyapunov,coupled Sylvester,and coupled second-order Sylvester matrix equations that appear in a wide range of applications in engineering,communications and scientific computations.In the first chapter of this thesis,we present a brief account of the numerous application problems where linear matrix equations arise.We also provide an overview of the major algo-rithmic developments that have taken place over the past few decades in the numerical solution of the linear matrix equations and the discrete-time periodic matrix equations.In the second chapter,by applying the Kronecker product and the vectorization operator,we develop a matrix form of the GPBiCOR method to approximate the solutions of the general ma-trix equation and the general discrete-time periodic matrix equations.We present the theoretical background of the extended GPBiCOR method and its main computational aspects.Further-more,several numerical examples of matrix equations are considered to exhibit the accuracy and efficiency of the proposed method as compared to other popular methods in the literature.In the third chapter,we generalize the GPBiCG(m,l)method to obtain the solutions of the general matrix equation and the general discrete-time periodic matrix equations.We present an overview of the GPBiCG(m,l)method,then by using the Kronecker product and the vector-ization operator we obtain two iterative algorithms to approximate the solutions of the general matrix equations.Several numerical results are stated to demonstrate the efficiency of the ex-tended method.In the fourth chapter,after a brief background of the GPBiCOR(m,l)method,we develop it to find the solutions for the general matrix equation and the general discrete-time periodic matrix equations.Some numerical experiments arising in different applications are included to verify the efficiency of the proposed method in comparison with some existing methods.In the fifth chapter,we extend the GPBiCOR method to solve the general coupled ma-trix equations and the general coupled discrete-time periodic matrix equations by means of the Kronecker product and the vectorization operator.The accuracy and efficiency of the extended GPBiCOR method assessed against some existing iterative methods are illustrated by several numerical experiments.In the sixth chapter,we develop the GPBiCG(m.,l)method to determine the solutions of the general coupled matrix equations and the general coupled discrete-time periodic matrix equation-s by using the Kronecker product and the vectorization operator.We examine the convergence property of the GPBiCG(m,l)method and investigate its effectiveness compared to some exist-ing iterative methods through some numerical experiments.Finally,we construct two iterative algorithms of the GPBiCOR(m,l)method to identify the solutions of the general coupled matrix equations and the general coupled discrete-time periodic matrix equations.Several numerical convergence results are shown to assess the performances of the proposed GPBiCOR(m,l)method,also against other popular extended Krylov subspace methods.