Theory And Iterative Algorithms Of Some Matrix Equation Problems

The contents of this thesis are divided into two parts:the first part is concerned with the iterative algorithm researches of coupled matrix equation problems, constrained coupled matrix equation problems and associated optimal approximation problems which are included in Chapter2, Chapter3and Chapter4; the second one is devoted to the nonlinear matrix equation Xs+∑(?)A*X-tiAi=Q, see Chapter5for details. The main content is given as follows.1. Iterative algorithms for coupled matrix equation problems and associ-ated optimal approximation problemsThe optimal approximation solution to a given matrix of the coupled matrix equations can be obtained by finding the minimum Frobenius norm (least squares) solution of new coupled matrix equations. Three iterative algorithms are proposed to investigate the (least squares) solutions to the coupled matrix equations in this chapter. The first iterative algorithm is the Bi-CGSTAB algorithm, which terminates in finite iteration steps in the absence of round-off errors when the coupled matrix equations have a unique solution. The second iterative algorithm is the CG algorithm. For any initial matrix, a (least squares) solution can be obtained within finite iteration steps in the absence of round-off errors. The minimum Frobenius norm (least squares) solution can be derived when a suitable initial matrix is chosen. The third iterative algorithm is the LSQR algorithm. For a suitable initial matrix, a (least squares) solution can be obtained within finite iteration steps in the absence of round-off errors. The minimum Frobenius norm (least squares) solution can be derived when the initial matrix is zero matrix.2. Iterative algorithms for coupled matrix equation problems with single constraint and associated optimal approximation problemsOn the view of operator, more than ten kinds of common constraints on the structure of matrices (such as symmetric constraint, centrosymmetric constraint, reflexive constraint and so on) are reduced to a kind of special operator constraint. The constraint to the solution can either be the same species of constraint or the different species of constraint. The optimal approximation solution to a given matrix of the constrained coupled matrix equations can be obtained by finding the minimum Frobenius norm constrained (least squares) solution of new constrained coupled matrix equations. Two iterative algorithms are proposed to investigate the constrained (least squares) solutions to the constrained coupled matrix equations in this chapter. The first iterative algorithm is the CG algo-rithm. For any initial matrix which satisfies the single constraint, a constrained (least squares) solution can be obtained within finite iteration steps in the absence of round-off errors. The minimum Frobenius norm constrained (least squares) solution can be derived when a suitable initial matrix is chosen. The second iterative algorithm is the LSQR algo-rithm. For a suitable initial matrix, a constrained (least squares) solution can be obtained within finite iteration steps in the absence of round-off errors. The minimum Frobenius norm constrained (least squares) solution can be derived when the initial matrix is zero matrix.3. Iterative algorithms for coupled matrix equation problems with two constraints and associated optimal approximation problemsThe constraints to the solution in this chapter can either be the same species of constraints or the different species of constraints. The CG and LSQR algorithms are proposed to investigate the constrained (least squares) solutions to the constrained coupled matrix equations in this chapter. By the proposed iterative algorithms, the constrained (least squares) solutions can be obtained in finite iteration steps in the absence of round-off errors. Moreover, the associated optimal approximation problems can also be solved.4. Hermitian positive definite solutions of nonlinear matrix equation Xs+∑(?)Ai*X-ti Ai=Q The solvability and numerical algorithms for the nonlinear matrix equation Xs+∑(?)Ai*X-tiAi=Q are investigated, where m is a positive integer and s,ti>0, i=1,2,...,m. Necessary and sufficient conditions for the existence of a Hermitian positive definite solution are derived by using matrix decomposition principle. Necessary condi-tions and sufficient conditions for the existence of the Hermitian positive definite solutions are studied. Besides, iterative methods for obtaining the Hermitian positive definite so-lutions with two cases:s>1,0<ti≤1(i=1,2,..., m) and s, ti∈Z+(i=1,2,...,m) are proposed.