The Iteration Methods For Solving The Tridiagonal Solution Of The Matrix Equation AXB=C

The problem of solving linear matrix equations and the corresponding least-squares problem have been a hot topic in the field of numerical algebra in recent years, and have been widely applied in many fields such as structural design, system identification, structural dynamics, automatics control theory, vibration theory.This master thesis is mainly concerned with the problem how to get the iterative tridiagonal solution and least squares tridiagonal solution of the matrix equation AXB=C and its optimal approximation by applying iterative method systematically. The problems are as.follow:problemⅠ. Given matrix A∈Rm×n, B∈Rn×p, and C∈Rm×p, find X∈TRn×n,such that AXB=C.problem II. Given matrix A∈Rm×n, B∈Rn×p, and C∈Rm×p, find X∈TRnxn, such thatproblemⅢ. Let SE denotes the solution set of problem I or problemⅡ, given X*∈Rn×n, find X∈SE such that where TRn×n denotes the set of all n×n tridiagonal matrices, and‖·‖denotes the Frobenius norm.The main works and results are as follows:1. For problemⅠ, many references have studied it and obtained its common solutions, symmetric solutions, skew-symmetric solutions and its optimal approxi-mation constrained solution, but the problems to find its tridiagonal solutions and its optimal approximation constrained solution haven't been solved. The second chapter using iterative method research for tridiagonal solution and the optimal approximate solution. The iterative method in finite steps can terminate in the absence of round-off errors, Through the iteration process can be judge the com-patibility of the matrix equation AXB=C over the tridiagonal set.2. An iterative method with short recurrence is presented to solve the least-squares problems associated with the above mentioned matrix equation. For any initial tridiagonal matrix, a least-squares solution of the matrix equation AXB=C over the tridiagonal set can be determined within finite iterative steps in the ab-sence of round-off errors. The corresponding least-squares solution with minimum norm can be also obtained by choosing a kind of special initial matrices. Moreover, ProblemⅢcan be transformed equivalently into a problem that finding the mini-mum norm least-squares solution of a new inconsistent matrix equation. we further prove that the iteration will not stop before getting the least-squares solution and the approximate solution generated by this iterative method minimizes the Frobe-nius norm of the residual sequence is strictly monotone decreasing. Finally, we give several numerical examples to verify the obtained theoretical results.3. The matrix-form LSQR method is presented for solving the least-squares problem of the inconsistent matrix equation AXB=C with tridiagonal matrix constraint without the employment of the Kronecker product. Firstly, a matrix-form bidiagonalization procedure is given to compute a set of orthonormal basis of a matrix Krylov subspace, and the solvability of the matrix equation AXB=C over tridiagonal matrices can be determined automatically by the matrix-form bidiagonalization procedure in exact arithmetic. Based on the matrix-form bidi-agonalization procedure, the least squares problem associated with the tridiagonal constrained matrix equation AXB=C reduces to a unconstrained least squares problem of linear system, which can be solved by using the classical LSQR algo-rithm. Furthermore, the preconditioned matrix-form LSQR method is adopted for solving the corresponding least squares problem. Finally, several numerical exper-iments are reported to illustrate the efficiency of the matrix-form LSQR method as well as the corresponding preconditioning methods.