Studies On The Iterative Algorithms For Matrix Equation Solutions Under Several Special Types Constraints

Constrained matrix equation problem is one of the hottest topics in the field of numerical algebra. It has been widely used in various subjects, such as structural design, parameter identi-fication, automatics control theory, biology, electricity, vibration theory, linear optimal control, etc. The constrained matrix equation problem is the problem to find a solution of a matrix equation from a matrices set satisfying some constraint. There are different constrained matrix equation problems and their least square solutions for different constraints or different types of matrix equations. This dissertation studies the matrix equations solution problems under differ-ent types of principle submatrices or inequality constraints, discusses the least square solutions and their optimal approximation for different types of matrices or principle submatrices con-straints, and introduces the construction problem of a kind of special matrices under spectral constraints. The main work and research results are as follows.1. Two iterative algorithms are constructed using conjugate gradient method and the idea of gradient projection to solve matrix equation AXB+CYD=E and the following system of equations under different types of principle submatrices constraints. Then the iterative idea is extended to solve the matrix equation AXB=C under any linear subspace constraint. Several numerical computations of examples demonstrate the feasibility of the proposed algorithms.2. Using the iterative idea above, several iterative algorithms are constructed to solve the least square problem and its optimal approximation for different types of matrices or principle submatrices constraints. We discuss the least square solution and its optimal approximation of ATXA=C under (I, M) symmetry constraint, and of i-1/Σ/l/AiXBi=C and i-1/Σ/lAiXiBi=C under central symmetry principle submatrices constraint where its rows and columns are inequal. Using the matrix equation AXB=C as an example, the least square solution and its optimal approximation are discussed under any linear subspace constraint. Without considering rounding errors, the iteration algorithm can obtain the solution in finite steps for any initial matrix. If a special initial matrix is selected, we also can obtain its minimal norm solution.3. This dissertation discusses the matrix equation solution of AX=B under matrix inequality CXD≥E constraint. Using the necessary and sufficient condition and general solution expression of the central symmetric solution of matrix equation AX=B, the ma-trix equation problem is equivalent to the minimal non-negative deviation problem. Then, an iterative algorithm is proposed to solve the matrix inequality minimal non-negative deviation problem. After that, the convergence of the proposed algorithm is proved using polar decompo-sition theorem and other matrix theories. Next, the necessary and sufficient condition of solution existence is given according to the results of iteration algorithm. If the solution exists, the solu-tion expression is given. Finally, several numerical computations of examples demonstrate the feasibility of the proposed algorithms.4. This dissertation studies the construction problem of Jacobi matrices under the spec-trum constraint of their principle submatrices. The necessary and sufficient condition of unique solution existence is given. Two numerical computation examples verify the conclusions.