Iterative Methods Of Preconditioning For Constrained Matrix Equation

The constraint matrix equation problems have been widely used in structural design, the modifying of dynamic models, vibration theory, and so on, the research about constraint matrix equation problems have become the most popular subject of computational mathematics, and there have been many achievements about constraint matrix equation problems, but the literature about the acceleration technology or the preconditioning technology of matrix equations is less.The preconditioning iteration methods of the following matrix equations will be studied in the M. S thesis:ProblemⅠGiven A, B∈Rm×n, find X∈Rn×n, such that AX＝B.ProblemⅡGiven A∈Rm×n, B∈Rp×q, C∈Rm×q, find X∈Rn×p, such that AXB=C.ProblemⅢGiven A, B∈Rm×n, find X∈SRn×n, such that ATXA=B.The polynomial preconditioning technology is systematic used to accelerate the orthogonal projection iteration methods of the above three kinds of matrix equations and theirs optimal approximation.The achievements are as follows:1.The general solution for the matrix equation AX=B. According to the ideas of polynomial preconditioning of linear equations, the preconditioning polynomial is obtained by using singular values and interpolation methods, then the preconditioning orthogonal projection iteration method is put forward by combining the preconditioning polynomial and the orthogonal projection iteration method, and the estimation of convergence rate is given. Numerical results show that the convergence rate of new method under certain conditions is faster than the orthogonal projection iteration method.2.The general solution for the matrix equation AXB=C. Similarly, the preconditioning polynomial can be obtained by using singulars value and interpolation methods, then the matrix equation is preconditioned with preconditioning polynomial, the convergence rate of estimation is given. Numerical results show that the convergence rate of new method under certain conditions is faster than the orthogonal projection iteration method.3. According to the polynomial preconditioning ideas, a new method for the symmetric solution to matrix equation AT XA=B can be obtained by combining the orthogonal projection iteration method, the estimation of convergence rate is given. Numerical results show that the convergence rate of new method under certain conditions is faster than the orthogonal projection iteration method.