The Research On Iterative Algorithms For The Solutions Of General Coupled Matrix Equations

Matrix theory and computing have been widely used in the field of modernscience research. In contemporary science and engineering issues, there are a lot ofproblems attributed to the mass matrix computation problems finally. But in practice,the matrix solutions we calculated usually have a special structure or special nature,which is called constrained solution problem of matrix equation. In practicalengineering design, we need to know is a particular solution of the equation, and assuch, iterative methods become a popular method for solving the problem. So it hasimportant applications in many areas, such as structural design, linear systems andautomatic control theory, remote sensing technology, and so on.The research on the constrained solution of matrix equations which have one ortwo variable is fruitful. This thesis studies particular solution of matrix equationswhich have more general form, including the famous coupled Sylvester matrixequations. It is a important result to improve and expand the existing results.By theextension of conjugate gradient method, we will give effective iterative algorithmsto solve the general coupled matrix equations over the generalized centralsymmetric, central anti-symmetric and generalized bisymmetric matrices.One advantage of our algorithms is that we can use the intermediate resultscalculated in the solution process to determine whether the matrix equations areconsistent over the corresponding constrained solutions or not directly. If the generalcoupled matrix equations are consistent over the constrained solutions, so withoutconsidering the machine errors and rounding errors, for any given initial constrainedmatrix, the constrained solutions can be obtained within limited iterative steps byusing the iterative algorithms. First, we will give the rigorous derivation of theseconclusions by using nature of the algorithms. Second, we prove that the leastFrobenius norm constrained solutions can also be derived by choosing a special kindof initial matrices. Furthermore, the optimal approximation problem for any givenmatrix is solved. Finally, as an application of the algorithms, some correspondingnumerical examples are given.