| Matrix equations often arise in areas of scientific and engineering computing.In the field of control theory, system theory, neural network, model reduction, image restoration and signal processing will involve numerical solution problem of matrix equations. Modified and constructing new iterative algorithm based on the hierarchical identification principle and gradient iterative algorithm to the solution of matrix equation. The main contents are described as follows.1. Based on the original algorithm, as the information of the first half iterative step is required to update the solution by the modified method to solve matrix equations AXB+CXHD= F and A1XB1+A2XHB2= F1, C1XD1+C2XHD2= F2. And the modified algorithm has been proved to be convergent to the true solution under any initial value. The same time, numerical results show that the propose algorithm is efficient than the existing numerical ones.2. In the process of construction of least squares iterative algorithm solve matrix equations (AX-YB,DX-YE)= (C,F) and (A1XB1-C1YD1,A2XB2-C2YD2)= (F1,F2). The symmetric positive definite matrix has special eigenvalues properties, according to the properties of the algorithm to determine the bounds of the convergence factor, and find the best convergence factor.3. Two iterative algorithms are proposed to investigate the iterative reflexive and Hermitian reflexive solutions to the coupled Sylvester conjugate matrix equations A1X+B1Y= E1XF1+C1,A2X+B2Y= E2XF2+C2. The next, we give to two modified algorithms by two-dimensional projection technique to improve the rate of convergence of the gradient-based iterative algorithms. Finally, two numerical examples are given to show the effectiveness of the proposed algorithm, and the modified algorithm is efficient than the before ones. |
