Studies On Iterative Algorithms For Matrix Equation(s)

In this paper, I studies an iterative method for several types of matrix equations and its optimal approximate problem, as well as gives the convergence of the iterative method. These large matrix equation(s) include AXB=C, One-sided and generalized coupled Sylvester matrix equations (AY-ZB,CY-ZD)=(E,F), Two-sided and generalized coupled Sylvester matrix equations (AXB-CYD,EXF-GYH)=(M,N). This paper contains four chapters. In the first chapter, I introduce the basis of the article topic and the present situation on matrix equation research at home and abroad. In the second chapter, I studies the skew-symmetric Solutions of linear equations AXB=C Firstly,an iterative method is constructed to solve the linear matrix equation and the convergence of the algorithm is proved. And then studies the equation of the optimal approximation problem Finally, numerical examples are given to verify the validity of the algorithm In the third chapter, I studies the One-sided and generalized coupled Sylvester matrix equations (AY-ZB,CY-ZD)=(E,F)and its optimal approximation over generalized reflexive iteration algorithm solutions, When the matrix equations are consistent, for any initial generalized reflexive matrix pair [Y1,Z1], the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors. Let initially iterative matrix pair Y1=ATK+CTG+PATKQ+PCTGQ, Z1=-KBT-GDT-MKBTN-MGDTN where K, G∈s×t are arbitrary matrices, or more especially, let Y1=0,Z1=0, then we can get the least Frobenius norm generalized reflexive solution pair. The unique optimal approximation generalized reflexive solution pair [Y0,Z0]can be derived by finding the least-norm generalized reflexive solution of two new corresponding generalized coupled Sylvester matrix equations. Lastly, several numerical examples are given to show the effectiveness of the presented iterative algorithm In the fourth chapter, I studies Two-sided and matrix equations (AXB-CYD,EXF-GYH)=(M,N) over reflexive iteration algorithm solutions. First of all, an iterative algorithm is constructed to solve the equations, then I prove the convergence of the algorithm, and discuss the best approximation of the equation. Several numerical examples are given to show the effectiveness of the presented iterative algorithm...