Research On The Multi-step Iterative Algorithm For Some Kinds Of Constrained Matrix Equation

The introduction of matrix equation theories and methods has become a common phenomenon when solving problems in many fields such as engineering technology,control theory,information and image processing,power system and correction,time series analysis,etc.[1-3].This not only promotes the development of modern engineering and science and technology,but also provides more research directions for the solution of constraint matrix equations in the field of numerical algebra.The main research work of this master thesis is:Multi-step iterative algorithm to solve the following types of constraint matrix equations:Problem ? Given matrices A,B?Rm×n,(m?n)S(?)Rm×n,solve X?S such that AX=B(or?AX-B?=min).Problem ? Given matrices A?Rp×m,B?Rn×q,C?Rp×q,(p?m,n?q),S(?)Rn×n,solve X?S such that AXB=C(or?AXB-C?=min).Problem ? Set problem ? and ? is compatible,and its solution set is SE,given matrices X0?S solve X?SE such that#12In the above three problems,?·? is the Frobenius norm,S is the set of matrices in matrix set Rm×n that satisfy certain constraint conditions,such as general matrix,symmetric matrix and anti-symmetric matrix.Based on the idea of fixed point iteration,the multi-step iterative algorithm for solving problems ? and ? is given,and the convergence conditions of the multi-step iterative algorithm are given and proved in this paper.The comparative experimental results show that the proposed multi-step iterative algorithm relative to the gradient based iterative algorithm has better convergence effect.Finally,the analysis shows that problem? can also be solved by the multi-step iterative algorithm,and numerical experiments have verified the effectiveness of the proposed algorithm.