Obtaining The Solutions Of The Large Matrix Equations (AX,XB)=(C,D) By The Recursive Blocked Method

The problems of solving constrained matrix equations and the correspondi-ng least-squares problems have been a very active field of recent research with a wide range of applications, such as structural design, system identification, structural dynamics and automatics control theory. By studying these problems, we can have better schemes to solve the small order matrix equations, furthermore it is of great significance for solving the large-scale matrix equations in a short time.This paper studies the solution of large-scale matrix equations (AX,XB)=(C,D) by recursive method and decomposing the matrix A and B, including the generalized solution, the least-squares solution and the least-squares symmetric solution. These problems are stated as follows.Problem Ⅰ:Given A,B,C,D D∈Rn×n, find X∈Rn×n such that the large-scale matrix equations satisfy (AX,XB)=(C,D),Problem Ⅱ:Given A,B,C,D∈Rn×n, find the least-squares solution X∈Rn×n of the large-scale matrix equations (AX,XB)=(C,D),Problem Ⅲ:Given A,B,C,D∈Rn×n, find the least-squares symmetric solution X∈Rn×n of the large-scale matrix equations (AX,XB)=(C,D).The paper is organized as follows:Section Ⅰ, we introduce the background of the large matrix equations and its research status.Section Ⅱ, we describe some prior knowledge, including some basic notations and lemmas that will be used throughout the paper.Section Ⅲ, we solve the problem Ⅰ. The matrix A and B are decomposed by making use of the generalized QR decomposition, hence the original large matrix equations of problem Ⅰ transformed into the four equivalent matrix equations,and we can obtain the general expressions of the corresponding solution for problem Ⅰ by finding the solutions of the four equivalent small order matrix equations;Furthermore, the corresponding numerical algorithms and examples are given.Bsaed on the section Ⅲ and by recursive method, in the section IV we solve the equivalent matrix equations by the classical conjugate gradient and orthogonal direct sum algorithm, and give the numerical algorithms and examples to solve problem II and problem III respectively.To solve the problem I, problem II and problemIII, we need to simulate thenumerical experimentation for the different order matrixs.The numerical testingshows that we spend less on computing time, hence recursive method has someadvantages to solve the large-scale matrix equations.