Research On Preconditioning Of Matrix Equations

Constrained matrix equations have very important applications in vibration theory,electrical engineering,control theory and nonlinear programming,and have made gratifying research results.This paper focuses on the preprocessing method of orthogonal projection iterative method.Problem I:Given A?R n×n,B?Rn×n,S(?)Rn×n,find X?S,such that AX=B where S areRn×n?SRn×n?ASRn×n?respectively.Problem II:Given A?Rn×n,B?Rn×n,C?Rn×n,S(?)Rn×n,find X?S,such that AXB = C where S is Rn×n.The main work of the paper:1.For problem I,for S,respectively,the case of Rn×n?SRn×n?ASRn×n.Firstly,the PSAI(tol)preprocessing algorithm for problem I is constructed by using the compatibility of PSAI(tol)technique based on F-norm minimization and the matching of vector 2-norm,and finding the approximate inverse of A.Then the algorithm is combined with orthogonal projection iter-ative method to obtain PSAI(tol)pretreatment orthogonal projection iterative method,and the convergence of this algorithm is proved.Finally,a numerical example is given to demonstrate the effectiveness of the corresponding algorithm.2.For Problem II,we first construct the PSAI(tol)preprocessing algorithm for problem II,and then combine the algorithm with orthogonal projection iterative method to obtain PSAI(tol)preprocessing orthogonal projection iterative method,and the convergence of this algorithm is proved.Finally,a numerical example is given to demonstrate the effectiveness of the corresponding algorithm.