Computing generalized inverses of matrices by iterative methods based on splittings of matrices

摘要

In this paper, we study the iterative methods for computing the generalized inverses of the form A(2)T,S. We prove the result that if A has the splitting A=M−N and if the generalized inverses A(2)T,S and M(2)T,S exist, then the iterationXj+1=M(2)T,SNXj+M(2)T,Sconverges to A(2)T,S for any X0 if and only ifT=T,S=S,andρM(2)T,SN<1,or equivalently,M(2)T,SexistsandρM(2)T,SN<1.The splitting A=M−N is called (T,S)-splitting of A if M satisfies MT⊕S=Cm where T and S are subspaces of Cn and Cm, respectively, with dimT=dimS⊥. We also present a specific method to construct convergent (T,S)-splitting of A.Apply these two results to the most of commonly used generalized inverses such as A+,A(d),Ad,w,A+HK,…, we can get criteria of convergence for the corresponding iterations, many old and new splittings, and specific choice of the related convergent splittings.