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

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 iterationX"j"+"1=M^(^2^)"T","SNX"j+M^(^2^)"T","Sconverges to A^(^2^)"T","S for any X"0 if and only ifT=T,S=S,[email protected]^(^2^)"T","SN<1,or equivalently,M^(^2^)"T","[email protected]^(^2^)"T","SN<1. The splitting A=M-N is called (T,S)-splitting of A if M satisfies [email protected]__ __S=C^m where T and S are subspaces of C^n and C^m, 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^),A"d","w,A^+"H"K,..., we can get criteria of convergence for the corresponding iterations, many old and new splittings, and specific choice of the related convergent splittings.