Limit representations of generalized inverses and related methods

摘要

We generalize the iterative method for implementation of the limit representation of the Moore–Penrose inverse, limα→0αI+A∗A−1A∗, introduced by Žukovski and Lipcer [Ž. Vicisl. Mat. i Mat. Fiz. 12 (1972) 843; 15 (1975) 489]. More precisely, we define an iterative process for implementing the general limit formula limα→0(αI+R∗S)−1R∗. In a certain case the presented method gives an iterative process which implements the known limit representation of the Drazin inverse, introduced by Meyer [SIAM J. Appl. Math. 26 (1974) 469]. Also, the known limit representation of the generalized inverse AT,S(2) is a special case of the general limit formula. In the case of a full rank matrix A, we define a limit representation and an iterative method for computation of the left or right inverses of A. Also, we introduce two alternative limit representations for the set of {2} and {1,2}-inverses, and develop the corresponding iterative processes. Moreover, we propose a finite algorithm for computation of the generalized inverses contained in the limit formula limα→0(αI+R∗S)−1R∗, using the method developed by Ji [Appl. Math. Comput. 61 (1994) 151].