An efficient method to compute different types of generalized inverses based on linear transformation

摘要

In this paper, we present functional definitions of all types of generalized inverses related to the {1}-inverse, which is a continuation of the work of Campbell and Meyer (2009). According to these functional definitions, we further derive novel representations for all types of generalized inverses related to the {1}-inverse in terms of the bases for R(A*), N(A) and N(A*). Based on these representations, we present the corresponding algorithm for computing various generalized inverses related to the {1}-inverse of a matrix and analyze the computational complexity of our algorithm for a constant matrix. Finally, we implement our algorithm and several known algorithms for symbolic computation of the Moore-Penrose inverse in the symbolic computational package MATHEMATICA and compare their running times. Numerical experiments show that our algorithm outperforms these known algorithms when applied to compute the Moore-Penrose inverse of one-variable rational matrices, but is not the best choice for two-variable rational matrices in practice. (C) 2018 Elsevier Inc. All rights reserved.