Computing {2,4} and {2,3}-inverses by using the Sherman–Morrison formula

A finite recursive procedure for computing {2,4} generalized inverses and the analogous recursive procedure for computing {2,3} generalized inverses of a given complex matrix are presented. The starting points of both introduced methods are general representations of these classes of generalized inverses. These representations are formed using certain matrix products which include the Moore–Penrose inverse or the usual inverse of a symmetric matrix product and the Sherman–Morrison formula for the inverse of a symmetric rank-one matrix modification. The computational complexity of the methods is analyzed. Defined algorithms are tested on randomly generated matrices as well as on test matrices from the Matrix Computation Toolbox.