Generalized Inverse Computations Using the Gradient Projection Method

A number of methods have been proposed for computing the (Moore-Penrose-Bjerhammar) generalized inverse A + of an arbitrary m X n complex matrix A of rank r < m _~ n. Boot, Ben~Israel and Wersan have recently published methods which require the formation either of AA* or BB*, where B is an r X n submatrix of A having rank r. In this paper a procedure for computing A + is given which consists of a variant of the gradient projection method. The procedure, which is equivalent to a Hestenes conjugate directions method in the special case r = m, may be applied to any complex matrix. The basic procedure requires the application of the Gram-Schmidt orthogonalization process, first to the column vectors of A*, then, if A is not of full row rank, to the column vectors of A. A computer program (FoRTRXN IV for the IBM 7090) using the method has been tested and used in connection with the generalized inverse-eigenvector method for solving linear programruing problems.