A variable metric method for approximating generalized inverses of matrices

摘要

We present two variable metric update methods that have some attractive features: They deal with the problem of finding a least-squares solution of a linear system Ax = b with an m x n-matrix of maximal rank, and can be viewed as a generalization of the DFP- and of the BFGS-methods, respectively, to nonsymmetric matrices A: The methods generate a sequence of n x m-matrices H-k so that the AH(k) are positive semidefinite and the H-k approximate a right-inverse of A if m less than or equal to n, and the Moore-Penrose pseudoinverse of A, if m greater than or equal to n. Thus for m = n the methods find approximations to A-1 that could be used as preconditioners in other methods for solving Ax = b when A is not positive definite. Both methods are related to cg-type algorithm minimizing the residual on Krylov-subspaces. [References: 25]