In 1965 Broyden introduced a family of algorithms called(rank-one) quasi-New-ton methods for iteratively solving systems of nonlinear equations. We show that when any member of this family is applied to an n x n nonsingular system of linear equations and direct-prediction steps are taken every...

We show that if a certain nondegeneracy assumption holds, it is possible to guarantee the existence of a solution to a system of nonlinear equations f(x) = 0 whose Jacobian matrix J(x) exists but maybe singular. The main idea is to modify small singular values of J(x) in such away that the...

Various quasi-Newton methods periodically add a symmetric "correction" matrix of rank at most 2 to a matrix approximating some quantity A of interest (such as the Hessian of an objective function). In this paper we examine several ways to express a symmetric rank 2 matrix [delta] as the sum of...