On the Local Convergence of Update Methods

This paper concerns a class of update methods, or, as they are also called, quasi-Newton methods, variable metric methods, or modification methods. A general theory of rank-one and symmetric rank-two update formulas is presented which covers many of the special methods proposed in the literature. Recently, Broyden, Dennis and More found a local convergence theorem for a class of these methods. A new and unified proof of this theorem is given here which uses a geometrically more intuitive and also more general convergence condition than the original theorem. The proof utilizes elliptic-norm estimates to derive a majorizing system of difference inequalities. Then more careful estimates involving a generalized Frobenius norm show that under rather general conditions the methods under consideration are superlinearly convergent.