Quasi-Newton Preconditioners for the Inexact Newton method

In this paper preconditioners for solving the linear systems of the Newton method in each nonlinear iteration are studied. In particular, we define a sequence of preconditioners built by means of Broyden-type rank-one updates. Optimality conditions are derived which guarantee that the preconditioned matrices are not far from the identity in a matrix norm. Some notes on the implementation of the corresponding inexact Newton method are given and some numerical results on two model problems illustrate the application of the proposed preconditioners.