Local convergence of inexact Newton methods under affine invariant conditions and hypotheses on the second Fréchet derivative

Volume 26 / 1999

Ioannis Argyros Applicationes Mathematicae 26 (1999), 457-465 DOI: 10.4064/am-26-4-457-465

Abstract

We use inexact Newton iterates to approximate a solution of a nonlinear equation in a Banach space. Solving a nonlinear equation using Newton iterates at each stage is very expensive in general. That is why we consider inexact Newton methods, where the Newton equations are solved only approximately, and in some unspecified manner. In earlier works [2], [3], natural assumptions under which the forcing sequences are uniformly less than one were given based on the second Fréchet derivative of the operator involved. This approach showed that the upper error bounds on the distances involved are smaller compared with the corresponding ones using hypotheses on the first Fréchet derivative. However, the conditions on the forcing sequences were not given in affine invariant form. The advantages of using conditions given in affine invariant form were explained in [3], [10]. Here we reproduce all the results obtained in [3] but using affine invariant conditions.

Authors

  • Ioannis Argyros

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