# The inexact, inexact perturbed and quasi-Newton methods are equivalent models

## Abstract

A classical model of Newton iterations which takes into account some error terms is given by the quasi-Newton method, which assumes perturbed Jacobians at each step. Its high q-convergence orders were characterized by Dennis and Moré [Math. Comp. 28 (1974), 549-560].

The inexact Newton method constitutes another such model, since it assumes that at each step the linear systems are only approximately solved; the high q-convergence orders of these iterations were characterized by Dembo, Eisenstat and Steihaug [SIAM J. Numer. Anal. 19 (1982), 400-408].

We have recently considered the inexact perturbed Newton method [J. Optim. Theory Appl. 108 (2001), 543-570] which assumes that at each step the linear systems are perturbed and then they are only approximately solved; we have characterized the high q-convergence orders of these iterates in terms of the perturbations and residuals.

In the present paper we show that these three models are in fact equivalent, in the sense that each one may be used to characterize the high q-convergence orders of the other two. We also study the relationship in the case of linear convergence and we deduce a new convergence result.

## Authors

Emil Cătinaş
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

## Keywords

nonlinear system of equations in Rn; inexact Newton method; perturbed Newton method; quasi-Newton methods; linear systems of equation in Rn; backward errors; GMRES; GMBACK; MINPERT; Newton-Krylov methods; residual; local convergence; convergence order.

## Cite this paper as:

E. Cătinaş, The inexact, inexact perturbed and quasi-Newton methods are equivalent models, Math. Comp., 74 (2005) no. 249, pp. 291-301
doi: 10.1090/S0025-5718-04-01646-1

## PDF

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##### Publisher Name

American Mathematical Society

0025-5718

1088-6842

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