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

Author: Emil Catinas
Journal: Math. Comp. 74 (2005), 291-301
MSC (2000): Primary 65H10
Published electronically: March 23, 2004
MathSciNet review: 2085412
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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 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 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 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 convergence orders of the other two. We also study the relationship in the case of linear convergence and we deduce a new convergence result.

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Additional Information

Emil Catinas
Affiliation: Romanian Academy, “T. Popoviciu” Institute of Numerical Analysis, P.O. Box 68–1, Cluj-Napoca 3400, Romania

Keywords: Inexact, inexact perturbed and quasi-Newton methods, convergence orders
Received by editor(s): July 23, 2001
Received by editor(s) in revised form: May 3, 2003
Published electronically: March 23, 2004
Additional Notes: This research has been supported by the Romanian Academy under grant GAR 97/1999, and by the National Agency for Science, Technology and Innovation under grant GANSTI 6100 GR/2000.
Article copyright: © Copyright 2004 American Mathematical Society