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Newton's method under mild differentiability conditions

Keller, Herbert B. (1970) Newton's method under mild differentiability conditions. Journal of Computer and System Sciences, 4 (1). pp. 15-28. ISSN 0022-0000. https://resolver.caltech.edu/CaltechAUTHORS:20170802-073135318

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Abstract

We study Newton's method for determining the solution of f(x) = 0 when f(x) is required only to be continuous and piecewise continuously differentiable in some sphere about the initial iterate, x^(0). First an existence, uniqueness and convergence theorem is obtained employing the modulus of continuity of the first derivative, f_x(x). Under the more explicit assumption of H6lder continuity several other such results are obtained, some of which extend results of Kantorovich and Akilov [1] and Ostrowski [5]. Of course, when Newton's method converges, it is now of order (1 + α), where a is the Hö1der exponent. Other results on Newton's method without second derivatives are given by Goldstein [2], Schroeder [3], Rheinboldt [6], and Antosiewicz [7], to mention a few. It seems clear that the error analysis for Newton's method given by Lancaster [4] can be extended to the present case.


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https://doi.org/10.1016/S0022-0000(70)80009-5DOIArticle
http://www.sciencedirect.com/science/article/pii/S0022000070800095?via%3DihubPublisherArticle
Additional Information:© 1970 Published by Elsevier Inc. Received 1 July 1968. This work was supported by the U.S. Army Research Office, Durham, N.C., under contract DAHC 04-68-C-0006.
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Army Research Office (ARO)DAHC 04-68-C-0006
Issue or Number:1
Record Number:CaltechAUTHORS:20170802-073135318
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170802-073135318
Official Citation:Herbert B. Keller, Newton's method under mild differentiability conditions, Journal of Computer and System Sciences, Volume 4, Issue 1, 1970, Pages 15-28, ISSN 0022-0000, http://dx.doi.org/10.1016/S0022-0000(70)80009-5. (http://www.sciencedirect.com/science/article/pii/S0022000070800095)
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ID Code:79734
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:02 Aug 2017 17:25
Last Modified:03 Oct 2019 18:23

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