A Caltech Library Service

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.

Full text is not posted in this repository. Consult Related URLs below.

Use this Persistent URL to link to this item:


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.

Item Type:Article
Related URLs:
URLURL TypeDescription
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.
Funding AgencyGrant Number
Army Research Office (ARO)DAHC 04-68-C-0006
Issue or Number:1
Record Number:CaltechAUTHORS:20170802-073135318
Persistent URL:
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, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:79734
Deposited By: Tony Diaz
Deposited On:02 Aug 2017 17:25
Last Modified:03 Oct 2019 18:23

Repository Staff Only: item control page