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Global Homotopies and Newton Methods

Keller, Herbert B. (1978) Global Homotopies and Newton Methods. In: Recent Advances in Numerical Analysis. Academic Press , New York, NY, pp. 73-94. ISBN 978-0-12-208360-0. https://resolver.caltech.edu/CaltechAUTHORS:20170802-105839555

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Abstract

This chapter describes the global homotopies and Newton methods. A key to devising global methods is to give up the monotone convergence and to consider more general homotopies. It turns out that singular matrices on the path cause no difficulties in the proof of Smales result. They cause trouble in attempts to implement this and most other global Newton methods numerically. Small steps must be taken in the neighborhood of vanishing Jacobians. This feature is not always pointed out in descriptions of the implementations but it is easily detected. These difficulties can be eliminated by using a somewhat different homotopy. The chapter discusses a pseudo-arc length continuation procedure in which the parameter is distance along a local tangent ray to the path. Using this parameter, this chapter discusses how to accurately locate the roots and the limit points on the path. These latter points are of great interest in many physical applications.


Item Type:Book Section
Related URLs:
URLURL TypeDescription
https://doi.org/10.1016/B978-0-12-208360-0.50009-7DOIArticle
http://www.sciencedirect.com/science/article/pii/B9780122083600500097PublisherArticle
Additional Information:© 1978 Academic Press, Inc. Published by Elsevier Inc.
Record Number:CaltechAUTHORS:20170802-105839555
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170802-105839555
Official Citation:Herbert B. Keller, Global Homotopies and Newton Methods, In Recent Advances in Numerical Analysis, edited by CARL DE BOOR and GENE H. GOLUB, Academic Press, 1978, Pages 73-94, ISBN 9780122083600, https://doi.org/10.1016/B978-0-12-208360-0.50009-7. (http://www.sciencedirect.com/science/article/pii/B9780122083600500097)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:79758
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:02 Aug 2017 18:36
Last Modified:03 Oct 2019 18:23

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