A global Newton-type scheme based on a simplified Newton-type approach

Amrein, Mario

doi:10.1007/s12190-020-01393-w

Please use this identifier to cite or link to this item: https://doi.org/10.21256/zhaw-20601

Publication type:	Article in scientific journal
Type of review:	Peer review (publication)
Title:	A global Newton-type scheme based on a simplified Newton-type approach
Authors:	Amrein, Mario
et. al:	No
DOI:	10.1007/s12190-020-01393-w 10.21256/zhaw-20601
Published in:	Journal of Applied Mathematics and Computing
Volume(Issue):	65
Issue:	1-2
Page(s):	321
Pages to:	334
Issue Date:	9-Jul-2020
Publisher / Ed. Institution:	Springer
ISSN:	1598-5865 1865-2085
Language:	English
Subjects:	Global Newton method; Simplified Newton method; A posteriori analysis; Newton path
Subject (DDC):	510: Mathematics
Abstract:	Globalization concepts for Newton-type iteration schemes are widely used when solving nonlinear problems numerically. Most of these schemes are based on a predictor/corrector step size methodology with the aim of steering an initial guess to a zero of f without switching between different attractors. In doing so, one is typically able to reduce the chaotic behavior of the classical Newton-type iteration scheme. In this note we propose a globalization methodology for general Newton-type iteration concepts which changes into a simplified Newton iteration as soon as the transformed residual of the underlying function is small enough. Based on Banach’s fixed-point theorem, we show that there exists a neighborhood around a suitable iterate xn such that we can steer the iterates—without any adaptive step size control but using a simplified Newton-type iteration within this neighborhood—arbitrarily close to an exact zero of f. We further exemplify the theoretical result within a global Newton-type iteration procedure and discuss further an algorithmic realization. Our proposed scheme will be demonstrated on a low-dimensional example thereby emphasizing the advantage of this new solution procedure.
URI:	https://digitalcollection.zhaw.ch/handle/11475/20601
Fulltext version:	Published version
License (according to publishing contract):	CC BY 4.0: Attribution 4.0 International
Departement:	School of Management and Law
Organisational Unit:	Institute for Risk & Insurance (IRI)
Appears in collections:	Publikationen School of Management and Law

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Amrein, M. (2020). A global Newton-type scheme based on a simplified Newton-type approach. Journal of Applied Mathematics and Computing, 65(1-2), 321–334. https://doi.org/10.1007/s12190-020-01393-w

Amrein, M. (2020) ‘A global Newton-type scheme based on a simplified Newton-type approach’, Journal of Applied Mathematics and Computing, 65(1-2), pp. 321–334. Available at: https://doi.org/10.1007/s12190-020-01393-w.

M. Amrein, “A global Newton-type scheme based on a simplified Newton-type approach,” Journal of Applied Mathematics and Computing, vol. 65, no. 1-2, pp. 321–334, Jul. 2020, doi: 10.1007/s12190-020-01393-w.

AMREIN, Mario, 2020. A global Newton-type scheme based on a simplified Newton-type approach. Journal of Applied Mathematics and Computing. 9 Juli 2020. Bd. 65, Nr. 1-2, S. 321–334. DOI 10.1007/s12190-020-01393-w

Amrein, Mario. 2020. “A Global Newton-Type Scheme Based on a Simplified Newton-Type Approach.” Journal of Applied Mathematics and Computing 65 (1-2): 321–34. https://doi.org/10.1007/s12190-020-01393-w.

Amrein, Mario. “A Global Newton-Type Scheme Based on a Simplified Newton-Type Approach.” Journal of Applied Mathematics and Computing, vol. 65, no. 1-2, July 2020, pp. 321–34, https://doi.org/10.1007/s12190-020-01393-w.