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Newton's method revisited: how accurate do we have to be?
Umeå University, Faculty of Science and Technology, Department of Computing Science.ORCID iD: 0000-0002-9158-1941
Barcelona Supercomputing Center, Barcelona, Spain.
Independent Scholar, Berlin, Germany.
2024 (English)In: Concurrency and Computation, ISSN 1532-0626, E-ISSN 1532-0634, Vol. 36, no 10, article id e7853Article in journal (Refereed) Published
Abstract [en]

We analyze the convergence of quasi-Newton methods in exact and finite precision arithmetic using three different techniques. We derive an upper bound for the stagnation level and we show that any sufficiently exact quasi-Newton method will converge quadratically until stagnation. In the absence of sufficient accuracy, we are likely to retain rapid linear convergence. We confirm our analysis by computing square roots and solving bond constraint equations in the context of molecular dynamics. In particular, we apply both a symmetric variant and Forsgren's variant of the simplified Newton method. This work has implications for the implementation of quasi-Newton methods regardless of the scale of the calculation or the machine.

Place, publisher, year, edition, pages
John Wiley & Sons, 2024. Vol. 36, no 10, article id e7853
Keywords [en]
approximation error, convergence, quasi-Newton methods, rounding error, stagnation, systems of nonlinear equations
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-212265DOI: 10.1002/cpe.7853ISI: 001020863100001Scopus ID: 2-s2.0-85164157230OAI: oai:DiVA.org:umu-212265DiVA, id: diva2:1783320
Funder
eSSENCE - An eScience CollaborationAvailable from: 2023-07-20 Created: 2023-07-20 Last updated: 2024-06-26Bibliographically approved

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Kjelgaard Mikkelsen, Carl Christian

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CiteExportLink to record
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Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
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Output format
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