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Journals / An International Journal of Optimization and Control: Theories & Applications (IJOCTA) / 2019 / Cilt: 9 - Sayı: 2

A New Broyden rank reduction method to solve large systems of nonlinear equations

Mostafa OUARİT
Ali SOUISSI
Mohammed ZIANI
Pages
176–185
Publication Date
2023-12-17
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Local superlinear convergence of themethod is proved and some numerical examples are displayed.

📌 3 Key Takeaways

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Objective & Hypothesis:We propose a modification of limited memory Broyden methods, called dy-namical Broyden rank reduction method, to solve high dimensional systems ofnonlinear equations.
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Key Findings & Data:Based on a thresholding process of singular values, theproposed method determines a priori the rank of the reduced update matrix.It significantly reduces the number of singular values decomposition calls ofthe update matrix during the iterations.
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Conclusion & Impact:Local superlinear convergence of themethod is proved and some numerical examples are displayed.
🔬Design:Makale
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters

Abstract

We propose a modification of limited memory Broyden methods, called dy-namical Broyden rank reduction method, to solve high dimensional systems ofnonlinear equations. Based on a thresholding process of singular values, theproposed method determines a priori the rank of the reduced update matrix.It significantly reduces the number of singular values decomposition calls ofthe update matrix during the iterations. Local superlinear convergence of themethod is proved and some numerical examples are displayed.

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Cite This Article

OUARİT, M., SOUISSI, A., ZIANI, M. (2023). A New Broyden rank reduction method to solve large systems of nonlinear equations. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), Cilt: 9 - Sayı: 2, 176–185. https://doi.org/10.11121/ijocta.01.2019.00743

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References (4)

  1. Dennis, J. E. and Schnabel, R. B. (1983). Nu- merical methods for unconstrained optimiza- tion and nonlinear equations. Prentice-Hall. Kelley, C. T. (1995). Iterative methods for linear and nonlinear equations. Volume 16 of Frontiers in Applied Mathematics, SIAM, Philadelphia. Ortega, J. M. and Reinboldt, W. C. (2000). Iterative solution of nonlinear equations in several variables. SIAM, Philadelphia. Broyden, C. G. (1965). A class of methods for solving nonlinear simultaneous equations. Mathematics of Computation, (19), 577–593.
  2. Broyden, C. G., Dennis, J. E. and Moré, J. J. (1973). On the local and superlinear conver- gence of quasi-Newton methods. IMA Journal of Applied Mathematics, (12), 223–245.
  3. Byrd, R. H., Nocedal, J. and Schnabel, R. B. (1994). Representations of quasi-Newton ma- trices and their use in limited memory meth- ods. Mathematical Programming, (63), 129– 156. Van De Rotten, B. and Verduyn Lunel, S. M. (2003). A limited memory Broyden method to solve high dimensional systems of nonlin- ear equations. Technical Report MI 2003-06,
  4. Mathematical Institute, University of Leiden, The Netherlands. Ziani M. and Guyomarc’h F. (2008). An autoadaptative limited memory Broyden’s method to solve systems of nonlinear equa- tions. Applied Mathematics and Computation, (205), 202–211.