Improved Convergence Analysis of Gauss-Newton-Secant Method for Solving Nonlinear Least Squares Problems
We study an iterative differential-difference method for solving nonlinear least squares problems, which uses, instead of the Jacobian, the sum of derivative of differentiable parts of operator and divided difference of nondifferentiable parts. Moreover, we introduce a method that uses the derivativ...
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| Published in | Mathematics (Basel) Vol. 7; no. 1; p. 99 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Basel
MDPI AG
01.01.2019
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| Online Access | Get full text |
| ISSN | 2227-7390 2227-7390 |
| DOI | 10.3390/math7010099 |
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| Abstract | We study an iterative differential-difference method for solving nonlinear least squares problems, which uses, instead of the Jacobian, the sum of derivative of differentiable parts of operator and divided difference of nondifferentiable parts. Moreover, we introduce a method that uses the derivative of differentiable parts instead of the Jacobian. Results that establish the conditions of convergence, radius and the convergence order of the proposed methods in earlier work are presented. The numerical examples illustrate the theoretical results. |
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| AbstractList | We study an iterative differential-difference method for solving nonlinear least squares problems, which uses, instead of the Jacobian, the sum of derivative of differentiable parts of operator and divided difference of nondifferentiable parts. Moreover, we introduce a method that uses the derivative of differentiable parts instead of the Jacobian. Results that establish the conditions of convergence, radius and the convergence order of the proposed methods in earlier work are presented. The numerical examples illustrate the theoretical results. |
| Author | Argyros, Ioannis Shakhno, Stepan Shunkin, Yurii |
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| Cites_doi | 10.1007/s10958-014-1971-3 10.1137/1.9781611971200 10.1016/j.amc.2003.12.025 10.1007/s11075-011-9470-9 10.1016/j.amc.2010.09.040 10.1201/9781315153469 10.1080/01630568708816254 10.1016/j.amc.2013.09.049 |
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| Copyright | 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
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| References | ref_14 Argyros (ref_4) 2011; 58 Shakhno (ref_6) 2005; 161 Shakhno (ref_9) 2017; 25 (ref_11) 1994; 23 Argyros (ref_8) 2013; 225 Ulm (ref_10) 1967; 16 ref_1 ref_3 ref_2 Ren (ref_5) 2010; 217 Zabrejko (ref_13) 1987; 9 Shakhno (ref_12) 2014; 201 ref_7 |
| References_xml | – ident: ref_7 – volume: 201 start-page: 32 year: 2014 ident: ref_12 article-title: Analysis of the Convergence of a Combined Method for the Solution of Nonlinear Equations publication-title: J. Math. Sci. doi: 10.1007/s10958-014-1971-3 – ident: ref_2 doi: 10.1137/1.9781611971200 – ident: ref_3 – volume: 161 start-page: 253 year: 2005 ident: ref_6 article-title: On an iterative algorithm of order 1.839... for solving the nonlinear least squares problems publication-title: Appl. Math. Comput. doi: 10.1016/j.amc.2003.12.025 – volume: 25 start-page: 38 year: 2017 ident: ref_9 article-title: One combined method for solving nonlinear least squares problems publication-title: Visnyk Lviv Univ. Ser. Appl. Math. Inform. – volume: 58 start-page: 555 year: 2011 ident: ref_4 article-title: A derivative free iterative method method for solving least squares problems publication-title: Numer. Algorithms doi: 10.1007/s11075-011-9470-9 – volume: 16 start-page: 13 year: 1967 ident: ref_10 article-title: On generalized divided differences publication-title: Proc. Acad. Sci. Estonian SSR. Phys. Mathe. – volume: 217 start-page: 3816 year: 2010 ident: ref_5 article-title: Local convergence of a secant type method for solving least squares problems publication-title: Appl. Math. Comput. doi: 10.1016/j.amc.2010.09.040 – ident: ref_14 doi: 10.1201/9781315153469 – ident: ref_1 – volume: 23 start-page: 47 year: 1994 ident: ref_11 article-title: On some iterative methods for solving nonlinear equations publication-title: Revue d’Analyse Numérique et de Theorie de l’Approximation – volume: 9 start-page: 671 year: 1987 ident: ref_13 article-title: The majorant method in the theory of Newton-Kantorovich approximations and the Pták error estimates publication-title: Numer. Funct. Anal. Optim. doi: 10.1080/01630568708816254 – volume: 225 start-page: 372 year: 2013 ident: ref_8 article-title: On an improved convergence analysis of Newton’s method publication-title: Appl. Math. Comput. doi: 10.1016/j.amc.2013.09.049 |
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| SubjectTerms | Approximation Convergence differential-difference method divided differences Hypotheses Iterative methods Least squares Mathematics Methods Nonlinear equations nonlinear least squares problem Operators (mathematics) order of convergence Parameter estimation residual |
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| Title | Improved Convergence Analysis of Gauss-Newton-Secant Method for Solving Nonlinear Least Squares Problems |
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