Derivative-free convergence analysis for Steffensen-type schemes for nonlinear equations.
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| Title: | Derivative-free convergence analysis for Steffensen-type schemes for nonlinear equations. |
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| Authors: | George, Santhosh1 (AUTHOR) sgeorge@nitk.edu.in, M, Muniyasamy2 (AUTHOR) muniyasamy.237ma004@nitk.edu.in, Grammont, Laurence1,2 (AUTHOR) laurence.grammont@univ-st-etienne.fr |
| Source: | Applied Numerical Mathematics. May2026, Vol. 223, p101-120. 20p. |
| Subjects: | Nonlinear equations, Difference operators, Iterative methods (Mathematics), Banach spaces, Numerical analysis |
| Abstract: | Steffensen schemes have been constructed to approximate the solution of an operator equation, with the goal of avoiding the use of its derivatives. It is the reason why these schemes involve the first order divided difference operator. Until now, results on convergence order have been provided using Taylor series expansion, which implies that the operator must be several times differentiable. To be consistent with the nature of the Steffensen schemes, we propose a proof of the convergence order under assumptions that involve only the first and second order divided difference operators. In addition, the convergence order analysis for these Steffensen schemes is done here for the general case of Banach spaces, while it has been done only for finite-dimensional spaces so far. Until now, the assumptions required for semi-local analysis and those required for local analysis have been of a very different nature. A new idea was to unify these hypotheses; hence, we give a single set of convergence conditions. Moreover, our local convergence analysis provides consistently explicit convergence balls that are computable. [ABSTRACT FROM AUTHOR] |
| Copyright of Applied Numerical Mathematics is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 191979737 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Derivative-free convergence analysis for Steffensen-type schemes for nonlinear equations. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22George%2C+Santhosh%22">George, Santhosh</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> sgeorge@nitk.edu.in</i><br /><searchLink fieldCode="AR" term="%22M%2C+Muniyasamy%22">M, Muniyasamy</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> muniyasamy.237ma004@nitk.edu.in</i><br /><searchLink fieldCode="AR" term="%22Grammont%2C+Laurence%22">Grammont, Laurence</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> laurence.grammont@univ-st-etienne.fr</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Applied+Numerical+Mathematics%22">Applied Numerical Mathematics</searchLink>. May2026, Vol. 223, p101-120. 20p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Nonlinear+equations%22">Nonlinear equations</searchLink><br /><searchLink fieldCode="DE" term="%22Difference+operators%22">Difference operators</searchLink><br /><searchLink fieldCode="DE" term="%22Iterative+methods+%28Mathematics%29%22">Iterative methods (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Banach+spaces%22">Banach spaces</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Steffensen schemes have been constructed to approximate the solution of an operator equation, with the goal of avoiding the use of its derivatives. It is the reason why these schemes involve the first order divided difference operator. Until now, results on convergence order have been provided using Taylor series expansion, which implies that the operator must be several times differentiable. To be consistent with the nature of the Steffensen schemes, we propose a proof of the convergence order under assumptions that involve only the first and second order divided difference operators. In addition, the convergence order analysis for these Steffensen schemes is done here for the general case of Banach spaces, while it has been done only for finite-dimensional spaces so far. Until now, the assumptions required for semi-local analysis and those required for local analysis have been of a very different nature. A new idea was to unify these hypotheses; hence, we give a single set of convergence conditions. Moreover, our local convergence analysis provides consistently explicit convergence balls that are computable. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Applied Numerical Mathematics is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1016/j.apnum.2026.01.003 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 20 StartPage: 101 Subjects: – SubjectFull: Nonlinear equations Type: general – SubjectFull: Difference operators Type: general – SubjectFull: Iterative methods (Mathematics) Type: general – SubjectFull: Banach spaces Type: general – SubjectFull: Numerical analysis Type: general Titles: – TitleFull: Derivative-free convergence analysis for Steffensen-type schemes for nonlinear equations. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: George, Santhosh – PersonEntity: Name: NameFull: M, Muniyasamy – PersonEntity: Name: NameFull: Grammont, Laurence IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 05 Text: May2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 01689274 Numbering: – Type: volume Value: 223 Titles: – TitleFull: Applied Numerical Mathematics Type: main |
| ResultId | 1 |
