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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.
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]
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Database: Engineering Source
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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]
ISSN:01689274
DOI:10.1016/j.apnum.2026.01.003