Abstract
In this paper, we introduce a numerical method for nonlinear equations, based on the Chebyshev third-order method, in which the second-derivative operator is replaced by a finite difference between first derivatives. We prove a semilocal convergence theorem which guarantees local convergence with R-order three under conditions similar to those of the Newton-Kantorovich theorem, assuming the Lipschitz continuity of the second derivative. In a subsequent theorem, the latter condition is replaced by the weaker assumption of Lipschitz continuity of the first derivative.
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Hernández, M.A. Second-Derivative-Free Variant of the Chebyshev Method for Nonlinear Equations. Journal of Optimization Theory and Applications 104, 501–515 (2000). https://doi.org/10.1023/A:1004618223538
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DOI: https://doi.org/10.1023/A:1004618223538