A family of high order derivative-free iterative methods for solving root-finding problems

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Abstract

In this paper, we derive a new family of high order derivative-free iteration methods for finding simple and multiple roots of nonlinear algebraic equations of the form f (x) = 0. Each scheme requires only one initial guess. Our proposed procedure can be viewed as an extension of the second-order Steffensen’s method. The idea is to modify the family of derivative-based methods, which were recently proposed and analyzed by the author, to obtain derivative-free methods. The modified iterative methods are shown to have the same order of convergence as the derivative-based methods. The approach consists of approximating all derivatives with suitable difference formulas. The pth-order method requires evaluation of the function f at p suitable arguments. The error equations and asymptotic convergence constants are obtained. We also describe how to obtain derivative-free methods to find roots with multiplicity. Several numerical examples are provided to validate the theoretical order of convergence for nonlinear functions with simple and multiple roots.

Original languageEnglish (US)
Article number56
JournalInternational Journal of Applied and Computational Mathematics
Volume5
Issue number3
DOIs
StatePublished - Jan 1 2019

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Keywords

  • High order derivative-free iterative methods
  • Nonlinear equations
  • Orderofconvergence
  • Root-finding problem
  • Simple and multiple roots
  • Steffensen-like methods

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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