A Family of High Order Numerical Methods for Solving Nonlinear Algebraic Equations with Simple and Multiple Roots

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1 Citation (Scopus)

Abstract

In this paper, we present, analyze, and test a family of high order iterative methods for finding simple and multiple roots of nonlinear algebraic equations of the form f(x) = 0. The proposed method can achieve convergence of order p, where p≥ 2 is a positive integer. The standard Newton–Raphson method (p= 2 ) and the Chebyshev’s method (p= 3 ) are both special cases of this family of methods. The pth order method requires evaluation of the function and its derivative up to order p- 1 at each step. Several numerical experiments are provided to validate the theoretical order of convergence for nonlinear functions with simple and multiple roots.

Original languageEnglish (US)
Pages (from-to)1119-1133
Number of pages15
JournalInternational Journal of Applied and Computational Mathematics
Volume3
DOIs
StatePublished - Dec 1 2017

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Multiple Roots
Nonlinear Algebraic Equations
High-order Methods
Nonlinear equations
Numerical methods
Numerical Methods
Order of Convergence
Iterative methods
Derivatives
Chebyshev's Method
Newton-Raphson
Nonlinear Function
Experiments
Numerical Experiment
Iteration
Derivative
Integer
Family
Evaluation

Keywords

  • High order iterative methods
  • Newton’s method
  • Nonlinear equations
  • Order of convergence
  • Root-finding problem
  • Simple and multiple roots

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

Cite this

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abstract = "In this paper, we present, analyze, and test a family of high order iterative methods for finding simple and multiple roots of nonlinear algebraic equations of the form f(x) = 0. The proposed method can achieve convergence of order p, where p≥ 2 is a positive integer. The standard Newton–Raphson method (p= 2 ) and the Chebyshev’s method (p= 3 ) are both special cases of this family of methods. The pth order method requires evaluation of the function and its derivative up to order p- 1 at each step. Several numerical experiments are provided to validate the theoretical order of convergence for nonlinear functions with simple and multiple roots.",
keywords = "High order iterative methods, Newton’s method, Nonlinear equations, Order of convergence, Root-finding problem, Simple and multiple roots",
author = "Mahboub Baccouch",
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AB - In this paper, we present, analyze, and test a family of high order iterative methods for finding simple and multiple roots of nonlinear algebraic equations of the form f(x) = 0. The proposed method can achieve convergence of order p, where p≥ 2 is a positive integer. The standard Newton–Raphson method (p= 2 ) and the Chebyshev’s method (p= 3 ) are both special cases of this family of methods. The pth order method requires evaluation of the function and its derivative up to order p- 1 at each step. Several numerical experiments are provided to validate the theoretical order of convergence for nonlinear functions with simple and multiple roots.

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