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On generalization based on bi et Al. Iterative methods with eighth-order convergence for solving nonlinear equations.

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics

Background:

  • Iterative methods are crucial for solving complex equations.
  • Optimality in iterative methods is often defined by efficiency and convergence rate.
  • Previous work by Bi et al. (2009) laid groundwork for such methods.

Purpose of the Study:

  • To develop a general class of optimal three-step iterative methods.
  • To achieve eighth-order convergence with minimal function evaluations.
  • To satisfy Kung and Traub's conjecture for memoryless optimal methods.

Main Methods:

  • Generalization of Bi et al.'s (2009) iterative schemes.
  • Construction of three-step iterative methods.
  • Analysis of convergence properties, specifically eighth-order.

Main Results:

  • A general optimal three-step class of iterative methods was established.
  • The methods require four functional evaluations per iteration.
  • Eighth-order convergence was achieved, meeting Kung and Traub's conjecture.
  • Numerical implementation demonstrated the methods' efficiency and applicability.

Conclusions:

  • The developed iterative methods are theoretically optimal and practically efficient.
  • The findings contribute to the advancement of numerical methods for equation solving.
  • The methods offer a robust alternative for applications requiring high accuracy and speed.