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

TheScientificWorldJournal·2014
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On a new efficient Steffensen-like iterative class by applying a suitable self-accelerator parameter.

Taher Lotfi1, Elahe Tavakoli1

  • 1Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan 68138, Iran.

Thescientificworldjournal
|April 15, 2014
PubMed
Summary

Researchers developed a new class of iterative methods for solving nonlinear equations. These methods achieve a convergence order of 12 without additional function evaluations, significantly enhancing computational efficiency.

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Area of Science:

  • Numerical analysis
  • Computational mathematics
  • Applied mathematics

Background:

  • Nonlinear equations are fundamental in various scientific and engineering disciplines.
  • Iterative methods are commonly used for solving nonlinear equations.
  • Improving the efficiency and convergence speed of these methods is an ongoing research challenge.

Purpose of the Study:

  • To introduce a novel, derivative-free class of Steffensen-like iterative methods.
  • To enhance the convergence order of existing iterative methods.
  • To improve the efficiency index of methods for solving nonlinear equations.

Main Methods:

  • Construction of an optimal eighth-order, three-step, uniparameter iterative method without memory.
  • Estimation of a self-accelerator parameter using Newton's interpolation.
  • Analysis of convergence properties and efficiency index.

Main Results:

  • Achieved a convergence order of 12 without requiring extra function evaluations.
  • Increased the efficiency index from 8^(1/4) to 12^(1/4).
  • Demonstrated the applicability of the proposed methods through numerical examples.

Conclusions:

  • The proposed derivative-free Steffensen-like methods offer a significant improvement in convergence speed and efficiency.
  • The novel parameter estimation technique effectively accelerates the iterative process.
  • These methods provide a valuable tool for efficiently solving nonlinear equations in various applications.