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A new iterative multi-step method for solving nonlinear equation.

Muhammad Usman1, Javed Iqbal2, Alamgir Khan3

  • 1Department of Mathematics, Government Postgraduate College Mardan, 23200 Khyber Pakhtunkhwa, Pakistan.

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Summary

A new iterative technique efficiently solves nonlinear equations with simple roots, achieving sixteen-order convergence using only five function evaluations. This derivative-free method enhances computational efficiency and accuracy compared to existing approaches.

Keywords:
A New Multi-Step Method for Non Linear EquationConvergence analysisNonlinear equationsNumerical resultsSimple roots

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics

Background:

  • Solving nonlinear equations is crucial in various scientific and engineering disciplines.
  • Existing iterative methods often require high computational costs or derivative evaluations.

Purpose of the Study:

  • To introduce a novel, high-order iterative technique for solving nonlinear equations with simple roots.
  • To enhance computational efficiency and accuracy in numerical root-finding.

Main Methods:

  • Development of an advanced iterative algorithm.
  • Incorporation of finite difference approximations to avoid second derivatives.
  • Theoretical analysis of the method's convergence properties.
  • Numerical experimentation and comparison with established methods.

Main Results:

  • The proposed method achieves a sixteenth-order convergence rate.
  • It requires only five functional evaluations per iteration.
  • The derivative-free approach demonstrates superior accuracy and efficiency over existing techniques.

Conclusions:

  • The new iterative technique offers a highly efficient and accurate solution for nonlinear equations.
  • Its derivative-free nature broadens its applicability.
  • Numerical results confirm its effectiveness and superiority.