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On a new iterative scheme without memory with optimal eighth order.

M Sharifi1, S Karimi Vanani1, F Khaksar Haghani1

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

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Summary
This summary is machine-generated.

Researchers developed a new derivative-free iterative method for solving nonlinear equations. This novel approach maintains the high convergence rate of existing optimal methods, validated by numerical experiments.

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics

Background:

  • Nonlinear equations are fundamental in science and engineering.
  • Existing iterative methods often require derivative computations, limiting their applicability.

Purpose of the Study:

  • To derive and discuss a novel three-step iterative expression for solving nonlinear equations.
  • To present a derivative-free variant of an optimal eighth-order method, preserving its convergence order.

Main Methods:

  • Derivation of a new iterative formula.
  • Analysis of convergence properties.
  • Implementation of numerical experiments to validate theoretical findings.

Main Results:

  • A new three-step iterative expression for solving nonlinear equations was successfully derived.
  • The derivative-free form of the eighth-order method retains its original convergence rate.
  • Numerical results confirm the theoretical efficiency and accuracy of the proposed method.

Conclusions:

  • The proposed derivative-free iterative method offers an efficient alternative for solving nonlinear equations.
  • This advancement expands the applicability of high-order iterative solvers in computational mathematics.