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<p>This paper presented a sixth-order iterative scheme for solving nonlinear equations. The method involves four function evaluations per iteration and possesses an efficiency index of nearly 1.56. For establishing the reliability and robustness of the method, a mathematical convergence analysis is provided. Comparisons with other methods of the same order of convergence available in the literature demonstrate the accuracy and reduced cost of the proposed method. Large-scale numerical experimentation on a variety of test functions also establishes the effectiveness of the method. Plots of residual fall demonstrate the scheme’s behaviour of rapid convergence, whereas large-scale stability analysis of the scheme highlights its insensitivity to different conditions. These findings in aggregate highlight the potential of the method for fast and reliable numerical problem-solving in real-world applications.</p>

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An efficient sixth-order numerical iterative scheme for solving nonlinear equations: residual error analysis and stability investigation

  • 期刊论文

  • 发表日期：

    2025年12月17日

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  • Muhammad Hassan,
  • Shajar Abbas,
  • Nashwan Adnan Othman,
  • Muhammad Waseem,
  • Dilsora Abduvalieva,
  • Aiedh Mrisi Alharthi

摘要

This paper presented a sixth-order iterative scheme for solving nonlinear equations. The method involves four function evaluations per iteration and possesses an efficiency index of nearly 1.56. For establishing the reliability and robustness of the method, a mathematical convergence analysis is provided. Comparisons with other methods of the same order of convergence available in the literature demonstrate the accuracy and reduced cost of the proposed method. Large-scale numerical experimentation on a variety of test functions also establishes the effectiveness of the method. Plots of residual fall demonstrate the scheme’s behaviour of rapid convergence, whereas large-scale stability analysis of the scheme highlights its insensitivity to different conditions. These findings in aggregate highlight the potential of the method for fast and reliable numerical problem-solving in real-world applications.