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<p>In this work, we present a novel optimal iterative solver that is highly effective and accurate, achieving fourth-order convergence. The proposed method adheres to the Kung–Traub conjecture, with an efficiency index of 1.5874. The solver utilizes a combination of local and semi-local analyses to improve its runtime and rate of convergence. The basins of attraction are drawn to investigate stability and to illustrate the implementation of the optimal fourth-order numerical root-solver for solving nonlinear systems. Numerical experiments demonstrate the versatility of the solver across various mathematical contexts, including polynomiography, and show superior performance compared to several existing optimal solvers. Application problems from different scientific fields, such as kinetics, heat transfer, neurophysiology, and particle physics, are numerically solved to further demonstrate the potential of the proposed optimal solver.</p>

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A stable optimal solver for nonlinear equations with semilocal convergence using majorizing sequences in banach spaces

  • 期刊论文

  • 发表日期：

    2025年7月4日

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  • Sania Qureshi,
  • Ioannis K. Argyros,
  • Higinio Ramos,
  • Amanullah Soomro,
  • Paras Nizamani,
  • G Thangkhenpau,
  • Sunil Panday

摘要

In this work, we present a novel optimal iterative solver that is highly effective and accurate, achieving fourth-order convergence. The proposed method adheres to the Kung–Traub conjecture, with an efficiency index of 1.5874. The solver utilizes a combination of local and semi-local analyses to improve its runtime and rate of convergence. The basins of attraction are drawn to investigate stability and to illustrate the implementation of the optimal fourth-order numerical root-solver for solving nonlinear systems. Numerical experiments demonstrate the versatility of the solver across various mathematical contexts, including polynomiography, and show superior performance compared to several existing optimal solvers. Application problems from different scientific fields, such as kinetics, heat transfer, neurophysiology, and particle physics, are numerically solved to further demonstrate the potential of the proposed optimal solver.