https://doi.org/10.65770/AZRP8354
ABSTRACT
Solving nonlinear equations is a fundamental problem in numerical analysis, as analytical solutions are generally unattainable for most practical cases. While multipoint iterative methods incorporating weight functions have proven effective in achieving higher convergence orders without additional derivative evaluations, existing methods such as the fifth-order scheme of Liu et al. (2021) still leave room for improvement in convergence order within the same computational cost. This study proposes a three-step iterative method constructed by modifying Liu’s scheme through the introduction of a frozen derivative strategy combined with a linear weight function governed by a free scalar parameter. Convergence analysis via Taylor series expansion proves that the proposed method achieves sixth-order convergence for general parameter values and seventh-order convergence for a specific parameter choice. Numerical simulations on five standard test functions, benchmarked against Liu’s fifth-order methods, Bi-Ren-Wu’s seventh-order method, and Tao-Wan’s seventh-order method, confirm that the proposed scheme converges rapidly, produces highly accurate approximations, and yields computational convergence orders consistent with the theoretical results. These findings establish the proposed method as an efficient and competitive alternative for solving nonlinear equations.
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