@article{Mittal2025, 
author = {Shubham Kumar Mittal and Sunil Panday and Lorentz Jäntschi and Liviu C. Bolunduţ},
title = {Two novel efficient memory-based multi-point iterative methods for solving nonlinear equations},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {3},
pages = {5421-5443},
keywords = {nonlinear equation, roots, efficiency index, iterative method, with-memory methods},
url = {https://www.sciopen.com/article/10.3934/math.2025250},
doi = {10.3934/math.2025250},
abstract = {In order to solve nonlinear equations, we introduce two new three-step with-memory iterative methods in this paper. We have improved the order of convergence of a well-known optimal eighth-order iterative method by extending it into two with-memory methods using one and two self-accelerating parameters, respectively. The self-accelerating parameters that increase the convergence order are computed using the Hermite interpolating polynomial. The newly proposed uni-parametric and bi-parametric with-memory iterative methods (IM) improved the R-order of convergence of the existing eighth-order method from    8 to    10 and    10.7446, respectively. Furthermore, the efficiency index has increased from    1.6818 to    1.7783 and    1.8105, respectively. In addition, this improvement in convergence order and efficiency index can be obtained without using any extra function evaluations. Extensive numerical testing on a wide range of problems demonstrates that the proposed methods are more efficient than some well-known existing methods.}
}