A New Adaptive Eleventh-Order Memory Algorithm for Solving Nonlinear Equations

In this article, we introduce a novel three-step iterative algorithm with memory for finding the roots of nonlinear equations. The convergence order of an established eighth-order iterative method is elevated by transforming it into a with-memory variant. The improvement in the convergence order is achieved by introducing two self-accelerating parameters, calculated using the Hermite interpolating polynomial. As a result, the R-order of convergence for the proposed bi-parametric with-memory iterative algorithm is enhanced from 8 to 10.5208. Notably, this enhancement in the convergence order is accomplished without the need for extra function evaluations. Moreover, the efficiency index of the newly proposed with-memory iterative algorithm improves from 1.5157 to 1.6011. Extensive numerical testing across various problems confirms the usefulness and superior performance of the presented algorithm relative to some well-known existing algorithms.