A robust family of optimal fourth-order iteration schemes for multiple roots with applications

This article aims to introduce and analyze a novel class of iterative methods designed specifically to ensure robust and accurate approximation of multiple roots. These methods are derived from Newton-type iterative techniques and are characterized by their higher-order convergence properties. Utilizing the weight function technique, the proposed methods require just three functional evaluations per computation stage to achieve fourth-order convergence. Notably, the theoretical convergence characteristics of the proposed family exhibit symmetrical properties in scenarios involving roots with varying multiplicities. This symmetry motivates a generalized presentation of results confirming the convergence order across different cases. Furthermore, by appropriately selecting free parameters, existing methods can be obtained as specific instances within this broader methodological framework. In contrast to traditional methods, the proposed methods are capable of converging even when $f^{\prime }\left(x\right)=0$ or $f^{\prime }\left(x\right)\to 0$ near the required root. This notable characteristic broadens the proposed methods’ applicability, enabling them to handle cases where the traditional approaches fail due to the critical points, like the zeros of $f^{\prime }\left(x\right)=0$. The article substantiates its claims through extensive numerical experiments on diverse benchmark problems including the isentropic supersonic flow, Redlich–Kwong equation of state, continuous stirred tank reactor, fractional conversion in a chemical reactor and oscillation problem. Comprehensive comparisons with several established algorithms underscore the superior performance of proposed methods in respect of both CPU efficiency and minimization of absolute residual errors. In addition, to validate the theoretical and numerical results, we compare and visualize the convergence regions of the presented iteration schemes with several existing iteration schemes using diverse nonlinear equations by plotting their basins of attraction in complex plane. The presented iteration methods produce attractor basins in shorter time and exhibit broader convergence regions which confirm their stability and superiority compared to previously known iteration methods.