A novel class of fourth-order derivative-free iterative methods to obtain multiple zeros and their basins of attraction

This article proposed a novel fourth-order class based on weight functions to locate multiple roots numerically, which did not require the evaluation of derivatives at any stage of computation. For particular instances of a multiplicity of zeros, the theoretical convergence behavior of the proposed family has been proven to be symmetrical. This inspired us to show the general results which endorsed the convergence order of the suggested scheme. In addition, some special cases were introduced by using different weight functions. The basins of attraction of the proposed techniques for various parametric values in the complex plane were showcased to verify the stability and convergence features. Finally, we have included a range of problems like Planck's radiation law, the Van der Waals equation, the trajectory of an electron, and a few academic problems. Numerical analyses were performed and compared with other existing algorithms to verify the efficacy and applicability of the proposed techniques.