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.

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.