Solving nonlinear equations by a derivative-free form of the King's family with memory
For researchers in numerical analysis, this method improves computational efficiency for solving nonlinear equations by increasing convergence order without extra function evaluations.
The paper introduces a derivative-free three-point iterative method with memory for solving nonlinear equations, achieving 12th order convergence with only four function evaluations per iteration, outperforming related methods in high-precision computations.
In this paper, we present an iterative three-point method with memory based on the family of King's methods to solve nonlinear equations. This proposed method has eighth order convergence and costs only four function evaluations per iteration which supports the Kung-Traub conjecture on the optimal order of convergence. An acceleration of the convergence speed is achieved by an appropriate variation of a free parameter in each step. This self accelerator parameter is estimated using Newton's interpolation polynomial of fourth degree. The order of convergence is increased from 8 to 12 without any extra function evaluation. Consequently, this method, possesses a high computational efficiency. Finally, a numerical comparison of the proposed method with related methods shows its effectiveness and performance in high precision computations.