Two novel efficient memory-based multi-point iterative methods for solving nonlinear equations

Abstract

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.

Keywords

• nonlinear equation
• roots
• efficiency index
• iterative method
• with-memory methods