Efficient Four-Parametric with-and-without-Memory Iterative Methods Possessing High Efficiency Indices

Abstract EN

We construct a family of derivative-free optimal iterative methods without memory to approximate a simple zero of a nonlinear function.

Error analysis demonstrates that the without-memory class has eighth-order convergence and is extendable to with-memory class.

The extension of new family to the with-memory one is also presented which attains the convergence order 15.5156 and a very high efficiency index 15.51561/4≈1.9847.

Some particular schemes of the with-memory family are also described.

Numerical examples and some dynamical aspects of the new schemes are given to support theoretical results.