Numerical Algorithms

, Volume 58, Issue 4, pp 513–527

# Accurate fourteenth-order methods for solving nonlinear equations

• Parviz Sargolzaei
• Fazlollah Soleymani
Original Paper

## Abstract

We establish new iterative methods of local order fourteen to approximate the simple roots of nonlinear equations. The considered three-step eighth-order construction can be viewed as a variant of Newton’s method in which the concept of Hermite interpolation is used at the third step to reduce the number of evaluations. This scheme includes three evaluations of the function and one evaluation of the first derivative per iteration, hence its efficiency index is 1.6817. Next, the obtained approximation for the derivative of the Newton’s iteration quotient is again taken into consideration to furnish novel fourteenth-order techniques consuming four function and one first derivative evaluations per iteration. In providing such new fourteenth-order methods, we also take a special heed to the computational burden. The contributed four-step methods have 1.6952 as their efficiency index. Finally, various numerical examples are given to illustrate the accuracy of the developed techniques.

## Keywords

Nonlinear equations Three-step methods Four-step methods Efficiency index Order of convergence Simple root

65H05 65B99

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