# A new optimal method of order four of Hermite–Steffensen type

## Abstract

We introduce a new method for solving nonlinear equations in R, which uses three function evaluations at each step and has convergence order four, being, therefore, optimal in the sense of Kung and Traub:
\begin{align*}
y_{n} & =x_{n}-\frac{f\left( x_{n}\right) }{f^{\prime}\left( x_{n}\right)
}\label{f.1.10}\\
x_{n+1} & =y_{n}-\frac{\left[ x_{n},x_{n},y_{n};f\right] f^{2}\left(
x_{n}\right) }{\left[ x_{n},y_{n};f\right] ^{2}f^{\prime}\left(
x_{n}\right) },\ \; \; \; \;n=0,1,\ldots
\end{align*}

The method is based on the Hermite inverse interpolatory polynomial of degree two.

Under certain additional assumptions, we obtain convergence domains (sided intervals) larger than the usual ball convergence sets, and, moreover, with monotone convergence of the iterates. The method has larger convergence domains than of the methods which use intermediate points of the type $$y_n=x_n+f(x_n)$$ (as the later may not yield convergent iterates when $$|f|$$ grows fast near the solution).

## Authors

Ion Păvăloiu
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy

Emil Cătinaș
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy

## Keywords

Nonlinear equations in R; inverse interpolation; Hermite-Steffensen type method; computational convergence order.

## Paper coordinates

I. Păvăloiu, E. Cătinaș, A new optimal method of order four of Hermite-Steffensen type, Mediterr. J. Math. (2022) 19:147
https://doi.org/10.1007/s00009-022-02030-5

## PDF

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##### Journal

Mediterranean Journal Matematics

Springer

16605454

16605446