Posts by Emil Cătinaş


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:
y_{n} & =x_{n}-\frac{f\left( x_{n}\right) }{f^{\prime}\left( x_{n}\right)
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

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).


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

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


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



About this paper


Mediterranean Journal Matematics

Publisher Name


Print ISSN


Online ISSN


google scholar link

Related Posts

How many steps still left to x*?

Abstract The high speed of \(x_{k}\rightarrow x^\ast\in{\mathbb R}\) is usually measured using the C-, Q- or R-orders: \begin{equation}\tag{$C$} \lim \frac…

Methods of Newton and Newton-Krylov type

Book summaryLocal convergence results on Newton-type methods for nonlinear systems of equations are studied. Solving of large linear systems by…