## Abstract

We study the convergence of an iterative method for solving the equation \(f\left( x\right) =0,\ f:A\rightarrow B\), \(A,B\subseteq \mathbb{R}\), \(f\) assumed bijective. Denote by \(\varphi _{a,b,\alpha,\beta}\) the set of functions of the form \[\varphi \left( x\right) =\frac{ax+b}{\alpha x+\beta},\] \(\alpha \neq0\) with \(\varphi :\mathbb{R}\backslash \{ -\textstyle\frac{\beta}{\alpha}\} \rightarrow \mathbb{R}\). If \(x_{1},x_{2},x_{3}\in I\) denote \(y_{i}=f\left( x_{i}\right) ,\ i=1,2,3\). We are interested in determining a function \(\varphi\) such that \(\varphi \left(y_{i}\right) =x_{i},i=1,2,3\), which is the inverse rational interpolation function. The value \(\varphi \left( 0\right)\) is a new approximation of the solution of equation \(f\left( x\right) =0\). We study the convergence of the iterative method generated above.

## Authors

Crăciun Iancu, Ion Păvăloiu

## Title

### Original title (in French)

*La resolution des équations par interpolation inverse de type Hermite*

### English translation of the title

*Solving equations with the aid of inverse rational interpolation functions*

## Keywords

rational function; nonlinear equation in R; iterative method; inverse interpolation

## References

[1] Imamov, A., *Resenie nelineinih urvnenii metodomi spalin-interpolirovanie*. Medodi splain-funktii, Akademia Nauk SSSR. Novosibirsk, 81 (1979) , 74-80.

[1] Pavaloiu, I., *Rezolvarea ecuatiilor prin interpolare*. Editura Dacia, Cluj, 1981.

[3] Iancu, C., Pavaloiu, I., *Resolutions des equations a l’aide des fonctions spline d’interpolation inverse*. Seminar of Functional analysis and Numerical Methods, “Babes-Bolyai” University, Faculty of Mathematics, Research Seminaries, Preprint Nr. 1, (1984), 97-1-4.

## About this paper

##### Cite this paper as:

C. Iancu, I. Păvăloiu, *Resolution des equations à l’aide des fonctions rationnelles d’interpolation invèrse*, Seminar on functional analysis and numerical methods, Preprint no. 1 (1985), pp. 71-78 (in French).

##### Journal

Seminar on functional analysis and numerical methods,

Preprint

##### Publisher Name

“Babes-Bolyai” University,

Faculty of Mathematics,

Research Seminars

##### DOI

Not available yet.