# Solving equations with the aid of inverse rational interpolation functions

## 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

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

 Pavaloiu, I., Rezolvarea ecuatiilor prin interpolare. Editura Dacia, Cluj, 1981.

 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.

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##### 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.