Solving equations with the aid of inverse rational interpolation functions

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.

Crăciun Iancu, Ion Păvăloiu

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

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

[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.

Scanned paper.

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).

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