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
(Tiberiu Popoviciu Institute of Numerical Analysis)
Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)
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
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).
About this paper
Journal
Seminar on functional analysis and numerical methods,
Preprint
Publisher Name
“Babes-Bolyai” University,
Faculty of Mathematics,
Research Seminars
DOI
Not available yet.
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.