An algorithm in the solving of equations by interpolation

Abstract

Consider the nonlinear equation in \(R\), \(f\left( x\right) =0\), where \(f:A\rightarrow B \), \((A,B\subseteq \mathbb{R})\) which is assumed bijective. The Lagrange inverse interpolation polynomial leads to an iterative method of the form \[x_{i+n+2}=L_{n}[ y_{i+1},y_{i+2},…,y_{i+n+1}:f_{|0}^{-1}],\ \quad x_{1},x_{2},…,x_{n+1}\in I.\] At each iterative step we need to compute the values \(\omega_{k}\left( x\right) \) and \(\omega_{k}^{\prime}\left( y_{i}\right) ,\ i=k,\ldots,k+n\), where \(\omega_{k}\left( y\right) =\Pi_{i=k}^{k_{n}}\left( y-y_{k}\right) \). We give an algorithm for computing \(\omega_{k+1}\left( 0\right) \) and \(\omega_{k+1}^{\prime}\left( y_{i}\right) \) using \(\omega_{k}\left(0\right) \) and \(\omega_{k}^{\prime}\left( y\right) \).

Authors

Ion Păvăloiu

Title

Original title (in French)

Un algorythme de calcul dans la résolution des equations par interpolation

English translation of the title

An algorithm in the solving of equations by interpolation

Keywords

nonlinear equation in \(\mathbb{R}\); inverse interpolation; iterative method; Lagrange polynomial; inverse Lagrange interpolation polynomial

References

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

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About this paper

Cite this paper as:

I. Păvăloiu, Un algorythme de calcul dans la résolution des equations par interpolation, Seminar on functional analysis and numerical methods. Preprint no. 1 (1987), pp. 130-134 (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.

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