An algorithm in the solving of equations by interpolation

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 \(\omega_{k}^{\prime}\left( y\right) \).

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

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

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

Scanned paper.

PDF-Latex version of the paper.

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).

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