An iterative method for algebraic equation by Padé approximation
type: Item
instance of: scholarly article
title: An iterative method for algebraic equation by Padé approximation
zbMATH Open document ID: 0728.65040
DOI: 10.1007/BF02239167
published in: Computing
publication date: 1991-01-01T00:00:00Z
review text: The authors consider iterative formulae with high order of convergence for solving polynomial equations \(f(z)=0\). The numerator of the Padé approximant for \(f(z)/f'(z)\) is derived by combining the algorithm of \textit{B. Viskovatoff} [De la méthode générale pour réduire toutes sortes des quantités en fractions continues. Mémoires de l'Académie Impériale des Sciences de St. Petersburg 1, 226-247 (1803-1806)] with the Euclidean algorithm. The new method has the same order whatever the multiplicity of the zero may be. Numerical examples show that the number of iterations is almost independent of problem. By choosing some parameters and functions, the algorithm can yield the methods of Newton-Raphson, Halley, \textit{I. Kiss} [Z. Angew. Math. Mech. 34, 68-69 (1954; Zbl 0058.337)], \textit{A. W. M. Nourein} [BIT 16, 291-297 (1976; Zbl 0336.65026)] and \textit{T. Pomentale} [Numer. Math. 18, 193-203 (1971; Zbl 0213.418)]. The proposed algorithm is applicable to transcendental equations.
Mathematics Subject Classification ID: 65H05
Mathematics Subject Classification ID: 41A21
zbMATH DE Number: 4203586
zbMATH Keywords: algebraic equations
zbMATH Keywords: iterative formulae
zbMATH Keywords: high order of convergence
zbMATH Keywords: polynomial equations
zbMATH Keywords: Padé approximant
zbMATH Keywords: Euclidean algorithm
zbMATH Keywords: Numerical examples
zbMATH Keywords: transcendental equations
author: Tetsuya Sakurai
author: Tatsuo Torii
author: Hiroshi Sugiura
reviewed by: S. M. Grzegórski
MaRDI profile type: Publication
cites work: Convergence of Multipoint Iterative Methods
cites work: Padésche Näherungsbrüche und Iterationsverfahren höherer Ordnung
cites work: Approximation by Sequences of Padé Approximants in Regions of Meromorphy
cites work: Root determination by use of Padé approximants
cites work: Comments on a root finding method using Padé approximation
cites work: Continued Fractions and Linear Recurrences
cites work: A class of algorithms for obtaining rational approximants to functions which are defined by power series
cites work: Certain continued fractions associated with the Padé table
cites work: Q3921474
cites work: Q3750018
cites work: The Padé table, the Frobenius identities, and the qd algorithm
cites work: A simplified algorithm for calculating the Padé table derived from Baker and Longman schemes
cites work: The Padé Table and Its Relation to Certain Algorithms of Numerical Analysis
cites work: A new look at the Padé table and the different methods for computing its elements
cites work: Computation of Padé approximants and continued fractions
cites work: Q4076249
cites work: A Property of Euclid’s Algorithm and an Application to Padé Approximation
cites work: Padé approximation and its applications. Proceedings of a conference held in Antwerp, Belgium, 1979
full work available at URL: https://doi.org/10.1007/bf02239167
OpenAlex ID: W2107401570
description: scientific article; zbMATH DE number 4203586