Chebyshev-Halley methods for analytic functions
type: Item
instance of: scholarly article
title: Chebyshev-Halley methods for analytic functions
zbMATH Open document ID: 1146.65043
author: Naoki Osada
published in: Journal of Computational and Applied Mathematics
publication date: 2008-06-19T00:00:00Z
review text: The Chebyshev-Halley iteration method \[ z^{n+1}=z^n - u_f(z^n)\left[1+\frac{L_f(z^n)}{2(1-\alpha L_f(z^n))}\right] \] is discussed for approximating zeros of an analytic function \(f(z)\). Here \(u_f(z)=\frac{f(z)}{f'(z)}\) and \(L_f(z)=\frac{f(z)f''(z)}{(f'(z))^2}\). \(\alpha\) is a real constant. This method is firstly modified for approximating a multiple zero of an analytic function. Locally cubic convergence is established if the multiplicity of the zero is known. Furthermore, the method is extended to a family of simultaneous methods for finding all distinct zeros of a polynomial. This family includes the Halley-like method, the Euler-Chebyshev-like method and the Schröder-like method. Numerical examples are used to illustrate efficiency of the proposed methods.
reviewed by: Zhen Mei
Mathematics Subject Classification ID: 65H05
Mathematics Subject Classification ID: 65E05
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 5290915
zbMATH Keywords: multiple zero
zbMATH Keywords: analytic function
MaRDI profile type: Publication
full work available at URL: j.cam.2007.06.020
OpenAlex ID: W1966457467
cites work: A construction of attracting periodic orbits for some classical third-order iterative methods
cites work: Q4325644
cites work: Q5839993
cites work: A local convergence theorem for the super-Halley method in a Banach space
cites work: Q3279563
cites work: An algorithm for the total, or partial, factorization of a polynomial
cites work: Further Applications of Circular Arithmetic: Schroeder-Like Algorithms with Error Bounds for Finding Zeros of Polynomials
cites work: A family of Chebyshev-Halley type methods in Banach spaces
cites work: A family of root finding methods
cites work: Some modifications of Laguerre's method
cites work: Q4072022
cites work: Q4304466
cites work: A family of chebyshev-halley type methods
cites work: On locating clusters of zeros of analytic functions
cites work: An optimal multiple root-finding method of order three
cites work: Asymptotic error constants of cubically convergent zero finding methods
cites work: A one parameter family of locally quartically convergent zero-finding methods
cites work: Q4060241
cites work: Q4928185
cites work: A new fourth-order family of simultaneous methods for finding polynomial zeros
cites work: Family of simultaneous methods of Hansen--Patrick's type
cites work: On the \(R\)-order of some accelerated methods for the simultaneous finding of polynomial zeros
cites work: Q3925805
cites work: Q3957862
DOI: 10.1016/J.CAM.2007.06.020
description: scientific article; zbMATH DE number 5290915