Some variant of Newton's method with third-order convergence.
type: Item
instance of: scholarly article
title: Some variant of Newton's method with third-order convergence.
zbMATH Open document ID: 1037.65051
DOI: 10.1016/S0096-3003(02)00238-2
author: E. Sormani
author: M. Frontini
published in: Applied Mathematics and Computation
publication date: 2003-09-09T00:00:00Z
review text: Newton's method for the approximation of the root \(x\) of a system of nonlinear equations can be interpreted as a computation of the indefinite integral arising from Newton's theorem. Using various quadrature formulas one can obtain various iterative processes, the so-called modifications of Newton method. An error analysis providing the higher order of convergence is proposed and the best efficiency, in the term of function evaluations, of two of these methods is provided. It is proved that if the order of the quadrature formula is at last one and if \(x\) is a simple root, then the order of the method is always three, independently of the order quadrature formula. Among this class of methods the most efficient ones are the iterative methods using the mid/point and trapezoidal quadrature rules.
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 30E20
zbMATH DE Number: 1978093
zbMATH Keywords: Newton method
zbMATH Keywords: Newton's theorem
zbMATH Keywords: quadrature formula
zbMATH Keywords: trapezoidal formula
zbMATH Keywords: convergence
zbMATH Keywords: system of nonlinear equations
zbMATH Keywords: indefinite integral
zbMATH Keywords: error analysis
reviewed by: Jiří Vaníček
MaRDI profile type: Publication
cites work: Q3702408
cites work: Q4354453
cites work: A variant of Newton's method with accelerated third-order convergence
description: scientific article; zbMATH DE number 1978093