Abstract and Applied Analysis
Volume 2011 (2011), Article ID 495312, 6 pages
http://dx.doi.org/10.1155/2011/495312
Research Article

On the Distance to a Root of Polynomials

Somjate Chaiya1,2

1Department of Mathematics, Faculty of Science, Silpakorn University, Nakorn Pathom 73000, Thailand
2Centre of Excellence in Mathematics, Commission on Higher Education, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 29 July 2011; Accepted 1 September 2011

Academic Editor: Natig Atakishiyev

Copyright © 2011 Somjate Chaiya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In 2002, Dierk Schleicher gave an explicit estimate of an upper bound for the number of iterations of Newton's method it takes to find all roots of polynomials with prescribed precision. In this paper, we provide a method to improve the upper bound given by D. Schleicher. We give here an iterative method for finding an upper bound for the distance between a fixed point 𝑧 in an immediate basin of a root 𝛼 to 𝛼 , which leads to a better upper bound for the number of iterations of Newton's method.