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Journal of Applied Mathematics
Volume 2015, Article ID 413816, 14 pages
http://dx.doi.org/10.1155/2015/413816
Research Article

The Polynomial Pivots as Initial Values for a New Root-Finding Iterative Method

Department of Continuum Mechanics and Theory of Structures, Universitat Politècnica de València, 46022 Valencia, Spain

Received 15 May 2014; Accepted 11 December 2014

Academic Editor: Luigi Muglia

Copyright © 2015 Mario Lázaro et al. 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

A new iterative method for polynomial root-finding based on the development of two novel recursive functions is proposed. In addition, the concept of polynomial pivots associated with these functions is introduced. The pivots present the property of lying close to some of the roots under certain conditions; this closeness leads us to propose them as efficient starting points for the proposed iterative sequences. Conditions for local convergence are studied demonstrating that the new recursive sequences converge with linear velocity. Furthermore, an a priori checkable global convergence test inside pivots-centered balls is proposed. In order to accelerate the convergence from linear to quadratic velocity, new recursive functions together with their associated sequences are constructed. Both the recursive functions (linear) and the corrected (quadratic convergence) are validated with two nontrivial numerical examples. In them, the efficiency of the pivots as starting points, the quadratic convergence of the proposed functions, and the validity of the theoretical results are visualized.