Tangent Graeffe Iteration
Numerical Analysis
2025-10-20 v1 Numerical Analysis
Abstract
Graeffe iteration was the choice algorithm for solving univariate polynomials in the XIX-th and early XX-th century. In this paper, a new variation of Graeffe iteration is given, suitable to IEEE floating-point arithmetics of modern digital computers. We prove that under a certain generic assumption the proposed algorithm converges. We also estimate the error after N iterations and the running cost. The main ideas from which this algorithm is built are: classical Graeffe iteration and Newton Diagrams, changes of scale (renormalization), and replacement of a difference technique by a differentiation one. The algorithm was implemented successfully and a number of numerical experiments are displayed.
Cite
@article{arxiv.math/9908150,
title = {Tangent Graeffe Iteration},
author = {Gregorio Malajovich and Jorge P. Zubelli},
journal= {arXiv preprint arXiv:math/9908150},
year = {2025}
}