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Journal Articles Journal of Computational and Applied Mathematics Year : 2004

Efficient isolation of polynomial's real roots

Abstract

This paper revisits an algorithm isolating the real roots of a univariate polynomial using Descartes' rule of signs. It follows work of Vincent, Uspensky, Collins and Akritas, Johnson, Krandick. Our first contribution is a generic algorithm which enables one to describe all the known algorithms based on Descartes' rule of sign and the bissection strategy in a unified framework. Using that framework, a new algorithm is presented, which is optimal in terms of memory usage and as fast as both Collins and Akritas' algorithm and Krandick variant, independently of the input polynomial. From this new algorithm, we derive an adaptive semi-numerical version, using multi-precision interval arithmetic. We finally show that these critical optimizations have important consequences since our new algorithm still works with huge polynomials, including orthogonal polynomials of degree $1000$ and more, which were out of reach of previous methods.

Dates and versions

inria-00099941 , version 1 (26-09-2006)

Identifiers

Cite

Fabrice Rouillier, Paul Zimmermann. Efficient isolation of polynomial's real roots. Journal of Computational and Applied Mathematics, 2004, 162 (1), pp.33-50. ⟨10.1016/j.cam.2003.08.015⟩. ⟨inria-00099941⟩
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