Almost tight recursion tree bounds for the Descartes method (2006)
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| Venue: | In Proc. Int. Symp. on Symbolic and Algebraic Computation |
| Citations: | 24 - 2 self |
BibTeX
@INPROCEEDINGS{Eigenwillig06almosttight,
author = {Arno Eigenwillig and Vikram Sharma and Chee K. Yap},
title = {Almost tight recursion tree bounds for the Descartes method},
booktitle = {In Proc. Int. Symp. on Symbolic and Algebraic Computation},
year = {2006},
pages = {71--78},
publisher = {ACM Press}
}
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Abstract
We give a unified (“basis free”) framework for the Descartes method for real root isolation of square-free real polynomials. This framework encompasses the usual Descartes ’ rule of sign method for polynomials in the power basis as well as its analog in the Bernstein basis. We then give a new bound on the size of the recursion tree in the Descartes method for polynomials with real coefficients. Applied to polynomials A(X) = P n i=0 aiXi with integer coefficients |ai | < 2 L, this yields a bound of O(n(L + log n)) on the size of recursion trees. We show that this bound is tight for L = Ω(log n), and we use it to derive the best known bit complexity bound for the integer case.
