Bounds on a polynomial

Lane, Jeff; Riesenfeld, Richard F.

doi:10.1007/bf01934076

Cited by 112 publications

(43 citation statements)

References 10 publications

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“…This paper is focused on the comparison of black-box implementations of state-of-the-art algorithms for isolating real roots of univariate polynomials over the integers, that is, the output is supposed to consists of intervals with rational endpoints, each containing exactly one real root of the polynomial We consider two classes of algorithms for real root isolation of integer polynomials. The first class consists of the subdivision algorithms [7,11,23,27,8,29,16], which exploit either Sturm's theorem or Descartes' rule of signs. The second class contains the Continued Fraction algorithms [1,32,30], which are based on the continued fraction expansion of the roots of the polynomial.…”

Section: Introductionmentioning

confidence: 99%

Experimental evaluation and cross-benchmarking of univariate real solvers

Hemmer

Tsigaridas

Zafeirakopoulos

et al. 2009

Proceedings of the 2009 Conference on Symbolic Numeric Computation

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Real solving of univariate polynomials is a fundamental problem with several important applications. This paper is focused on the comparison of black-box implementations of state-of-the-art algorithms for isolating real roots of univariate polynomials over the integers. We have tested 9 different implementations based on symbolic-numeric methods, Sturm sequences, Continued Fractions and Descartes' rule of sign. The methods under consideration were developed at the GALAAD group at INRIA, the VEGAS group at LO-RIA and the MPI-Saarbrücken. We compared their sensitivity with respect to various aspects such as degree, bitsize or root separation of the input polynomials. Our datasets consist of 5 000 polynomials from many different settings, which have maximum coefficient bitsize up to bits 8 000, and the total running time of the experiments was about 50 hours. Thereby, all implementations of the theoretically exact methods always provided correct results throughout this extensive study. For each scenario we identify the currently most adequate method, and we point to weaknesses in each approach, which should lead to further improvements. Our results indicate that there is no "best method" overall, but one can say that for most instances the solvers based on Continued Fractions are among the best methods. To the best of our knowledge, this is the largest number of tests for univariate real solving up to date.

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Section: Introductionmentioning

confidence: 99%

Experimental evaluation and cross-benchmarking of univariate real solvers

Hemmer

Tsigaridas

Zafeirakopoulos

et al. 2009

Proceedings of the 2009 Conference on Symbolic Numeric Computation

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“…The connection between root isolation in the power basis using the Descartes method, and in the Bernstein basis using de Casteljau's algorithm and the variation-diminishing property of Bézier curves was already pointed out by Lane and Riesenfeld [13], but this connection is often unclear in the literature. In Section 2, we provide a general framework for viewing both as a form of the Descartes method.…”

Section: Introductionmentioning

confidence: 99%

Almost tight recursion tree bounds for the Descartes method

Eigenwillig

Sharma²,

Yap³

2006

Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation

136

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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 aiX i 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.

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“…This rule is traditionally stated for the power basis and the interval (0, ∞); see [16] for a proof with historical references. The Bernstein formulation appears in [5][6][7][8].…”

Section: The Descartes Methods In the Bernstein Basismentioning

confidence: 99%

“…The Descartes method can be formulated for polynomials in the usual power basis [2,1,3,4] and for polynomials in the Bernstein basis [5][6][7][8]. The early work concentrated on polynomials with integer coefficients.…”

Section: Comparison To Related Workmentioning

confidence: 99%

A Descartes Algorithm for Polynomials with Bit-Stream Coefficients

Eigenwillig

Kettner

Krandick

et al. 2005

Computer Algebra in Scientific Computing

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Abstract. The Descartes method is an algorithm for isolating the real roots of square-free polynomials with real coefficients. We assume that coefficients are given as (potentially infinite) bit-streams. In other words, coefficients can be approximated to any desired accuracy, but are not known exactly. We show that a variant of the Descartes algorithm can cope with bit-stream coefficients. To isolate the real roots of a square-free real polynomial q(x) = q n x n + . . . + q 0 with root separation ρ, coefficients |q n | ≥ 1 and |q i | ≤ 2 τ , it needs coefficient approximations to O(n(log(1/ρ) + τ)) bits after the binary point and has an expected cost of O(n 4 (log(1/ρ) + τ) 2 ) bit operations.

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Bounds on a polynomial

Abstract: Methods for computing th e maximum and minimum of a polynomial with real coefficients in the inte rval [0 , 1] are de scribed, and certain bounds are given.

Cited by 112 publications

References 10 publications

Experimental evaluation and cross-benchmarking of univariate real solvers

Experimental evaluation and cross-benchmarking of univariate real solvers

Almost tight recursion tree bounds for the Descartes method

A Descartes Algorithm for Polynomials with Bit-Stream Coefficients

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