An implementation of Vincent's theorem

Akritas, Alkiviadis G.

doi:10.1007/bf01395988

Cited by 23 publications

(34 citation statements)

References 10 publications

(12 reference statements)

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“…Their analysis relies on a conjecture regarding the continued fractions expansions of non-quadratic algebraic irrationals (see p.161 of [20]). They also discuss an earlier analysis by Akritas [2,3].…”

Section: Introductionmentioning

confidence: 75%

“…For the third claim, suppose a complex root α ∈ C(f ) then under I it is mapped to 1/α − 1 = α/|α| 2 − 1. From Lemma 2.1 we have Re(α) ≤ |α| 2 . It follows that Re(1/α − 1) ≤ 0 and so 1/α − 1 ∈ C(f T ) ∪ C(f I ).…”

Section: Definitionsmentioning

confidence: 89%

“…For the same reasons we will assume that a 0 = 0. In practice transformations of the first type are replaced by z → z + c where c is a good integer lower bound on the smallest positive root of f , see [2,3]. Without this optimisation the algorithm is provably exponential, see [6].…”

Section: Definitionsmentioning

confidence: 99%

See 2 more Smart Citations

Tree Breadth of the Continued Fractions Root Finding Method

Kalorkoti

2020

New Trends in Algebras and Combinatorics

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The continued fractions algorithm for isolating the real roots of polynomials with certainty is one of the most efficient known and widely used. It can be viewed as exploring a tree created by a sequence of simple transformations. In this paper we produce new upper bounds for the breadth of the tree that are significantly smaller than the degree of the input polynomial. We also consider the expected breadth under a reasonable distribution and derive a bound, subject to a plausible assumption, that grows logarithmically with the degree and coefficient size.

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Section: Introductionmentioning

confidence: 75%

Section: Definitionsmentioning

confidence: 89%

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Tree Breadth of the Continued Fractions Root Finding Method

Kalorkoti

2020

New Trends in Algebras and Combinatorics

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“…Using Vincent's theorem, Collins and Akritas [13] derived a polynomial subdivision-based algorithm using Descartes' rule of sign. Akritas [3,1] dealt with the exponential behavior of the CF algorithm, by computing the c i 's in the transformations as positive lower bounds of the positive real roots, via Cauchy's bound (for details, see sec. 3).…”

Section: Previous Work and Our Resultsmentioning

confidence: 99%

“…4 holds. Akritas [1,3] replaced a series of X → X + 1 transformations by X → X + b, where b is the positive lower bound (PLB) on the positive roots of the tested polynomial. This was computed by Cauchy's bound [3,26,42].…”

Section: The Cf Algorithmmentioning

confidence: 99%

Univariate Polynomial Real Root Isolation: Continued Fractions Revisited

Tsigaridas

Emiris

2006

Lecture Notes in Computer Science

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We present algorithmic, complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real algebraic numbers. One motivation is to explain the method's good performance in practice. We improve the previously known bound by a factor of dτ , where d is the polynomial degree and τ bounds the coefficient bitsize, thus matching the current record complexity for real root isolation by exact methods. Namely, the complexity bound is OB(d 4 τ 2 ) using the standard bound on the expected bitsize of the integers in the continued fraction expansion. We show how to compute the multiplicities within the same complexity and extend the algorithm to non square-free polynomials. Finally, we present an efficient open-source C++ implementation in the algebraic library synaps, and illustrate its efficiency as compared to other available software. We use polynomials with coefficient bitsize up to 8000 and degree up to 1000.

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Computing in the Field of Complex Algebraic Numbers

Strzeboński

1997

Journal of Symbolic Computation

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In this paper we present two methods of computing with complex algebraic numbers. The first uses isolating rectangles to distinguish between the roots of the minimal polynomial, the second method uses validated numeric approximations. We present algorithms for arithmetic and for solving polynomial equations, and compare implementations of both methods in Mathematica.c 1997 Academic Press Limited Representation of Complex Algebraic NumbersLet K denote the field of complex algebraic numbers, i.e. the field of all complex numbers that are algebraic over the rationals. For every complex algebraic number a there is a unique monic polynomial f with rational coefficients (called the minimal polynomial of a) such that f (a) = 0 and for every polynomial g with rational coefficients if g(a) = 0 then f divides g. The minimal polynomial is necessarily irreducible.To specify an algebraic number we give its minimal polynomial, and also we need to tell which root of this polynomial we have in mind. To distinguish between the roots of a given polynomial we find disjoint isolating sets in the complex plane, such that each set contains exactly one root of the polynomial.The first method described here uses one-point isolating sets for rational roots, open intervals with rational endpoints for real roots, and open rectangles in the complex plane (Cartesian products of open intervals with rational endpoints) for complex roots. For real root isolation we use the method described by Akritas et al. (1994), based on the Descartes' Rule of Signs (see also Akritas and Collins (1976), and Akritas (1980)). Our complex root isolation algorithm is based on Collins and Krandick (1992). An algebraic number is represented by the minimal polynomial and the standard isolating rectangle given by our root isolation procedure-so that the representation of a given algebraic number is unique. (This makes checking equality of two algebraic numbers an easy task, otherwise to check equality we would have to count roots of the minimal polynomial in the intersection of isolating rectangles.)The second approach is to identify roots of a given polynomial based on their variableprecision floating-point approximations (which corresponds to giving circular isolating

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An implementation of Vincent's theorem

Cited by 23 publications

References 10 publications

Tree Breadth of the Continued Fractions Root Finding Method

Tree Breadth of the Continued Fractions Root Finding Method

Univariate Polynomial Real Root Isolation: Continued Fractions Revisited

Computing in the Field of Complex Algebraic Numbers

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