Real Algebraic Numbers: Complexity Analysis and ... · Real Algebraic Numbers: Complexity Analysis and Experimentations Ioannis Z. Emiris, Bernard Mourrain, Elias P. Tsigaridas To

Real Algebraic Numbers: Complexity Analysis and

Experimentations

Ioannis Z. Emiris, Bernard Mourrain, Elias P. Tsigaridas

To cite this version:

Ioannis Z. Emiris, Bernard Mourrain, Elias P. Tsigaridas. Real Algebraic Numbers: ComplexityAnalysis and Experimentations. P. Hertling, C. Hoffmann, W. Luther and N. Revol. ReliableImplementations of Real Number Algorithms: Theory and Practice, 2008, Dagsthul, Germany.Springer, 5045, pp.57-82, 2008, Lecture Notes in Computer Science. <inria-00071370>

HAL Id: inria-00071370

https://hal.inria.fr/inria-00071370

Submitted on 23 May 2006

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Thème SYM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Real Algebraic Numbers: Complexity Analysis andExperimentations

I.Z. Emiris — B. Mourrain — E. Tsigaridas

N° 5897

Avril 2006

Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France)

Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65

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LhfFRqSHqGH

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β, βu¥mp[_

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2√

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αI , βI ∈ I¶

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Iu=k

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2 x± y2 = 0

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RI′

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d

2−|J |√3

|J |,

¿YÀ

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I ∈ I !

N ≤ ∑I∈I lg |b−a|

|I|≤ |I| lg |b − a| − ∑

I∈I lg |I |≤ |I| lg |b − a| + |I| − ∑

I∈I lg |αI − βI | Prop.QÖ

´&i ¿FtÀ>&_/[Öt_§ −∑I∈I′ lg |αI − βI | ≤ (d − 1) lg(M(f)) + ( d

2 + |I ′|) lg d − |I ′| lg√

3. kuv^ù¥s=|z

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N < |I| + |I| lg |b − a| − ∑I∈I lg |αI − βI |

≤ d + d(τ + 1) + (d − 1)(2τ + lg(d + 1)) + 2d lg d= O (dτ + d lg d) .

2

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αI , βI

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1 ≤ i ≤ m¶

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α ∈ Ralg

α ∼=

(P (X), I) r uwqG P (X) ∈ Z[X ] W0pR ohSE

P (α) = 0 r I = [a, b] r a, b,∈ QE

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2

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Ai(γ) > 0§Bj(γ) < 0

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