InforLorV4, PascalFrancis, Corpus, bibRecord, 000314

On the complexity of real root isolation using continued fractions

Identifieur interne : 000314 ( PascalFrancis/Corpus ); précédent : 000313; suivant : 000315

On the complexity of real root isolation using continued fractions

Auteurs : Elias P. Tsigaridas ; Ioannis Z. Emiris

Source :

Theoretical computer science [ 0304-3975 ] ; 2008.

RBID : Pascal:08-0241602

Descripteurs français

Pascal (Inist)
- Complexité algorithme, Fraction continue, Algorithmique, Implémentation, Nombre entier, Polynôme, Développement fraction continue, Expansion, Nombre réel, Nombre algébrique, Performance, Multiplicité, Algorithme, Complétude, Efficacité, Logiciel, Informatique théorique, 58A25, Pire cas, 13H15, 68Wxx, 68N01.

English descriptors

KwdEn :
- Algebraic number, Algorithm, Algorithm complexity, Algorithmics, Completeness, Computer theory, Continued fraction expansion, Continued fractions, Efficiency, Expansion, Implementation, Integer, Multiplicity, Performance, Polynomial, Real number, Software.

Abstract

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 derive an expected complexity bound of ========Otilde;_B (d⁶ + d⁴τ²), where d is the polynomial degree and τ bounds the coefficient bit size, using a standard bound on the expected bit size of the integers in the continued fraction expansion, thus matching the current worst-case complexity bound for real root isolation by exact methods (Sturm, Descartes and Bernstein subdivision). Moreover, using a homothetic transformation we improve the expected complexity bound to OB (d³τ). We compute the multiplicities within the same complexity and extend the algorithm to non-square-free polynomials. Finally, we present an open-source C++ implementation in the algebraic library SYNAPS, and illustrate its completeness and efficiency as compared to some other available software. For this we use polynomials with coefficient bit size up to 8000 bits and degree up to 1000.

Notice en format standard (ISO 2709)

Pour connaître la documentation sur le format Inist Standard.

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C01	`01`		`ENG`	@0 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 derive an expected complexity bound of ========Otilde;_B (d⁶ + d⁴τ²), where d is the polynomial degree and τ bounds the coefficient bit size, using a standard bound on the expected bit size of the integers in the continued fraction expansion, thus matching the current worst-case complexity bound for real root isolation by exact methods (Sturm, Descartes and Bernstein subdivision). Moreover, using a homothetic transformation we improve the expected complexity bound to OB (d³τ). We compute the multiplicities within the same complexity and extend the algorithm to non-square-free polynomials. Finally, we present an open-source C++ implementation in the algebraic library SYNAPS, and illustrate its completeness and efficiency as compared to some other available software. For this we use polynomials with coefficient bit size up to 8000 bits and degree up to 1000.
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A30	`01`	`1`	`ENG`	`@1 Computational Algebraic Geometry and Applications. Conference @3 Nice FRA @4 2006-06-02`

Format Inist (serveur)

NO :	PASCAL 08-0241602 INIST
ET :	On the complexity of real root isolation using continued fractions
AU :	TSIGARIDAS (Elias P.); EMIRIS (Ioannis Z.); BUSE (L.); ELKADI (M.); MOURRAIN (B.)
AF :	INRIA -LORIA Lorraine, project VEGAS, 615, rue du Jardin Botanique, B.P. 101/54602 Villers-des-Nancy/France (1 aut.); Department of Informatics and Telecommunications, National Kapodistrian University of Athens/Hellas/Grèce (2 aut.); GALAAD, INRIA Sophia Antipolis-Méditerranée/France (1 aut., 3 aut.); University of Nice/France (2 aut.)
DT :	Publication en série; Congrès; Niveau analytique
SO :	Theoretical computer science; ISSN 0304-3975; Coden TCSCDI; Pays-Bas; Da. 2008; Vol. 392; No. 1-3; Pp. 158-173; Bibl. 60 ref.
LA :	Anglais
EA :	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 derive an expected complexity bound of ========Otilde;_B (d⁶ + d⁴τ²), where d is the polynomial degree and τ bounds the coefficient bit size, using a standard bound on the expected bit size of the integers in the continued fraction expansion, thus matching the current worst-case complexity bound for real root isolation by exact methods (Sturm, Descartes and Bernstein subdivision). Moreover, using a homothetic transformation we improve the expected complexity bound to OB (d³τ). We compute the multiplicities within the same complexity and extend the algorithm to non-square-free polynomials. Finally, we present an open-source C++ implementation in the algebraic library SYNAPS, and illustrate its completeness and efficiency as compared to some other available software. For this we use polynomials with coefficient bit size up to 8000 bits and degree up to 1000.
CC :	001D02A08; 001A02G04; 001A02C04; 001D02A05
FD :	Complexité algorithme; Fraction continue; Algorithmique; Implémentation; Nombre entier; Polynôme; Développement fraction continue; Expansion; Nombre réel; Nombre algébrique; Performance; Multiplicité; Algorithme; Complétude; Efficacité; Logiciel; Informatique théorique; 58A25; Pire cas; 13H15; 68Wxx; 68N01
ED :	Algorithm complexity; Continued fractions; Algorithmics; Implementation; Integer; Polynomial; Continued fraction expansion; Expansion; Real number; Algebraic number; Performance; Multiplicity; Algorithm; Completeness; Efficiency; Software; Computer theory
SD :	Complejidad algoritmo; Algorítmica; Implementación; Entero; Polinomio; Desarrollo fracción continua; Expansión; Número real; Número algebraico; Rendimiento; Multiplicidad; Algoritmo; Completitud; Eficacia; Logicial; Informática teórica
LO :	INIST-17243.354000175110240100
ID :	08-0241602

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Pascal:08-0241602

Le document en format XML

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<front><div type="abstract" xml:lang="en">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 derive an expected complexity bound of ========Otilde;<sub>B</sub>
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 + d<sup>4</sup>
τ<sup>2</sup>
), where d is the polynomial degree and τ bounds the coefficient bit size, using a standard bound on the expected bit size of the integers in the continued fraction expansion, thus matching the current worst-case complexity bound for real root isolation by exact methods (Sturm, Descartes and Bernstein subdivision). Moreover, using a homothetic transformation we improve the expected complexity bound to OB (d<sup>3</sup>
τ). We compute the multiplicities within the same complexity and extend the algorithm to non-square-free polynomials. Finally, we present an open-source C++ implementation in the algebraic library SYNAPS, and illustrate its completeness and efficiency as compared to some other available software. For this we use polynomials with coefficient bit size up to 8000 bits and degree up to 1000.</div>
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<fC01 i1="01" l="ENG"><s0>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 derive an expected complexity bound of ========Otilde;<sub>B</sub>
 (d<sup>6</sup>
 + d<sup>4</sup>
τ<sup>2</sup>
), where d is the polynomial degree and τ bounds the coefficient bit size, using a standard bound on the expected bit size of the integers in the continued fraction expansion, thus matching the current worst-case complexity bound for real root isolation by exact methods (Sturm, Descartes and Bernstein subdivision). Moreover, using a homothetic transformation we improve the expected complexity bound to OB (d<sup>3</sup>
τ). We compute the multiplicities within the same complexity and extend the algorithm to non-square-free polynomials. Finally, we present an open-source C++ implementation in the algebraic library SYNAPS, and illustrate its completeness and efficiency as compared to some other available software. For this we use polynomials with coefficient bit size up to 8000 bits and degree up to 1000.</s0>
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</fC03>
<fC03 i1="06" i2="X" l="ENG"><s0>Polynomial</s0>
<s5>22</s5>
</fC03>
<fC03 i1="06" i2="X" l="SPA"><s0>Polinomio</s0>
<s5>22</s5>
</fC03>
<fC03 i1="07" i2="X" l="FRE"><s0>Développement fraction continue</s0>
<s5>23</s5>
</fC03>
<fC03 i1="07" i2="X" l="ENG"><s0>Continued fraction expansion</s0>
<s5>23</s5>
</fC03>
<fC03 i1="07" i2="X" l="SPA"><s0>Desarrollo fracción continua</s0>
<s5>23</s5>
</fC03>
<fC03 i1="08" i2="X" l="FRE"><s0>Expansion</s0>
<s5>24</s5>
</fC03>
<fC03 i1="08" i2="X" l="ENG"><s0>Expansion</s0>
<s5>24</s5>
</fC03>
<fC03 i1="08" i2="X" l="SPA"><s0>Expansión</s0>
<s5>24</s5>
</fC03>
<fC03 i1="09" i2="X" l="FRE"><s0>Nombre réel</s0>
<s5>25</s5>
</fC03>
<fC03 i1="09" i2="X" l="ENG"><s0>Real number</s0>
<s5>25</s5>
</fC03>
<fC03 i1="09" i2="X" l="SPA"><s0>Número real</s0>
<s5>25</s5>
</fC03>
<fC03 i1="10" i2="X" l="FRE"><s0>Nombre algébrique</s0>
<s5>26</s5>
</fC03>
<fC03 i1="10" i2="X" l="ENG"><s0>Algebraic number</s0>
<s5>26</s5>
</fC03>
<fC03 i1="10" i2="X" l="SPA"><s0>Número algebraico</s0>
<s5>26</s5>
</fC03>
<fC03 i1="11" i2="X" l="FRE"><s0>Performance</s0>
<s5>27</s5>
</fC03>
<fC03 i1="11" i2="X" l="ENG"><s0>Performance</s0>
<s5>27</s5>
</fC03>
<fC03 i1="11" i2="X" l="SPA"><s0>Rendimiento</s0>
<s5>27</s5>
</fC03>
<fC03 i1="12" i2="X" l="FRE"><s0>Multiplicité</s0>
<s5>28</s5>
</fC03>
<fC03 i1="12" i2="X" l="ENG"><s0>Multiplicity</s0>
<s5>28</s5>
</fC03>
<fC03 i1="12" i2="X" l="SPA"><s0>Multiplicidad</s0>
<s5>28</s5>
</fC03>
<fC03 i1="13" i2="X" l="FRE"><s0>Algorithme</s0>
<s5>29</s5>
</fC03>
<fC03 i1="13" i2="X" l="ENG"><s0>Algorithm</s0>
<s5>29</s5>
</fC03>
<fC03 i1="13" i2="X" l="SPA"><s0>Algoritmo</s0>
<s5>29</s5>
</fC03>
<fC03 i1="14" i2="X" l="FRE"><s0>Complétude</s0>
<s5>30</s5>
</fC03>
<fC03 i1="14" i2="X" l="ENG"><s0>Completeness</s0>
<s5>30</s5>
</fC03>
<fC03 i1="14" i2="X" l="SPA"><s0>Completitud</s0>
<s5>30</s5>
</fC03>
<fC03 i1="15" i2="X" l="FRE"><s0>Efficacité</s0>
<s5>31</s5>
</fC03>
<fC03 i1="15" i2="X" l="ENG"><s0>Efficiency</s0>
<s5>31</s5>
</fC03>
<fC03 i1="15" i2="X" l="SPA"><s0>Eficacia</s0>
<s5>31</s5>
</fC03>
<fC03 i1="16" i2="X" l="FRE"><s0>Logiciel</s0>
<s5>32</s5>
</fC03>
<fC03 i1="16" i2="X" l="ENG"><s0>Software</s0>
<s5>32</s5>
</fC03>
<fC03 i1="16" i2="X" l="SPA"><s0>Logicial</s0>
<s5>32</s5>
</fC03>
<fC03 i1="17" i2="X" l="FRE"><s0>Informatique théorique</s0>
<s5>33</s5>
</fC03>
<fC03 i1="17" i2="X" l="ENG"><s0>Computer theory</s0>
<s5>33</s5>
</fC03>
<fC03 i1="17" i2="X" l="SPA"><s0>Informática teórica</s0>
<s5>33</s5>
</fC03>
<fC03 i1="18" i2="X" l="FRE"><s0>58A25</s0>
<s4>INC</s4>
<s5>70</s5>
</fC03>
<fC03 i1="19" i2="X" l="FRE"><s0>Pire cas</s0>
<s4>INC</s4>
<s5>71</s5>
</fC03>
<fC03 i1="20" i2="X" l="FRE"><s0>13H15</s0>
<s4>INC</s4>
<s5>72</s5>
</fC03>
<fC03 i1="21" i2="X" l="FRE"><s0>68Wxx</s0>
<s4>INC</s4>
<s5>73</s5>
</fC03>
<fC03 i1="22" i2="X" l="FRE"><s0>68N01</s0>
<s4>INC</s4>
<s5>74</s5>
</fC03>
<fN21><s1>154</s1>
</fN21>
<fN44 i1="01"><s1>OTO</s1>
</fN44>
<fN82><s1>OTO</s1>
</fN82>
</pA>
<pR><fA30 i1="01" i2="1" l="ENG"><s1>Computational Algebraic Geometry and Applications. Conference</s1>
<s3>Nice FRA</s3>
<s4>2006-06-02</s4>
</fA30>
</pR>
</standard>
<server><NO>PASCAL 08-0241602 INIST</NO>
<ET>On the complexity of real root isolation using continued fractions</ET>
<AU>TSIGARIDAS (Elias P.); EMIRIS (Ioannis Z.); BUSE (L.); ELKADI (M.); MOURRAIN (B.)</AU>
<AF>INRIA -LORIA Lorraine, project VEGAS, 615, rue du Jardin Botanique, B.P. 101/54602 Villers-des-Nancy/France (1 aut.); Department of Informatics and Telecommunications, National Kapodistrian University of Athens/Hellas/Grèce (2 aut.); GALAAD, INRIA Sophia Antipolis-Méditerranée/France (1 aut., 3 aut.); University of Nice/France (2 aut.)</AF>
<DT>Publication en série; Congrès; Niveau analytique</DT>
<SO>Theoretical computer science; ISSN 0304-3975; Coden TCSCDI; Pays-Bas; Da. 2008; Vol. 392; No. 1-3; Pp. 158-173; Bibl. 60 ref.</SO>
<LA>Anglais</LA>
<EA>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 derive an expected complexity bound of ========Otilde;<sub>B</sub>
 (d<sup>6</sup>
 + d<sup>4</sup>
τ<sup>2</sup>
), where d is the polynomial degree and τ bounds the coefficient bit size, using a standard bound on the expected bit size of the integers in the continued fraction expansion, thus matching the current worst-case complexity bound for real root isolation by exact methods (Sturm, Descartes and Bernstein subdivision). Moreover, using a homothetic transformation we improve the expected complexity bound to OB (d<sup>3</sup>
τ). We compute the multiplicities within the same complexity and extend the algorithm to non-square-free polynomials. Finally, we present an open-source C++ implementation in the algebraic library SYNAPS, and illustrate its completeness and efficiency as compared to some other available software. For this we use polynomials with coefficient bit size up to 8000 bits and degree up to 1000.</EA>
<CC>001D02A08; 001A02G04; 001A02C04; 001D02A05</CC>
<FD>Complexité algorithme; Fraction continue; Algorithmique; Implémentation; Nombre entier; Polynôme; Développement fraction continue; Expansion; Nombre réel; Nombre algébrique; Performance; Multiplicité; Algorithme; Complétude; Efficacité; Logiciel; Informatique théorique; 58A25; Pire cas; 13H15; 68Wxx; 68N01</FD>
<ED>Algorithm complexity; Continued fractions; Algorithmics; Implementation; Integer; Polynomial; Continued fraction expansion; Expansion; Real number; Algebraic number; Performance; Multiplicity; Algorithm; Completeness; Efficiency; Software; Computer theory</ED>
<SD>Complejidad algoritmo; Algorítmica; Implementación; Entero; Polinomio; Desarrollo fracción continua; Expansión; Número real; Número algebraico; Rendimiento; Multiplicidad; Algoritmo; Completitud; Eficacia; Logicial; Informática teórica</SD>
<LO>INIST-17243.354000175110240100</LO>
<ID>08-0241602</ID>
</server>
</inist>
</record>

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On the complexity of real root isolation using continued fractions

On the complexity of real root isolation using continued fractions

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