Experimental evaluation and cross-benchmarking of univariate real solvers

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univariate real solvers

Ioannis Z. Emiris, Michael Hemmer, Menelaos Karavelas, Bernard Mourrain,

Elias P. Tsigaridas, Zafeirakis Zafeirakopoulos

To cite this version:

Ioannis Z. Emiris, Michael Hemmer, Menelaos Karavelas, Bernard Mourrain, Elias P. Tsigaridas, et al..

Experimental evaluation and cross-benchmarking of univariate real solvers. International Workshop

SNC, ACM, Aug 2009, Kyoto, Japan. pp.85-94, �10.1145/1577190.1577202�. �inria-00340887v2�

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ISSN 0249-6399

a p p o r t

d e r e c h e r c h e

Thème SYM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Experimental evaluation and cross-benchmarking

of univariate real solvers

Ioannis Emiris — Michael Hemmer — Menelaos Karavelas — Bernard Mourrain — Elias

P. Tsigaridas — Zafeirakis Zafeirakopoulos

N° 6954

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Unité de recherche INRIA Sophia Antipolis

2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France)

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

of univariate real solvers

Ioannis Emiris

∗

, Mi hael Hemmer

†

,Menelaos Karavelas

‡

, Bernard Mourrain

§

, EliasP. Tsigaridas

§

, ZafeirakisZafeirakopoulos

∗

ThèmeSYMSystèmessymboliques ProjetGALAAD

Rapportdere her he n°6954O tomber200817pages

Abstra t: Real solvingofunivariatepolynomialsisafundamental problem withseveral importantappli ations. Thispaperfo usesonthee ientandgeneri bla k-box implemen-tationsofstate-of-the-artalgorithmsforisolatingallrealrootsofpolynomialswith integer oe ients, motivated by geometri appli ations and the re ent need to developsoftware thathandlesexa tly omplexgeometri obje ts,parti ularlyinthe aseoftheCGALlibrary. Wesummarizealargesetofexperimentalresultsfrom ross-ben hmarkingthreeunivariate algebrai kernels developed at the GALAAD groupat INRIA, MPI-Saarbrü ken,and the VEGASgroupatLORIA.Wehavetested6solversfromtheINRIAkernel,whi harebased onSturmsequen es,symboli -numeri methods,andContinuedFra tions(CF);twosolvers from theMPI kernel,namely Des artesand Bitstream Des artes; and onesolverfrom theLORIAkernel,relyingontheDes artes-basedRSsolverdevelopedattheSALSAgroup ofINRIA.Weusedatotalof5000polynomialsof5typesandvariousdegreesandbitsizes, distributed in 150datasets. TheCFfamilyof solvers,Des artes,Bitstream Des artes, and RS are numeri ally and ombinatorially robust, i.e. they always provide the orre t results. With respe tto speed, theresultsare notde isivein most ases; overall,two CF methods seemfaster,even though urrentlythey donotusesymboli -numeri te hniques, while for very large bitsizes Bitstream Des artes and RS seem more e ient, provided thedegreeis notveryhigh. For polynomialsof degreeupto 4and moderatebitsize, spe- ial algorithms give better results. It is important to note that the implementations of thetheoreti allyexa tmethodsare omplete,e ientandtheyalwaysprovided orre t re-sultsthroughoutthisextensiveben hmarkingpro ess. Lastly,we ommentonthedierent

Partiallysupportedby ontra tANR-06-BLAN-0074"De otes".

∗

NationalKapodistrianUniversityofAthens,Athens,Gree e{emiris, grad0783}di.uoa.gr

†

Max-Plan k-InstitutfürInformatik,Saarbrü ken,Germany,hemmermpi-inf.mpg.de

‡

UniversityofCrete&FO.R.T.H.,Heraklion,Gree e,mkaraveltem.uo .gr

§

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programmingdesign approa hes adopted by the dierent kernels, in luding the degree of interoperabilitybetweendierentalgorithmi omponents,aswellasarithmeti datatypes.

Key-words: univariate polynomials, real solving, algebrai kernel, exa t omputation, robustimplementation

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équations polynomiales en une variable

Résumé:

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1 Introdu tion

Real solving of univariate polynomials is a fundamental algebrai question with several importantappli ations. This paperfo uses onthee ientandgeneri implementationsof state-of-the-artalgorithmsforisolatingallrealrootsofpolynomialswithinteger oe ients, motivated bygeometri appli ationsand there entneed todevelopsoftwarethat handles exa tly omplexgeometri obje ts,parti ularly in the aseof theCGAL

1

library. We sum-marizetheresultsof extensive ross-ben hmarksforthree univariatealgebrai kernels(see Se tion 3,[9, 12, 14, 16℄) developed at theGALAAD groupat INRIA, MPI-Saarbrü ken, andtheVEGAS groupatLORIA.All tablesand graphsprodu ed duringtheben hmarks areavailablethrough

http://a s. s.rug.nl/index.php?page=p_ben hmarks.

Previousworkin ludes[13℄whi hdes ribesben hmarksbetweenaSturm-sequen ebased solver developed within ACS

2

, and MPI's solvers Des artes and Bitstream Des artes. Themajor on lusionfor realrootisolationwasthatfordegrees

≥ 20

thelatterarefaster ormu hfaster. For smallerdegrees,allmethodsare omparable.

Wetested9methodson5000polynomialsdistributedin150datasets(250000solve alls, over50htotalruntime). Weused5dierenttypesofpolynomialsofdegreesbetween3and 184andbitsizesbetween10and8000, see Se tion4. Tothebestof ourknowledge, thisis thelargestnumberoftestsforunivariaterealsolvingupto date.

The CF family of solvers, Des artes, Bitstream Des artes, and RS are numeri ally and ombinatorially robust, i.e. they always provide the orre tresults. With respe t to speed,theresultsarenotde isivein most ases. Overall,twoCFmethods,namelyCFand NCF2 seem faster, while for very large bitsizes Bitstream Des artes and RS seem more e ient,providedthedegreeisnotveryhigh (seeSe tion5,Figure3). Forpolynomialsof degreeupto4andmoderatebitsize,spe ialalgorithmsgivebetterresultsasexpe tedfrom theirtheoreti al ompexity,seeFigure 4. It isimportantto notethat theimplementations of the theoreti ally exa t methods are omplete, e ient and they never provided wrong resultsthroughout this extensiveben hmarking pro edure. We omment on the dierent programmingdesign approa hes adopted by the dierent kernels, in luding the degree of interoperabilitybetweendierentalgorithmi omponents,aswellasarithmeti datatypes, seeSe tion3.

Ourresultsareusefulforvalidating variousalgorithmi approa hesto thefundamental problem ofreal solving, aswellas for evaluating dierentimplementationstrategies of al-gebrai operations aimed at supporting robustand e ient omputing in a widerange of appli ations.

Inwhatfollows

O

B

meansbit omplexityandthe

O

e

B

-notationmeansthatweareignoring logarithmi fa tors. Moreover,

d

willbethedegreeofthepolynomialsand

τ

themaximum oe ientbitsize.

1

http://www. gal.org/ 2

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Therestofthepaperisstru turedasfollows. InSe tion2wedes ribethesetupofthe ben hmarks. Se tion 3des ribes themethods wetested. Se tion 4 des ribesthe datasets weused. Se tion5presentstheresultsoftheben hmarks. InSe tion6wepresentthe on- lusionsofthe ross-ben hmarkingpro ess fortheunivariatesolvers. Finally, Appendix A ontainsadditionalplotsprodu edbytheben hmarksandinAppendixBweprovidetables showingthebest methodforvarious ombinationsofpolynomialtype,degreeandbisize.

2 Ben hmark setup

Weusedfortheben hmarksa32-bitPentiumIIIwith256MBRAM.All ompilationswere done using the optimization ags -O3 and -DNDEBUG.We used the internal release 271of CGAL(released03-Apr-2008),aswellasthesoftwarelibrariesGMP

3

(version4.2)andNTL 4

(version5.4.1). The ompilerusedwasg++version4.1.2,onaDebianet hLinuxplatform. Wemeasuredthetimeneededbyamethodtoisolatealltherealrootsofapolynomial, usingthe lo k()fun tionofthe timelibrary(measurementsareinmse ). In aseamethod takesmorethan30se ondsforsomeinstan e,weignorethemeasurement. Theaverageover all polynomials in ea h datasetis reported. Forinstan es that took less than 1mse , we reportedtheaveragetimeoveranumberofiterationstoex eed1mse totaltimein order toredu enoise.

3 Real Root Isolation methods

Analgebrai kernel,see[2℄, onsistsofthreemajor omponents.

1. Polynomialsandbasi operations. (g d, squarefreefa torizationet .)

2. Isolationwhi histhe oreofthefun torSolve_1.

3. Representation of algebrai real roots, omparison of algebrai reals, approximation renement.

Although the ben hmarking was only on erned for univariate real root isolation, we impli itlytestedtheimplementationofsquarefreefa torizationandrealalgebrai numbers. The real root isolation methods we tested are part of theAlgebrai Kernels developed in INRIA, MPIandLORIA. TheINRIAAlgebrai Kernel isavailable throughmathemagix

5 . TheMPI andLORIAAlgebrai Kernelsareavailable throughtheCGAL'ssvnserver.

Thealgebrai kernelbyINRIAreliesonSYNAPS 6

,whi hinturnisnowbeenintegrated in mathemagix. It is anopensour e eortthatprovidesfundamental algebrai operations su h asalgebrai numbermanipulationtools,dierenttypesofunivariateandmultivariate

3

http://gmplib.org/ 4

http://www.shoup.net/ntl 5

http://www-sop.inria.fr/g alaa d/m athe mag ix/ 6

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polynomialrootsolvers,resultantandg d omputations, et . Themain motivation behind thisproje t,istheneedto ombinesymboli andnumeri omputations,whi hisubiquitous inmanyproblems. Wereferthereaderto[15℄formoredetails. Anotable ommentaboutthe polynomialsolversin thislibraryisthattheyexploitspe ializedalgorithmsforpolynomials ofdegreeupto 4,that have onstantarithmeti omplexity,i.e.

O(1)

e

,and bit- omplexity

e

O

B

(τ )

,where

τ

isthe oe ientbitsize. Thealgorithmsarebasedonpre- omputedSturm sequen esand they omputerationalpointsthat isolatethe realroots, asfun tionsin the oe ientsofthepolynomial. Wereferthereaderto[7,17℄formoredetails. Inparti ular, theSturm,Sleeve,Symbnumand CFsolvers(seebelow),use thespe ialized algorithmsfor realsolvingpolynomialsofdegree

≤ 4

. Hereafter,werefertothisalgorithmasIDS(Isolation andDis riminationSystems).

Thealgebrai kernelprovidedbytheMPIhasbeendesignedsu hthatexploitsgeneri ity for real root isolation methods and the way the algebrai real roots are represented, that is, the kernel has two template arguments. This is the prin ipal advantage of the MPI kernel, sin eit anpotentially ombine thebest rootisolatorwith thebest representation ofalgebrai realsduetoitsgeneri design. Inthe ontextofthispaperthemostimportant templateargumentistherst,namelytherootisolator. Formoredetailsaswellastheexa t denitionoftherootisolator on eptsee[12℄. UptodatetheMPIhasprovidedtwo lasses thataremodelsoftherootisolator on ept,theDes artesandtheBitstream Des artes. However,inprin ipaltheMPIkernel oulduseanyrootisolationmethodpresentedhereas longasitprovidestheproperinterfa e. Moreover,itisnotrestri tedtoa ertain oe ient typesin ethisisimpli itlydenedbythegivenrootisolator.

Des riptionof themethods

Sturm A subdivision method based on Sturm's theorem. In order to isolate the real roots,thealgorithmemployspolynomialremaindersequen esasarealroot ountingquery. Thepolynomialremaindersequen eisevaluatedattheendpointsoftheintervalatquestion, to determineifaroot isisolated in thegiven interval. The omplexity ofthealgorithm is

e

O

B

(d

6 + d

4 τ

2 )

. See[4,9℄andreferen estherein. Ingeneral,itisexpe tedtobetheslowest method. Weuseitmoreasareferen e,thana tually omparingitin a ompetitiveway.

Sleeve An approximate real root isolation method that is based on omputing an "upper" and "lower" (i.e. a sleeve) approximation of the polynomial. A Des artes-like subdivision algorithm is applied to the sleeve. The approximation is rened, if possible, until thema hine pre isionisrea hed. Theisolatingintervals,that thealgorithmreturns, may ontainmorethanonereal root. Insome aseswe an ertifythe resultusingasign testwiththerstderivativeandintervalarithmeti .

CFfamily TherearethreevariantsoftheContinuedFra tions(CF)algorithmstested. Thealgorithm omputesthe ontinuedfra tionsexpansionoftherealrootsofthepolynomial

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andthen omputesanisolatingintervalwithrationalend-pointsforea hroot,asdes ribed in[18℄. Thealgorithm's omplexityis

O

e

B

(d

6 + d

4 τ

2 )

.

NCFis basedon thesamealgorithm, but the implementation employstheNTL library forsquarefreefa torizationand fa torizationovertheintegers. Integerarithmeti in NTL isbasedonGMP, whi hisused throughaSYNAPS-NTL onversionsinterfa e,see[11℄.

Thedieren ebetweenNCFandNCF2isthatNCF2employssquarefreefa torization,but not fa torization over the integers. We have to note that the urrent implementation of thesemethodsusesonlyexa tarithmeti anddoesnotexploitapproximationte hniques.

Symbnum It isanexperimental symboli -numeri algorithm,whi his a ombinationof the Sleeveand the Continued Fra tion algorithm. Initially the Sleeve algorithm runs and produ es someintervals, that may ontain more than one real root. In the sequel, using exa tarithmeti ,we omputethe ontinuedfra tionsexpansionoftherealroot(s)thatare in the interval, the resultingintervalis transformedto (0, 1) and the Sleeve algorithm is appliedagain(possibleafters alingthe oe ients).

Des artes Areal root isolationmethod basedonDes artes'sruleof sign. This algo-rithmhasbeenintegrated in CGALfrom the NumeriXlibraryof EXACUS and was imple-menteda ordingto[3℄. Giventhenewtreeboundin[5℄the omplexityofthealgorithmis

e

O

B

(d

5 τ

2 )

.

Bitstream Des artes The oe ientsofthepolynomialare onvertedto(potentially innite) bitstreams. A variantof Des artes'smethodis used to ndisolating intervalsfor the real roots. The worst ase omplexity is

O

e

B

(d

5 τ

2 )

. The advantage of this method is that it isadaptivein the number ofbits used, seealso [5, 6℄. A major advantageof both Des artes implementations isthat theyaregeneri with respe t tothe oe ient type. In parti ular the Bitstream Des artes is intended to work over algebrai extensions, as it is reported in [5, 6, 12℄, for example, it anisolate roots of

f

(x, y)|x = α

, where

α

is an algebrai number. Thisisthea tualstrengthoftheBitstream Des artes.

RS-solver TherealrootisolationisbasedontheRSsolver 7

fromtheSALSAgroupat INRIA-Ro quen ourt,see [14,16℄. TheRSsolverisbasedonintervalDes artes algorithm, approximatingthetransformationof

[a, b]

to

[0, inf]

,whenthisissu ientforsign determi-nation. Theorder ofthetransformationsisimportantfor thememory onsumptionofthe method.

4 Data des ription

Weuseddatasetsofpolynomialswithdierent hara teristi s,inordertohaveapi tureof theadvantagesordisadvantagesofthesolversin various ases. This workwasdonewithin

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theACSproje t,thusanin reasedinterestinpolynomialsderivedfromgeometri problems isevident.

[rnd℄aredatasets ofrandompolynomialsofdegreebetween3and100andbitsizefrom 10to50. Thesepolynomialsaretheeasiesttosolveamongtheonestested.

[bts℄Randompolynomialsareimportant,sin etheyo urnaturallyinmanyproblems. Using odeforgeneratingpolynomialsfortestleswithCGAL,we reatedthe[bts℄datasets ofdegreebetween3and 100andbitsizeinfrom2000to8000.

[mgn℄datasets ontain polynomials of the form

x

d

− 2(kx − 1)

2

and degree between 3 and100and bitsizefrom 10 to 50. These polynomialshavetworeal roots very loseea h other,andarethemost"di ult"polynomialsforsubdivision-basedalgorithms.

[int℄/[rat℄datasets ontainpolynomialswithinteger/rationalroots. Whilethe hoi e of roots for the onstru tion of the polynomials is random, there are aseswith multiple sele tions of the same root. (e.g. small bitsize, high degree). The datasets onsist of ombinationsofdegreefrom 3to100andbitsizefrom 10to50.

[3darr℄in ludeunivariatepolynomialsinbformatasdes ribedin[1℄. Thepolynomials aremotivated by3D-arrangementproblems. Ea hdataset ontains100univariate polyno-mialsofbitsize200,400,600,800,1000,1200,1400,1600,1800or2000bitsanddegree12. The polynomialshave several real roots within the interval

[−10, 10]

. Ea h polynomialis theprodu tofsixparabolas,where ea hparabola hasat leastoneroot withintheinterval

[−10, 10]

. Consequently,allpolynomialshave12realroots.

[vorell℄ ontains10polynomialsofdegree184and bitsizeofapproximately3500bits. Thesepolynomialsareobtainedfrom resultantsanddes ribealltritangent ir lesto three ellipses,see[8,10℄. TheellipsesaregivenrandomlythroughaGraphi alUserInterfa e. The inputbitsizeoftheellipsesisfrom11to13bits,whilethebitsizeoftheresultantpolynomial rangesfrom 3100to3800bits.

The[3darr℄datasetsareavailablethroughCGAL'ssvnserver. The[rnd℄,[int℄,[rat℄ and[mgn℄datasetsareavailableathttp://synaps.inria.fr/ben h/upol_root.tgz.

5 Results and dis ussion

Isolation and Dis rimination Systems Methods using Isolation and Dis rimination Systemshaveanadvantagein asesofdegreeupto4andsmalltomoderatebitsize,butare outperformedbyDes artesandBitstream Des artesforbitsizesoverathreshold(2000 bits). Thisisexpe ted,sin etheIsolationandDis riminationSystemsarebitsizesensitive. The low theoreti al omplexity of spe ial algorithms for polynomialsof degreeup to 4 is onsistentwiththeben hmarkresults,seeFigure4.

Randompolynomials

[rnd℄ Sleeve and Symbnum are lose to NCF2 and faster than all other methods on polynomialsofdegree100andbitsizeupto50. Therandomnessofthe oe ientsisinfavor

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Bitstream Des artesareoutperformedbyNCF2,CFandthetwoapproximationmethods. Duetofa torizationovertheintegers,NCFisae tedbythebitsizein rease.

Figure1: [bts℄Randompolynomialsofdegree50andbigbitsize

5

10

15

20

25

30 1000

2000

3000

4000

5000

6000

7000

8000

msec

Type: random Degree: 50 Bitsize: VAR

ST

SV

SN

CF

NCF

NCF2

D

BD

RS

[bts℄ When thebitsizeis over4000, Des artesand Bitstream Des artesare per-formingbetterthantherestofthemethodsprovidedthatthedegreeisupto10. Whilethe degreein reases, NCF2and RSare faster. Forbitsize 8000 bits and degree50, Bitstream Des artes is equivalent to CF, while for bitsize 2000 bits CF is faster than Bitstream Des artes. SeeFigure1.

Inany ase,NCF2andRSarefasterthanBitstream Des artesfordegreeshigherthan 10. The break-even point for RS and NCF2 for degree50 is 6000 bits. For bitsizesbigger thanthat,RSisfasterthanNCF2.

Bitstream Des artesisnotae tedbybitsizein rease. Forpolynomialsofdegree100 andbitsizefrom2000to8000bits,Bitstream Des artesshowsno hangeinperforman e. Whilethedegreeofthepolynomialin reases,Bitstream Des artesrequiresmorebitsdue tointernal boundsthat dependonthe degree. The onstantoverhead isbigin the aseof high degrees, thus up to 8000 bits Bitstream Des artes is not better than NCF2 orRS. Thisresultis onsistentwiththeresultsin [14℄.

Polynomials with integer or rational roots All methods a eptaag to determine whether a polynomial is known to be square-free. The [int℄ and [rat℄ sets, ontain both square-free and non square free polynomials, thus the methods were alled asking for multipli ities. This means that impli itly, we are timing square-free fa torization as well. Thedegree in reaseisae ting the squarefree fa torizationand inuen es thetime measured. Bythetimeoftheben hmarkstheMPIkerneldidnotemploymodulararithmeti

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to ompute the GCD during square-free fa torization 8

, this aused a serious drawba k. Hen e,itwas notpossibleto ompareDes artesandBitstream Des arteswithrespe t to the other methods based on these results. The fa t that these sets (espe ially [int℄) havemanypolynomialswithmultiplerootsgivesanadvantagetothemathemagixmethods, whi h usethe NTL library for fa torization. Therefore, NCF and NCF2are equivalent and learlythefastestmethods. TheLORIAkernelimplementsitsownmodularg d. Thismay explainwhyRSisinall asesinbetweenofthetwogroups,slowerthanNCFandNCF2,but fasterthantherestofthemethods,seeFigure5.

Everymethod hassimilarbehaviorfor[int℄andtherespe tive[rat℄ ases.

Mignotte polynomials The [mgn℄ polynomials are the most di ult to solve among thepolynomialstested. Insome ases, espe ially whenthebitsizewasbig,approximation methods(SleeveandSymbnum),failedtoisolatetheroots orre tly,providingonly3roots insteadof 4. Even for moderate degrees,only methods based onContinued Fra tionsare apableofsolvingMignotte polynomials. Forexample,thetimes arenoteven omparable for[mgn℄ofdegree30,as shownin Figure2.

RSis onsiderablyslowin[mgn℄. Theproblem isduetothehighmemory onsumption ofthealgorithm.In asesofdegreehigherthan30themethod wouldrun outofmemory.

Figure2: [mgn℄Mingottepolynomialsofdegree 30

20

40

60

80

100

120

140

160

180

200

10

15

20

25

30

35

40

45

50 msec

Type: mignotte Degree: 30 Bitsize: VAR

ST

SV

SN

CF

NCF

NCF2

D

BD

RS

Geometri problems The [3darr℄ polynomialshaverelativelybig bitsize and we an observethe break-even point of Bitstream Des artes. Forsmall bitsizesthe CFfamily methods perform onsiderablybetter. Bitstream Des artesoutperformsDes artesand theCF-familyforbitsizesbiggerthan1200. Thisobservationis onsistentwiththeresults

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presentedin[12℄. RSisthemostee tivemethodforbitsizesover600bits. Thebehaviorof themethod isverygoodin this set,sin eit isfastenoughforpolynomialsofbitsizelower than600,and thefastest oneforbitsizesoverthis threshold. Thebitsizein reasehasless ee tonBitstream Des artesandRSthanonCF,NCF2andDes artes. SeeFigure3. The lowdegree(inthis ase12)isinfavorofBitstream Des artesforhighbitsizes. Moreover, NCFdue tothefa torizationovertheintegersis slowerthanCFandNCF2inthedatasetsof bigbitsizes.

Figure3: [3darr℄Polynomialsofdegree12,derivedfrom3Darrangementproblems

4

6

8

10

12

14

16

18

200

400

600

800 1000

1200

1400

1600

1800

2000

msec

Type: Random Degree: 12 Bitsize: VAR

ST

SV

SN

CF

NCF

NCF2

D

BD

RS

The [vorell℄ dataset ontainspolynomials having the highestdegreeof all the poly-nomialstested. Moreover,thebitsizeis onsiderablyhigh(approx. 3500bits). Thefastest methods inthis aseareSleeveandSymbnum. Sin e thepolynomialsare random,the ap-proximation su es for theroot isolation. NCF2 is very lose to these methods. NCF and Des artesare equivalent, while Bitstream Des artesis not ompetitive. The degreeof thesepolynomialsslowsdownBitstream Des artes,whileNCF'sfa torizationis slowdue tothebitsize. RSisequivalentto CF,whi harebetterthanNCFandDes artes,but worse thanNCF2andtheapproximationmethods.

6 Con lusions

From the general pi ture of the ben hmarks, it is lear that there is no best method overall. Depending on the hara teristi sof the problem the performan e of any method may hange. A remarkable observation is that the implementations of the theoreti ally exa t methods are omplete, e ient and they never provided wrong results throughout thisextensiveben hmarkingpro edure.

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Inthispaperwesummarizedtheperforman eoftheunivariatesolversinseveraltypi al problem ases. Tables showing the method with the best performan e for [rnd℄/[bts℄, [int℄and[mgn℄arepresentedin Figure6,Figure 7andFigure8respe tively.

Robustness In terms of robustness, the CF-family, Des artes, Bitstream Des artes andRSneverfailedtoprovidethe orre tresult( onsideringthenumberofrootsreturned). Insome asesthough, CF, Des artes orBitstream Des artes mayneed toomu h time to nish. Nevertheless, given the time, they would nish by isolating all real roots. The implementationsofthese methods omplywiththeirtheoreti exa tness.

The sets that produ ed themost problems in approximationmethods are [mgn℄. The usage of inexa t types leads to false results, sin e the roots are too lose to be isolated orre tly.

Performan e TheSleevemethod isveryee tivein smallproblems,wherean approxi-mationisabletoisolatetheroots orre tly. Evenfor[mgn℄,whi histhemostdi ult ase forapproximationmethods. Whenit omestohigherdegreesandbitsizes,itiseithermu h slowerthan theother methods orfails to isolateall thereal roots orre tly. Nevertheless, forrandompolynomialsofmoderatebitsizeapproximationmethodsperformwell.

Inthe aseof [3darr℄and[bts℄notevenapproximationmethodsfailed toisolatethe roots. As longas thedegree is low and theroots are well separated, thebitsize does not auseapproximationmethods to fail in isolatingallroots, orexa tmethodsto ex eedthe timelimit.

Thedi ultyof rootisolation, asshownin [mgn℄,is due tothe separationbound and not to the oe ient bitsize. It is lear that Continued Fra tions based methods always provide the orre t resultmu h faster than anyother method in asesof small separation bound.

Summary Overallone ansaythatfornearlyallinstan estheCFandtheNCF2solversare amongthebestmethods,eventhoughthe urrentimplementationusesonlyexa tarithmeti and doesnot exploit symboli -numeri te hniques that would speedup theirperforman e evenmore. Theonlyex eptionsare asesofpolynomialshavingsimultaneouslylowdegree andhighbitsize,asshowninthetablesinAppendixB. Inthese asesBitstream Des artes orRSshowabetterperforman e,whereRSisslightlymoreee tedbythebitsizethanthe Bitstream Des artes. Bitstream Des artesisabletohandle oe ienttypesotherthan integers(rationals),e.g. oe ientsfromalgebrai extensions. Currentlyistheonlymethod re ommendedfornonrational oe ienttypes.

FutureWork Giventheadvantagesofthebitstreammethodforhugebitsizesaswellas theadvantagesofthemethodsbasedonContinuedFra tionsforsmallseparationbound,it mightbeworthtryingtoexploitthebitstreamideainthe ontextofCFmethods. Moreover, itshouldbepossibletoimprovetheperforman eofthesymboli -numeri te hniqueswhi h

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A knowledgments Allauthorsa knowledgepartialsupportbyISTProgrammeoftheEUasa Shared- ostRTD(FETOpen) Proje tunderContra tNoIST-006413-2 (ACS-AlgorithmsforComplexShapes). B.MourrainandE.Tsigaridasalsoa knowledgepartialsupportby ontra tANR-06-BLAN-0074"De otes". TheauthorsaregratefultoLuisPeñarandaforhishelpwiththeLORIAkernelandtoSebastianLimba h forhishelpwiththeMPIkernel.

Referen es

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[2℄ Eri Berberi h,Mi haelHemmer,MenelaosI.Karavelas,andMoniqueTeillaud. Revi-sionoftheinterfa espe i ationofalgebrai kernel.Te hni alReport ACS-TR-243301-01, INRIA Sophia-Antipolis,Max Plan kInstitut fürInformatik, NationalUniversity of Athens,2007.

[3℄ G.E.CollinsandA.G.Akritas. Polynomialrealrootisolationusingdes arte'sruleof signs. InPro .

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ACMSymp.onSymboli andAlgebrai Comp.,pages272275,NY, USA,1976. doi: http://doi.a m.org/10.1145/800205.806346.

[4℄ Z. Du, V. Sharma, and C. K. Yap. Amortized bound for root isolation via Sturm sequen es. InD.WangandL.Zhi,editors, Int.Workshopon Symboli Numeri Com-puting, pages 113129, S hool of S ien e, Beihang University, Beijing, China, 2005. Birkhauser.

[5℄ A. Eigenwillig. Real Root Isolation for Exa t and Approximate Polynomials Using Des artes' Ruleof Signs. PhDthesis,UniversitätdesSaarlandes,Saarbrü ken,2008.

[6℄ A. Eigenwillig, L. Kettner, W. Krandi k, K. Mehlhorn, S. S hmitt, and N. Wolpert. A Des artes algorithm for polynomials with bit-stream oe ients. In Pro . 8th Int. Workshop onComputerAlgebrainS ient.Comput.(CASC),LNCS.Springer,2005. to appear.

[7℄ I. Z. Emiris and E. P. Tsigaridas. Computing with real algebrai numbers of small degree. InS.AlbersandT.Radzik,editors,Pro .12th EuropeanSymp. ofAlgorithms (ESA),volume3221ofLNCS,pages652663,Bergen,Norway,Sep14172004.Springer Verlag.

[8℄ I. Z. Emiris, E. P. Tsigaridas, and G.M. Tzoumas. Predi ates forthe exa tVoronoi diagram ofellipsesunder theeu lideanmetri . Intern. J. ComputationalGeometry & Appli ations, Spe ialIssue. A epted.Available fromhttp://www.di.uoa.gr/geotz/.

[9℄ I. Z. Emiris, B. Mourrain, and E. P. Tsigaridas. Real Algebrai Numbers: Com-plexity Analysis andExperimentation. In P. Hertling,C. Homann, W. Luther, and

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N. Revol, editors, Reliable Implementations of Real Number Algorithms: Theory and Pra ti e, volume5045 ofLNCS, pages5782.SpringerVerlag, 2008. also available in www.inria.fr/rrrt/rr-5897.html.

[10℄ Ioannis Z. Emiris and George M. Tzoumas. A real-time and exa t implementa-tion of the predi ates for the Voronoi diagram of parametri ellipses. In Pro . of the 2007 ACM symposium on Solid and physi al modeling (SPM), pages 133 142, Beijing, China, June 2007. ACM Press. ISBN 978-1-59593-666-0. doi: http://doi.a m.org/10.1145/1236246.1236266.

[11℄ I.Z.Emiris,E.PTsigaridas,andG.M.Tzoumas. VoronoidiagramofellipsesinCGAL. InPro . Europ.Works. Comp. Geom.,pages8790,Nan y,Fran e,2008.

[12℄ Mi haelHemmerandSebastianLimba h. Ben hmakrsonageneri univariatealgebrai kernel. Te hni al Report ACS-TR-243306-03, Algorithms for Complex Shapes with ertied topologyandnumeri s, Max Plan k Institut f¼r Informatik,Saarbr¼ ken, GERMANY, 2007.

[13℄ Menelaos I.Karavelas. Preliminary ross-ben hmarksoftheMPIandNUA univariate algebrai kernels.Te hni alReportACS-TR-243306-05,NationalUniversityofAthens, 2008.

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[15℄ B. Mourrain, P. Pavone, P. Trébu het, E. P. Tsigaridas, and J. Wintz. synaps, a library fordedi ated appli ations in symboli numeri omputations. In M.Stillman, N.Takayama,andJ.Vers helde,editors,IMAVol.inMath.anditsAppli ations,pages 81110.Springer, 2007.

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[17℄ E. P. Tsigaridas. Algebrai algorithms and appli ations to geometry. PhD thesis, National Kapodistrian University of Athens, Aug 2006. (http://www-sop.inria.fr/galaad/elias).

[18℄ EliasP.TsigaridasandIoannisZ.Emiris.Onthe omplexityofrealrootisolationusing ontinuedfra tions. Theor.Comput.S i.,392(1-3):158173,2008.ISSN0304-3975.doi: http://dx.doi.org/10.1016/j.t s.2007.10.010.

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A Plots

Figure4: [rnd℄Randompolynomialsofdegree3.WeobservetheIDSbehaviour.

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Figure5: [int℄Polynomialsofdegree30withintegerroots.

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B Best performan e tables

Inthefollowingtableswepresentthefastestsolverforea h ombinationofpolynomialtype, degreeandbitsizetested. Thelegendgivesthe orrespondan eoftheabbreviationsusedin thetablesandthenamesofthemethodsasdes ribedinSe tion3.

Table1: Legend SN Symbnum CF CF NCF NCF NCF2 NCF2 CFF CFFamily

IDS IsolationandDis riminationSystems BD Bitstream Des artes

RS RSSolver n/a Notde isive

Figure6: Bestmethodsforrandompolynomials

Figure7: Bestmethodsforpolynomialswithintegerroots

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Figure8: BestmethodsforMignottepolynomials

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