Parallel computation of real solving bivariate polynomial systems by zero-matching method

0.11 MB·English

by  Xiaolin Qin

Parallel computation of real solving bivariate polynomial systems by
I
zero-matching method
XiaolinQina,b,c,∗∗,YongFenga,∗,JingweiChenb,c,JingzhongZhanga,b
aLaboratoryofComputerReasoningandTrustworthyComputation,UniversityofElectronicScienceandTechnologyofChina,Chengdu610054,
China
bLaboratoryforAutomatedReasoningandProgramming,ChengduInstituteofComputerApplications,CAS,Chengdu610041,China
cGraduateUniversityofChineseAcademyofSciences,Beijing100049,China
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1 Abstract
0
2 WepresentanewalgorithmforsolvingtherealrootsofabivariatepolynomialsystemΣ={f(x,y),g(x,y)}withafinite
numberofsolutionsbyusingazero-matchingmethod.Themethodisbasedonalowerboundforbivariatepolynomial
n
a system whenthe system isnon-zero. Moreover,themultiplicitiesof therootsofΣ = 0 canbe obtainedby a given
J neighborhood. Fromthis approach,the parallelizationof the methodarises naturally. By usinga multidimensional

8 matchingmethodthisprinciplecanbegeneralizedtothemultivariateequationsystems.
1
Keywords: Bivariatepolynomialsystem;Zero-matchingmethod;Realroots;Symbolic-numericalcomputation;

] Parallelcomputation
C
S
.
s
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[ 1. Introduction

1 Consideringthefollowingsystem:
v
Σ={f(x,y),g(x,y)}, (1)
0
4 weassumethat f(x,y),g(x,y)∈ Q[x,y],whereQisthefieldofrationalnumbers. Wecallthezero-dimensionifthe
9
bivariatepolynomialsystem(1)hasafinitenumberofsolutions.
2
Real solvingbivariate polynomialsystem in a real field is an active area of research. It is equivalentto finding
.
1
the intersections of f(x,y) and g(x,y) in the real plane. The problem is closely related to computing the topology
0
ofaplanerealalgebraiccurveandotherimportantoperationsinnon-linearcomputationalgeometryandComputer-
0
1 AidedGeometricDesign[1, 15, 13, 10, 18]. Anotherfield of applicationis the quantifierelimination[7, 17]. There
: areseveralalgorithmsthattacklethisproblemsuchastheGro¨berbasismethod[19,23],theresultantmethod[26],the
v
characteristicsetmethod[5],andthesubdivisionmethod[3,21]. However,theprocedureofthesetechniquesisvery
i
X complicated.Inthispaper,weproposeanefficientapproachtoremedythesedrawbacks.
r Inthispaper,weproposeazero-matchingmethodtosolvetherealrootsofanequationsystemlike(1). Thebasic
a
ideaofzero-matchingmethodisasfollows: FirstprojectingtherootsofΣtothe x-axis,givestheroots{x ,··· ,x },
1 u
and the y-axis, gives the roots {y ,··· ,y }, respectively. Subsequently, for every root x, and for every y is back-
1 v i j
substitutedin f(x,y)andg(x,y). Tothatend,forsomeroot x thereisthecorrespondingoneormorerootsy tobe
i j
determinedsatisfyingΣ. Themaincontributionofourmethodisthathowtodeterminethe realrootsof Σ = 0and
themultiplicitiesoftheroots. Moreover,ourapproachthathasgivensolutionstothissituationcanbethedesignof
parallelizedalgorithms.
IThis research was partly supported by China 973 Project NKBRPC-2004CB318003, the Knowledge Innovation Program of the Chinese
AcademyofSciencesKJCX2-YW-S02,andtheNationalNaturalScienceFoundationofChina(GrantNO.10771205).
∗Correspondingauthor
∗∗Principalcorrespondingauthor
Emailaddresses:[email protected](XiaolinQin),[email protected](YongFeng)
PreprintsubmittedtoElsevier January18,2010
In[9],Diochnosetalpresentedthreealgorithmstorealsolvingbivariatesystemsandanalyzedtheirasymptotic
bitcomplexities.Amongthethreealgorithms,thedifferenceisthewaytheymatchsolutions.Themethodofspecial-
izedRationalUnivariateRepresentation(rur)basedonfastgcdcomputationsofpolynomialswithcoefficientsinan
extensionfieldtoachieveefficiency(hencethenameg rur)hasthelowestcomplexityandperformsbestinnumerous
experiments. The grur method projects the roots to the x−axis and y−axis, for each x-coordinate α computes the
gcdh(α,y)ofthesquare-freepartsof f(α,y)andg(α,y),andisolatestherootsofh(α,y)= 0basedoncomputations
of algebraic numbers and the rur techniques. Our algorithm only uses resultant computation and real solving for
univariatepolynomialequationswithrationalcoefficients.
ThehybridmethodproposedbyHongetal[16]thatprojectstherootsofΣtothex-axisandy-axisrespectivelyand
usestheimprovedslope-basedHansen-Senguptatodeterminewhethertheboxesformedbytheprojectionintervals
containarootofΣ. ThenumericalmethodonlyworksforsimplerootsofΣ. Whenthesystemhasmultipleroots,the
rurtechniqueisusedtoisolatetheroots. Comparedwiththismethod,ourapproachalsocomputestworesultantsof
thesametotaldegrees.However,ourmethodisacompleteone,theirnumericaliterationmethodneedstousetherur
techniquetofindmultipleroots.
In[2], Bekker et al presented a Combinatorial Optimization Root Selection method (hence the name cors) to
match the roots of a system of polynomial equations. However, the method is only suitable for solving a small
systemofpolynomialequations,anddoesnotworkforthemultipleroots. Recently,Chengetal[4]proposedalocal
genericpositionmethodtosolvethebivariatepolynomialequationsystem. Themethodcanbeusedtorepresentthe
roots of a bivariate equation system as the linear combination of the roots of two univariate equations. Moreover,
the multiplicities of the roots of the bivariate polynomial system are also derived. However, the method is very
complicatedtoextendtosolvethemultivariateequationsystems. Ourmethodcansolvethelargersystemsandeasily
generalizetothemultivariateequationsystems.
Therestofthispaperisorganizedasfollows. InSection2,wegivesomenotations,alowerboundforbivariate
polynomial equation if it is non-zero, and how to determine the root multiplicity. In Section 3, we propose the
algorithmtorealsolvingthebivariatepolynomialsystemandgiveadetailedexample. Insection4,wepresentsome
comparisonsofouralgorithm.Thefinalsectionconcludesthispaper.
2. Notationsandmainresults
2.1. Notations
InwhatfollowsDisaring,FisacommutativefieldofcharacteristiczeroandFitsalgebraicclosure. Typically
D=Z,F=QandF=Q.
Inthispaper,weconsiderthezero-dimensionalbivariatepolynomialsystemasfollows:
 f(x,y)= X X ai,jxiyj =0
g(x,y)=00X≤≤ii≤≤np00X≤≤jj≤≤mqbi,jxiyj =0 . (2)
Throughoutthis paper, note that deg = max(n,p), deg = max(m,q), N = max(||f|| ,||g|| ), where the ||f|| and
x y 1 1 1
||g|| are the one normof the vector(a ,a ,··· ,a ,··· ,a ,··· ,a ) and (b ,b ,··· ,b ,··· ,b ,··· ,b ), so
1 00 01 0m n0 nm 00 01 0q p0 pq
||f|| =ΣΣ |a |,and||g|| =ΣΣ |b |,respectively. M =max(||t|| ,||T|| ),wherethet(x)andT(y)arethenoextraneous
1 i j ij 1 i j ij 1 1
factorsinresultantpolynomialofΣ. |Σ|denotesthatthe bivariatepolynomialsystemΣ hasbeenassignedvaluesto
twovariables.
LetπbetheprojectionmapfromtheΣtothex-axis:
π:R2 →R, suchthatπ(x,y)= x. (3)
Forazero-dimensionalsystemΣdefinedin(2),lett(x)∈Q[x]betheresultantof f(x,y)andg(x,y)withrespecttoy:
t(x)=Res (f(x,y),g(x,y)). (4)
y
2
SinceΣiszero-dimensional,wehavet(x) . 0. Thenπ(V(Σ)) ⊆ V(t(x)), whereV(f ,··· , f ) isthesetofcommon
1 m
realzerosof f =0. Ift(x)isirreducible,thendenotethehighestdegreebydeg. Lettherealrootsoft(x)=0be
i t
α <α <···<α . (5)
1 2 u
Byusingthesamemethod,letT(y)∈Q[y]betheresultantof f(x,y)andg(x,y)withrespecttox:
T(y)=Res (f(x,y),g(x,y)). (6)
x
IfT(y)isirreducible,thendenotethehighestdegreebydeg . LettherealrootsofT(y)beasfollows:
T
β <β <···<β . (7)
1 2 v
We observe thatthe aboveprojectionmap may generateextraneousroots. Fortunately,we can easily discard these
extraneousfactorsbycomputingthedeterminantofthesub-matrixofthecoefficientmatrix.Moreover,iftheresultant
is irreducible, then it is no extraneous factors. However, when the resultant is reducible, it may suffer from the
extraneous factors. The method of removing extraneous factors mentioned can be adapted to the resultant for the
bivariatepolynomialsystem[27]. Itisthefollowingtheoremtoremovetheextraneousroots.
Theorem2.1. Σ is definedas in (2). If the resultants of Σ for one variable is reducible, denotedby tem, then the
resultantofbivariatepolynomialsystemistheonlysomeirreduciblefactorsinwhichtheothervariableappear.
Proof. TheproofcanbegivensimilarlytothatinProposition4.6ofChapter3of[8].
2.2. Alowerboundfor|Σ|,if Σ,0
Thepurposeofthissubsectionistoprovethefollowingtheorem.
Theorem 2.2. Σ is defined as in (2). Let α,β be two approximate real algebraic numbers. Denote by the integer
s=deg ·deg ,andN asabove.If |Σ|,0,then
t T
|Σ|≥ N1−sM−c·s, (8)
wherecistheconstantsatisfyingcertainconditions,|Σ|isthefollowingtwocases:
(a)If f(α,β)=0org(α,β)=0,then|Σ|=max{|f(α,β)|,|g(α,β)|};
(b)If f(α,β),0andg(α,β),0,then|Σ|=min{|f(α,β)|,|g(α,β)|}.
Beforegivingtheproofoftheorem2.2,werecalltwolemmas:
Lemma2.1. ([20],lemma3)Letα ,...,α bealgebraicnumbersofexactdegreeofd ,...,d respectively. Define
1 q 1 q
D=[Q(α ,...,α ):Q]. LetP∈Z[x ,...,x ]havedegreeatmostN inx (1≤h≤q). IfP(α ,...,α ),0,then
1 q 1 q h h 1 q
q
|P(α1,...,αq)|≥kPk11−D YM(αh)−DNh/dh,
h=1
wheretheM(α )istheMahlermeasureofα .
h h
Proof. SeetheLemma4of[20].
Lemma 2.2. Letαbe analgebraicnumber. Denoteby the M(α) ofthe Mahlermeasureofα. If P isapolynomial
overZ,then
M(α)≤||P|| .
1
Proof. For anypolynomialP = d p ∈ Z[x] of degreed with theall rootsσ(1),··· ,σ(d), we definethe measure
Pi=0 i
M(P)by
M(P)=|p |Πd max{1,|σ(i)|}.
d i=1
TheMahlermeasureofanalgebraicnumberisdefinedtobetheMahlermeasureofitsminimalpolynomialoverQ.
WeknowfromLandau([14],p. 154,Thm.6. 31)thatforeachalgebraicnumberα
M(α)≤||P|| ,
2
where||P|| =( d |p|2)1/2. Itisveryeasytogetthat||P|| ≤||P|| . Thiscompletestheproofofthelemma.
2 Pi=0 i 2 1
3
NowweturntogivetheproofofTheorem2.2.
Proof. Fromtheassumptionofthetheorem,sinceΣisdefinedasin(2). Letthepair(α,β)becorrespondingvalueto
thevariablexandyforΣrespectively.Wehavethefollowingequations:
f(α,β)= X X ai,jαiβj (9a)
0≤i≤n0≤j≤m
g(α,β)= X X bi,jαiβj. (9b)
0≤i≤p0≤j≤q
Atfirst,weconsiderthelowerboundfortheequation(9a). Definek =[Q(α,β):Q]. Denoteby|f|=|f(α,β)|,andr,t
bytheexactdegreeofalgebraicnumbersα,βrespectively.FromLemma(2.1),if|f|,0,then
|f|≥||f||1−kM(α)−kn/rM(β)−km/t.
1
WeobservethatM(α)andM(β)derivefromt(x)andT(y)respectively.FromLemma(2.2),wecangetthefollowing
inequality:
M(α)≤||t|| ,M(β)≤||T|| .
1 1
Sowecanobtainthat
|f|≥||f||1−k||t||−kn/r||T||−km/t. (10)
1 1
Byusingthesametechniqueasabove,wecanobtainthelowerboundfortheequation(9b). Denoteby|g|=|g(α,β)|.
If|g|,0,then
|g|≥||g||1−k||t||−kn/r||T||−km/t. (11)
1 1
Sincewehavethefollowingtwocases:
(a)If f(α,β)=0org(α,β)=0,then|Σ|=max{|f(α,β)|,|g(α,β)|};
(b)If f(α,β),0andg(α,β),0,then|Σ|=min{|f(α,β)|,|g(α,β)|}.
Henceweareabletoobtainthelowerboundforthebivariatepolynomialsystem. Fromtheaboveassumption,wecan
getthefollowingparameters:
k=[Q(α,β):Q]≤deg{t(x)}·deg{T(y)}=deg ·deg , (12a)
t T
N =max{||f|| ,||g|| },M =max{||t|| ,||T|| },r=deg,t=deg . (12b)
1 1 1 1 t T
Combinedwiththeequation(12a)and(12b),itisobviousthats=kandtheconstantc= degt +degT +1.Finally,note
r t
thattheconstantcsatisfiesbothcases. Thisprovesthetheorem.
AsthecorollaryofTheorem2.2,wehave
Corollay2.1. UnderthesameconditionofTheorem2.2,if |Σ|< N1−sM−c·s,then|Σ|=0. Wesaythatαisassociated
withβfortherealrootof Σ. Denotebytheε= N1−sM−c·s fortherestofthispaper.
Proof. Theproofisveryeasybycontradiction.
2.3. Rootmultiplicity
TheresultsofthissubsectioncanbeprovidedfortherootmultiplicityofΣ. Wefollowtheapproachandterminol-
ogyof[8]and[11].
LetC ,C be f,gcorrespondingaffinealgebraicplanecurves,definedbytheequationsΣ. LetI =< f,g>bethe
f g
idealthattheygeneratein F[x,y],andsotheassociatedquotientringisA = F[x,y]/I. Letthedistinctintersection
points,whicharethedistinctrootsof(Σ),beC ∩C ⊂{S =(α,β )} .
f g ij i j 1≤i≤u,1≤j≤v
ThemultiplicityofapointS is
ij
mult(S :C ∩C )=dim A <∞,
ij f g F Sij
4
whereA isthelocalringobtainedbylocalizingAatthemaximalidealI =< x−α,y−β >.
Sij i j
IfA isafinitedimensionalvectorspace,thenS =(α,β )isanisolatedzeroofI anditsmultiplicityiscalled
Sij ij i j
theintersectionnumberofthetwocurves.ThefiniteAcanbedecomposedasadirectsumA=A A ···
S11L S12L
A andthusdim A= uv mult(S :C ∩C ).
L Suv F Pi=1 ij f g
2
Proposition2.1. ([11],Proposition1)Let f,g∈F[x,y]betwocoprimecurves,andlet p∈F beapoint.Then
mult(p: fg)≥mult(p: f)mult(p:g),
whereequalityholdsifandonlyifC andC havenocommontangentsat p.
f g
Proposition 2.2. Let us obtain the real roots of Σ = 0 in (5) and (7). If the two matching pairs (α,β ) and
i j
(α ,β )(for 1 ≤ i ≤ u,1 ≤ j ≤ v) are satisfying Σ = 0, |α −α | < ε and |β −β | < ε, then the (α,β )
i+1 j+1 i i+1 j j+1 i j
ismultiplerootof Σ=0.
Proof. FromTheorem2.2andCorollary2.1,itisobviousthatΣ=0ifandonlyif
|Σ|<ε.
Therefore,theerrorcontrollingislessthanεinnumericalcomputation.Undertheassumptionoftheproposition,we
get|α −α |<εand|β −β |<ε. Soweareabletoobtainthat|α −α |=0and|β −β |=0intthetruncated
i i+1 j j+1 i i+1 j j+1
error.Thisprovestheproposition.
FromCorollary2.1,thetwo-tuple(α,β)istherealrootofΣ=0. Thismethodiscalledazero-matchingmethod.
Thetechniqueisa posteriorimethodtomatchthesolutionsforthe bivariatesystem. Itcanbe generalizedeasilyto
realsolvingthemultivariatepolynomialsystems.
3. DerivationoftheAlgorithm
Theaimofthissectionistodescribeanalgorithmforrealsolvingbivariatepolynomialequationsbyusingzero-
matchingmethod.WefirstfindtheparametersN,cands,thenobtainthenoextraneousfactorst(x)andT(y)withthe
resultanteliminationmethods, and real solvingtwo univariatepolynomials, and finally match the realroots for the
systems.
3.1. Descriptionofalgorithm
Algorithm1istodiscardtheextraneousfactorsfromtheresultantmethod,algorithm2istoobtainthesolutions
ofbivariatepolynomialsystems.
Algorithm1NoExtrRes(Σ,var)
Input: {f(x,y),g(x,y)},varisonevariable.
Output: NoextraneousfactorsresultantofΣ.
1: tem←Resvar{f(x,y),g(x,y)};
2: if temisirreduciblethen
3: return tem;
4: else
5: tem←Res·extraneousfacotrs;
6: return Res.
7: endif
Nowwecangivethealgorithm2tocomputetherealrootsforΣ=0.
Theparallelizationofthealgorithmthatwehavejustdescribedcanbeeasilydonebecauseitperformsthesame
computationsondifferentstepsofdatawithoutthenecessityofcommunicationbetweentheprocessors.Observethat
theStep1andStep2,Step6andStep7ofthealgorithmcanbeeasilyparalleled,respectively.
Nowwegetatheoremaboutthecomputationalcomplexityofthewholealgorithm.
5
Algorithm2zmm(Σ)
Input: Σ={f(x,y),g(x,y)}isazero-dimensionalbivariatepolynomialsystem.
Output: AsetfortherealrootsofΣ=0.
1: Projectonthe x-axissuchthatt(x)=Resy(f(x,y),g(x,y));
2: Projectonthey-axissuchthatT(y)=Resx(f(x,y),g(x,y));
3: Discardtheextraneousfactorsfromt(x)andT(y)byusingAlgorithm1;
4: FindtheparametersN ands,andComputecaccordingtotheTheorem2.2;
5: ObtainthelowerboundεbyCorollary2.1;
6: Solvetherealrootsoftheresultantt(x)forthesetSx ={α1,α2,··· ,αu};
7: SolvetherealrootsoftheresultantT(y)forthesetSy ={β1,β2,··· ,βv};
8: MatchtherealrootpairtogetthesolvingsetS={(αi,βj),1≤i≤u,1≤ j≤v}byCorollary2.1;
9: ChecktherootmultiplicityofthesetSbyProposition2.2.
Theorem3.1. Algorithm2workscorrectlyasspecifiedanditscomplexityincludesasfollows:
(a) O(dτ + dlgd) for computation of real solving univariate polynomial, where d is the degree of corresponding
polynomial,τ=1+max lg|a|anda isthecoefficients.
i≤d i i
(b)O(uv)formatchingthesolutionsofbivariatepolynomialsystem.
Proof. Correctnessofthealgorithmfollowsfromtheorem2.2.
(a)Thenumberofarithmeticoperationsrequiredto isolateallrealrootsisthe numberofrealrootisolationofuni-
variatepolynomialbyusingsubdivision-basedDescartes’ruleofsign. Usingexactlythesameargumentswe know
thattheyperformthesamenumberofsteps,thatisO(dτ+dlgd).
(b) As indicated before, the problemof matching the realroots of polynomialsystem mainly relies on the scale of
solutionsofeveryvariable,respectively.
3.2. Asmallexampleindetail
Example3.1. Weproposeasimpleexample f(x,y)= x2−y2−3andg(x,y)=3x2−2y3−1toillustrateouralgorithms.
Step1: t(x)=4∗x6−45∗x4+114∗x2−109;
Step2: T(y)=(−2∗y3+8+3∗y2)2;
Step3: DiscardtheextraneousfactorsT(y)=−3∗y2−8+2∗y3;
Step4: ObtaintheparametersN =5,c=2,s=4;
Step5: Obtainthelowerboundε=.1280×10−4;
Step6: Solvetherealrootsoftheresultantt(x)forthesetS ={−2.858288520,2.858288520};
x
Step7: SolvetherealrootsoftheresultantT(y)forthesetS ={2.273722337};
y
Step8: CombinethethepairsfromS andS respectively,SubstitutethepairsintoΣforvariablesxandy,
x y
determinewhetherlessthanthelowerboundε,finallywefindthatthepairsS={{x=−2.858288520,y=
2.273722337},{x=2.858288520,y=2.273722337}}arethesolutionsforΣ;
Step9: Themultiplicityoftherootofthesystemisone.
3.3. Generalizationandapplications
Asforthegeneralizationofthealgorithmtorealsolvingthemultivariateequationsystemscase,wehavetosaythat
thesituationiscompletelyanalogoustothebivariatecase. However,itskeytechniqueistotransformthemultivariate
polynomialequations to the correspondingunivariate polynomialequations. We can consider the Dixon Resultant
Method to break this problem [6]. However, we observe that how to improve the projection algorithm in resultant
methodsisthesignificantchallenge.
6
Moreover,ouralgorithmisapplicableforrapidlycomputingtheminimumdistancebetweentwoobjectscollision
detection[25]. Thisalso enablesusto improvethecomplexityofcomputingthe topologyof a realplanealgebraic
curve[9].
4. Somecomparisons
WehaveimplementedtheabovealgorithmsasasoftwarepackagezmminMaple12. Forproblemsofsmallsize
liketheexampleofSection3,anymethodcanobtainthesolutionsinlittletime. Butwhenthesizeoftheproblemsis
notsmallthedifferencesappearclearly. Extensiveexperimentswiththispackageshowthatthisapproachisefficient
andstable,especiallyforlargerandmorecomplexbivariatepolynomialsystems.
We compareourmethodwithlgp[4],Isolate[23], discoverer[24], andgrur [9]. lgpisasoftwarepackagefor
rootisolation of bivariatepolynomialsystemswith localgenericpositionmethod. Isolate is a toolto solve general
equation systems based on the Realsolving c library by Rouillier. discoverer is a tool for solving semi-algebraic
systems. grur is a tool to solve bivariate equation systems. The following examples run in the same platform of
Maple12underWindowsandamdAthlon(tm)2.70ghz,2.00gbofmainmemory. Wedidthreesetsofexperiments.
Theprecisionintheseexperimentsissettobehigh. Inthreetables,where’?’ representsthatthecomputationisnot
finished.
InTable1theresultsaregivenboth f andgarerandomlygenerateddensepolynomialswiththesamedegreeand
withintegercoefficientsbetween−20and20.ThecommandofMapleisasfollows:
randpoly([x,y],coeffs=rand(−20..20),dense,degree=10).
Table1:timeforcomputingdensebivariatepolynomialswithnomultipleroots
s
n
m o
yste deg oluti AverageTime(sec)
s s
f g zmm lgp Isolate discoverer grur
S1 4 7 2 0.031 0.031 0.047 0.313 2.734
S2 6 8 6 0.415 1.328 0.500 1.828 247.203
S3 7 8 6 1.204 2.734 1.500 7.047 382.640
S4 8 9 6 4.211 8.906 4.672 20.437 2714.438
S5 9 10 2 4.070 8.485 4.687 89.235 1645.312
S6 10 7 6 1.805 3.860 2.109 22.250 978.421
S7 10 11 4 21.078 43.734 22.828 ? ?
S8 12 11 2 26.945 54.969 29.094 ? ?
S9 12 13 4 118.266 241.734 123.469 ? ?
S10 13 11 1 15.446 31.485 17.796 ? ?
S11 14 10 8 63.914 200.828 68.594 ? ?
InTable2theresultsaregivenboth f andgarerandomlygeneratedsparsepolynomialsinthesamedegree,with
sparsitydefault,andwithintegercoefficientsbetween−20and20. ThecommandofMapleisasfollows:
randpoly([x,y],coeffs=rand(−20..20),sparse,degree=10).
7
Table2:timeforcomputingsparsebivariatepolynomialswithnomultipleroots
s
n
m o
yste deg oluti AverageTime(sec)
s s
f g zmm lgp Isolate discoverer grur
S1 5 6 1 0.015 0.032 0.015 0.141 1.032
S2 6 7 3 0.040 0.062 0.047 0.188 5.375
S3 7 5 3 0.024 0.047 0.047 0.265 2.688
S4 8 6 5 0.031 0.031 0.047 0.094 1.031
S5 9 8 2 0.047 0.172 0.078 1.828 51.000
S6 10 11 3 0.063 0.297 0.125 0.656 11.110
S7 11 9 2 0.164 0.609 0.375 3.938 877.875
S8 12 13 2 1.141 2.593 1.453 6.703 1607.719
S9 13 11 4 2.508 5.344 2.969 ? ?
S10 15 17 1 0.532 1.234 1.266 ? ?
S11 20 17 4 18.180 39.688 20.235 ? ?
In Table 3 the results are given is done with polynomialsystems with multiple roots. We randomlygenerate a
polynomialh(x,y,z)andtake f(x,y) = Res (h,h ),g(x,y) = f (x,y). Since f(x,y)istheprojectionofaspacecurve
z z y
tothe xy-plane,itmostprobablyhassingularpointsand f = g = 0isanequationsystemwith multipleroots. The
commandofMapleisasfollows:
h:=randpoly([x,y,z],coeffs=rand(−5..5),degree=5); f :=resultant(h,diff(h,z),z);g:=diff(f,y).
Table3:timeforcomputingbivariatepolynomialswithmultipleroots
s
n
m o
yste deg oluti AverageTime(sec)
s s
f g zmm lgp Isolate discoverer grur
S1 3 2 2 0.016 0.016 0.016 0.016 0.062
S2 4 3 2 0. 0.032 0.031 0.016 0.094
S3 4 6 7 0.024 0.016 0.047 0.109 1.109
S4 5 4 3 0. 0.016 0. 0.016 0.109
S5 6 5 2 0.015 0. 0. 0.016 0.063
S6 9 8 2 0.016 0.046 0.032 0.015 0.063
S7 12 11 3 0.109 0.234 0.187 0.063 0.094
S8 13 12 7 2.875 137.641 3.141 1.328 207.094
S9 14 13 4 0.860 2.891 0.953 0.141 0.3110
S10 19 18 1 0.672 1.547 0.797 22.156 1520.812
S11 16 15 5 7.945 27.047 9.000 ? ?
FromtheTable1,2and3,wehavethefollowingobservations.
Inthefirsttwocases, theequationsarerandomlygeneratedandhencemayhavenomultipleroots. Forsystems
withoutmultipleroots,zmmisthefastestmethod,whichissignificantlyfasterthanlgpandIsolate. Bothzmmandlgp
computetworesultantsandisolatetheirrealroots.lgpisslow,becausethepolynomialsobtainedbytheshearmapare
usuallydenseandwithlargecoefficients[4]. discovererandgrurgenerallyworkforequationsystemswithdegrees
nothigherthantenwithinreasonabletime.
Forsystemswith multipleroots, in thesparse andlow degreecases, allmethodsarefast. Note thatourmethod
8
isquitestableforequationsystemswithandwithoutmultipleroots. lgpandIsolatearealsoquitestable,butslower
thanzmmforbivariateequationsystems.
We also observe that all methods spend more time with sparse and dense polynomials than polynomials with
multiplerootsinthesamehighdegree.Thisphenomenonneedsfurtherexploration.
Remark 4.1. Of course, we should mention that discoverer and Isolate can be used to solve general polynomial
equationsandeveninequalities. Hereourcomparisonislimitedtothebivariatecase. Infurtherwork,wewouldlike
toconsidersolvingmultivariatepolynomialequations.
Remark4.2. Asiswellknown,theparallelalgorithmiswellsuitedfortheimplementationonparallelcomputersthat
allowsthe increase ofthe calculationspeed. Ifouralgorithmhavebeenfully parallelizedby using a large enough
numberof processors for each case, the real solutions of all the examples will have been computed in a couple of
seconds.
5. Conclusion
In this paper, we propose a zero-matching method to real solving bivariate polynomialequation systems. The
basic idea of this method is to find the lower bound for bivariate polynomialsystem when the system is non-zero.
Moreover, we provide an algorithm for discarding extraneous factors with resultant and show how to construct a
parallelizedalgorithmforrealsolvingthebivariatepolynomialsystem. Anefficientmethodformultiplicitiesofthe
rootsis alsoderived. Thecomplexityofourmethodhasincreasedsteadilywiththe growthofbivariatepolynomial
system.Extensiveexperimentsshowthatourapproachisefficientandstable. Theresultofthispapercanbeextended
to real solving of bivariate polynomial equations with more than two polynomials by using the resultant method.
Furthermore,ourmethodcanbegeneralizedeasilytomultivariatepolynomialsystems.
6. Acknowledgement
WewouldliketothankProf. XiaoshanGaoandDr. JinsanChengforprovidingtheMaplecodeoftheirmethod,
availableat: http://www.mmrc.iss.ac.cn/∼xgao/software.html.
ThefirstauthorisalsogratefultoDr. ShizhongZhaoforhisvaluablediscussionsaboutdiscardingtheextraneous
factorsinresultant.
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