From Approximate Factorization to Root Isolation with Application to Cylindrical Algebraic Decomposition

0.3 MB·English

by  Kurt Mehlhorn

From Approximate Factorization to Root Isolation
with Application to Cylindrical Algebraic Decomposition
3 KurtMehlhornand MichaelSagraloffand PengmingWang∗
1
0 January 22,2013
2

n
a
J
Abstract
1
2 Wepresentanalgorithmforisolatingtherootsofanarbitrarycomplexpolynomial pthatalsoworks
for polynomials with multiple roots provided that the number k of distinct roots is given as part of

] the input. It outputs k pairwise disjoint disks each containing one of the distinct roots of p, and its
C
multiplicity.Thealgorithmusesapproximatefactorizationasasubroutine.
S
Inaddition,weapplythenewrootisolationalgorithmtoarecentalgorithmforcomputingthetopol-
.
s ogyof areal planaralgebraiccurvespecifiedasthezerosetof abivariateintegerpolynomial andfor
c isolatingtherealsolutionsofabivariatepolynomialsystem. Forinputpolynomialsofdegreenandbit-
[ sizet ,weimprovethecurrentlybestrunningtimefrom O˜(n9t +n8t 2)(deterministic)toO˜(n6+n5t )
1 (randomized)fortopologycomputationandfromO˜(n8+n7t )(deterministic)toO˜(n6+n5t )(random-
v ized)forsolvingbivariatesystems.
0
7
8 1 Introduction
4
1. Inthemainpartofthispaper,wegiveanalgorithmforisolatingtherootsofapolynomialp(x)=(cid:229) ni=0pixi
0 witharbitrarycomplexcoefficientsthatworksforpolynomialswithmultipleroots,providedthatthenum-
3 berkofdistinctrootsisgivenaspartoftheinput. Moreprecisely,letz ,...,z bethedistinctrootsof p,
1 k
1 letm :=mult(z,p)bethemultiplicityofz,andlets :=s (z,p):=min z z betheseparationofz
v: fromitheotherriootsof p. Then,thealgoriithmoutputisisolatiingdisksD j=6=iD (iz˜−,Rj )toD =D (z˜ ,R )fori
1 1 1 k k k
i (cid:12) (cid:12) s
X tthheercoeonttserozf˜poafsD waepllparsoxthimecaotersrezsptoonadninegrrmorulotfiplleiscsititehsanm1stio.mIfk.thTehneurmadbiei(cid:12)rsaotfisdfyi(cid:12)stRinic<tr6o4ointsfoorfaplldi,iftfheurss
r i i i 64n
a fromk,wemakenoclaimsaboutterminationandoutput.
The coefficients of p are provided by oracles. On input L, such an oracle essentially returns binary
fractionapproximationsp˜ ofthecoefficientsp suchthat p (cid:229) n p˜xi 2 L p . Here, p := p =
i i − i=0 i ≤ − k k k k k k1
p +...+ p denotestheone-normof p. ThedetailsaregiveninSection2.1.
0 n
| | | | (cid:13) (cid:13)
Thealgorithmhasasimplestructure.Wefirstuseany(cid:13)algorithm(e.g(cid:13).[4,18,16,21])forapproximately
factorizingtheinputpolynomial.Itisrequiredthatitcanberunwithdifferentlevelsofprecisionandthat,
foranygivenintegerb,itreturnsapproximationszˆ tozˆ suchthat
1 n
p p (cid:213) (x zˆ ) 2 b p . (1)
− n 1 j n − j ≤ − k k
≤ ≤
(cid:13) (cid:13)
Inasecondstep,wepartitionther(cid:13)ootapproximationszˆ to(cid:13)zˆ intokclustersC ,...,C basedongeometric
(cid:13) 1 (cid:13)n 1 k
vicinity.WeencloseeachclusterC inadiskD =D (z˜,r)andmakesurethatthedisksarepairwisedisjoint
i i i i
and that the radii r are not“too small” comparedto the pairwise distances of the centers z˜.1 In a third
i i
step, we verify that the n-times enlarged disks D =D (z˜,R)=D (z˜,n r) are disjoint and that each of
i i i i i
·
themcontainsexactlythesamenumberofapproximationsasrootsof pcountedwithmultiplicity.2 Ifthe
MaxPlanckInstituteforInformatics,Germany
∗
1Thiswillturnouttobecrucialtocontrolthecostforthefinalverificationstep.Fordetails,werefertoSections2.2.2and2.2.3.
2ItseemstobeartificialtoconsiderenlargeddisksD iinsteadofthedisksDi.ThisismainlyduetotheusageofRouche´’sTheorem
toverifyroots.
1
clusteringandtheverificationsucceed,wereturnthedisksD ,...,D andthenumberofapproximationszˆ
1 k
∈
zˆ ,...,zˆ inthediskasthemultiplicityoftherootisolatedbythedisk.Ifeitherclusteringorverification
1 n
{ }
doesnotsucceed,werepeatwithahigherprecision.
IfPan’salgorithm[16]isusedfortheapproximatefactorizationstep,thentheoverallalgorithmhasbit
complexity3
k k
O˜ n3+n2i(cid:229)=1logM(zi)+ni(cid:229)=1log M(s i−mi)·M(Pi−1) ! (2)
(cid:16) (cid:17)
iwnhaerevePriy:=str(cid:213)onjg6=is(ezin−se,zjn)mamj =elyp,(mmtihi!)pe(nzia),boavnedbMo(uxn)d:=dirmecatxl(y1,d|exp|)e.ndOsbosenrvtheethaacttuoaulrmaluglotirpitlhicmitiiessaadnadpttivhee
geometry(i.e.theactualmodulusoftherootsandtheirdistancestoeachother)oftheroots. Thereisalso
no dependencyon the size or the type (i.e. whetherthey are rational, algebraic or transcendental)of the
coefficientsof p.
Ouralgorithmcanalsobeusedtofurtherrefinetheisolating diskstoasizeof2 k orless,wherek is
−
agiveninteger. Thebitcomplexityfortherefinementisgivenbytheboundin(2)plusanadditionalterm
O˜(n k max m). Inparticularforsquare-freepolynomials,theamortizedcostperrootandbitofprecision
i i
· ·
isone.
Forthebenchmarkproblemofisolatingallrootsofapolynomialpwithintegercoefficientsofabsolute
value bounded by 2t , the bound in (2) becomes O˜(n3+n2t ). The bound for the refinement becomes
O˜(n3+n2t +nk ),evenifthereexistmultipleroots.
Forasquare-freeintegerpolynomialp,weareawareofonlyonemethod[9,Theorem3.1]thatachieves
a comparable complexity bound for the benchmark problem. That is, based on the gap theorem from
Mahler,onecancomputeatheoreticalworstcaseboundb ofsizeQ (nt )withthepropertythatifnpoints
0
z˜ Cfulfilltheinequality(1)fora b b ,thentheyapproximatethecorrespondingroots z toanerror
j 0 j
les∈sthans /(2n);cf. Lemma4foran≥adaptiveversion. Hence,forb b ,Pan’sfactorizationalgorithm
j 0
alsoyieldsisolatingdisksfortherootsof pusingO˜(n2t )bitoperations≥.Notethatthisapproachachievesa
goodworstcasecomplexity,however,forthepriceofrunningthefactorizationalgorithmwithb=Q (nt ),
evenif therootsarewell conditioned. Incontrast, ouralgorithmturnsPan’sfactorizationalgorithminto
a highly adaptive method for isolating and approximating the roots of a general polynomial. Also, for
generalpolynomials, there exist theoreticalworst case bounds[18, Section 19] forthe distance between
therootsof pandcorrespondingapproximationsfulfilling(1). Theyareoptimalforrootsofmultiplicity
W (n) but they constitute strong overestimationsif all rootshave considerablysmaller multiplicities. For
thetaskofrootapproximation,thecomplexityofourmethod adaptstothehighestoccurringmultiplicity
(with bit complexity O˜(nmax m k ) for k dominating), whereas this is not given for the currently best
i i
method[16](withcomplexityO˜(n·2k )).
Wewouldalsoliketoremarkthatweareawareofonlyonepreviousrootisolationalgorithm[15]that
cancopewithmultipleroots,however,withatmostonemultipleroot. Inaddition,thenumberofdistinct
complexrootsaswellasthenumberofdistinctrealrootsmustbegivenasanadditionalinput.
Finally, we aim to stress the importanceof our root isolation method by applying it to the problems
of computingthe topology(in terms of a cylindricalalgebraic decomposition)of a real planar algebraic
curve specified as the zero set of an integerpolynomialand of isolating the real solutions of a bivariate
polynomial system. Both problems are well-studied [1, 11, 12, 19, 7, 5, 3, 8, 14]. More specifically,
we apply our method to a recent algorithm [3] for computing the topology of a planar algebraic curve.
Thisyieldsboundsontheexpectednumberofbitoperationswhichimprovethecurrentlybest(whichare
bothdeterministic)bounds[8,14]fromO˜(n9t +n8t 2)toO˜(n6+n5t )fortopologycomputationandfrom
O˜(n8+n7t )toO˜(n6+n5t )forsolvingbivariatesystems.
3O˜indicatesthatweomitlogarithmicfactors.
2
2 Root Approximation
2.1 Setting andBasicProperties
Weconsiderapolynomial
p(x)=p xn+...+p C[x] (3)
n 0
∈
ofdegreen 2,where p =0. Wefixthefollowingnotations:
n
≥ 6
M(x):=max(1, x),forx R,
• | | ∈
• t pdenotestheminimalnon-negativeintegerwith |ppni| ≤2tp foralli=0,...,n−1,
| |
p := p := p +...+ p denotesthe1-normof p,
• k k k k1 | 0| | n|
z ,...,z Carethedistinctcomplexrootsof p,withk n,
1 k
• ∈ ≤
m :=mult(z,p)isthemultiplicityofz,
i i i
•
s :=s (z,p):=min z z istheseparationofz,
i i j=i i j i
• 6 | − |
G :=M(max log z )thelogarithmicrootboundof p,
p i i
• | |
Mea(p)= pn (cid:213) iM(zi)mi theMahlerMeasureof p.
• | |·
Thequantitiest ,G , p andMea(p)arecloselyrelated.
p p n
| |
Lemma1. G 1+t andt n 1 logMea(p) nG .
p≤ p p− − ≤ pn ≤ p
| |
Proof. ByCauchy’srootboundmax z 1+max p / p ,andthusmax log z 1+t .Sincet 0,
i i i i n i i p p
by definition, we have G 1+t .|T|h≤e i-th coeffi|ci|en|t of| p is smaller than|or|e≤qual to n Mea(p≥)
p ≤ p i ≤
2nMea(p)≤2n(G p+1). Thus,fromthedefinitionoft p,eithert p=0or2nMepan(p) ≥maxi |ppni| (cid:0)≥(cid:1)2tp−1
| | | |
Weassumetheexistenceofanoraclewhichprovidesarbitrarygoodapproximationsofthepolynomial
p. LetL 1beaninteger. Wecallapolynomial p˜= p˜ xn+...+p˜ ,with p˜ =s 2 ℓ ands, ℓ Z, an
n 0 i i − i
approxim≥ationof precision L of p if p˜ p 2 L log(n+1) p , ℓ L+ log(n+·1) log p∈ , and
i i − −
log s L+ log(n+1) +1foralli.| W−hen|c≤onsideringp askinkfinit≤ebitstr⌈ingp =sgn⌉(−p⌊) (cid:229) +k¥ k⌋b 2k,
| i|≤ ⌈ ⌉ i i i · k= ¥ k
b 0,1 , then we can obtain p˜ from the partial string which starts at index k = log p a−nd ends
k i 1
∈{ } ⌊ k k⌋
at index k = log p L log(n+1) , that is, s :=2l sgn(p) (cid:229) k1 b 2k, and l =L+ log(n+
2 ⌊ k k⌋− −⌈ ⌉ i · i · k=k2 k ⌈
1) log p . We assume thatwe can ask for an approximationof precision L of p at cost O(n(L+
log⌉n−))⌊=O˜k(nkL⌋). ThisisthecostofreadingtheapproximationofprecisionL. ThenextLemmastatessome
elementarypropertiesofapproximationsofprecisionL.
Lemma2. Let p˜beanapproximationofprecisionLof pandL 1.
≥
p˜ /2 p 2 p˜ .
• k k ≤k k≤ k k
IfL t +4,then2 L log(n+1) p˜ p˜ /4.
p − − n
• ≥ k k≤| |
If2 L log(n+1) p˜ p˜ /4,then p˜ /2 p 2 p˜ .
− − n n n n
• k k≤| | | | ≤| |≤ | |
Proof. Fromtheinequality p˜ p (cid:229) p˜ p ,weconcludethat p˜ p 2 L p p /2.
i i i −
|k k−k k|≤ | − | |k k−k k|≤ k k≤k k
Thisestablishesthefirstclaim. Forthesecondclaim,weobservethat
p˜ p 2 L log(n+1) p 2 L+tp p p /16.
n n − − − n n
| − |≤ k k≤ | |≤| |
Thus, p˜ p /2and
n n
| |≥| |
2 L log(n+1) p˜ 2 2 L log(n+1) p p /8 p˜ /4.
− − − − n n
k k≤ · k k≤| | ≤| |
3
Thethirdclaimfollowsfrom
p˜n pn 2−L−log(n+1) p 2 2−L−log(n+1) p˜ p˜n /2.
| − |≤ k k≤ · k k≤| |
Lemma2suggestsanefficientmethodforestimating p . Weaskforapproximations p˜ ofprecisionL
n
ofpforL=1,2,4,...untiltheinequality2 L log(n+1) p˜ p˜ /4holds.Then, p˜ /2 p 2 p˜ by
− − n n n n
part3oftheLemma.AlsoL 2(t +4)bypart2ofthkekab≤ov|eL|emma.Thecost|isO|˜(nt≤)|=|O˜≤(n2|G )|bit
p p p
operations,whereweusedth≤eupperboundfor t fromLemma1. Observethatthisbounddependsonly
p
on the geometryof the roots(i.e. the actualroot bound G ) and the degree but not(directly) on the size
p
ofthecoefficientsof p. Wenextshowthata“good”integerapproximationG ofG canalsobecomputed
p
withO˜(n2G )bitoperations.
p
Theorem1. AnintegerG Nwith
∈
G G <8logn+G (4)
p p
≤
canbecomputedwithO˜(n2G )bitoperations.ThecomputationusesanapproximationofprecisionLof p
p
withL=O(nG ).
p
Proof. Wefirstcomputeanapproximation p˜ ofprecisionLof pwith p˜ /2 p 2 p˜ asdescribed
n n n
above. Letk :=2⌊log|p˜n|⌋−1. Thenk p˜n /2 pn andk 2⌊log|pn/|2|⌋−|1 ≤p|n /|8≤. Co·|nsid|erthescaled
polynomialq:= p/k . Its leading co≤effi|cie|ntl≤ies|be|tween 1≥and 8. Moreo≥ver|,G | =G , t =t , and an
q p q p
arbitraryapproximationofprecision Lof palsoyieldsanapproximationofprecisionLofqsincedivision
byl isjustashiftbylogk bitsand q = p /k .Hence,wemayassumethat1 p 8.
n
k k k k ≤| |≤
ConsidertheCauchypolynomial
n 1
p¯(x):= p xn (cid:229)− p xi
n i
| | − | |
i=0
of p. Then,accordingto[6,Proposition2.51], p¯hasauniquepositiverealrootx R+,andthefollowing
∈
inequalityholds:
n
max z x < max z <2n max z .
i i i
i | |≤ ln2· i | | · i | |
It follows that p¯(x)>0 for all x x and p¯(x)<0 for all x<x . Furthermore, since p¯ coincides with
its own Cauchy polynomial, each≥complex root of p¯ has absolute value less than or equal to x . Let
k be the smallest non-negativeinteger k with p¯(2k)>0 (which is equal to the smallest k with|2k|>x ).
0
Our goal is to compute an integer G with k G k +1. Namely, if G fulfills the latter inequality,
0 0
then M(max z ) M(x ) 2G <4M(x )<8n≤M(≤max z ), and thus G fulfills inequality(4). In order
i i i i
to compute a|G |w≤ith k ≤G k +1, we use ·exponenti|al|and binary search (try k =1,2,4,8,... until
0 0
≤ ≤
p¯(2k)>0 and, then, performbinarysearch on the interval k/2 to k) and approximateevaluationof p¯ at
thepoints2k: Moreprecisely,weevaluate p¯(2k)usingintervalarithmeticwithaprecision r (usingfixed
pointarithmetic)whichguaranteesthatthewidthwofB(p¯(2k),r )issmallerthan1,whereB(E,r )isthe
intervalobtainedbyevaluatingapolynomialexpressionE viaintervalarithmeticwithprecisionr forthe
basicarithmeticoperations;see[13,Section4]fordetails. Weuse[13,Lemma3]toestimatethecostfor
eachsuchevaluation:Since p¯hascoefficientsofsizelessthan2tp pn <2tp+3,wehavetochooser such
| |
that
2 r +2(n+1)22tp+3+nk<1
−
in order to ensure that w<1. Hence, r is bounded by O(t +nk) and, thus, each interval evaluation
p
needsO˜(n(t +nk))bitoperations.Wenowuseexponentialplusbinarysearchtofindthesmallestksuch
p
thatB(p¯(2k),r )containsonlypositivevalues. Thefollowingargumentthenshowsthatk k k +1:
0 0
Obviously,wemusthavek k since p¯(2k)<0and p¯(2k) B(p¯(2k),r )forallk<k . Fur≤therm≤ore,the
0 0
pointx=2k0+1 hasdistance≥morethan1toeachoftheroot∈sof p¯,andthus p¯(2k0+1) pn 1. Hence,
itfollowsthatB(p¯(2k0+1),r )containsonlypositivevalues.Forthesearch|,weneed|≥| |≥
O(logk )=O(loglogx )=O(log(logn+G ))
0 p
4
iterations,andthecostforeachoftheseiterationsisboundedbyO˜(n(t +nk ))=O˜(n2G )bitoperations.
p 0 p
2.2 Algorithm
Wepresentanalgorithmforisolatingtherootsofapolynomialp(x)=(cid:229) ni=0pixi=pn(cid:213) ki=1(x−zi)mi,where
the coefficients p are givenas describedin the previoussection. We may assume k>1; the problemis
i
trivialotherwise.Ifk=1, p /(np )istherootofmultiplicityn.Thealgorithmusessomepolynomial
n 1 n
factorizationalgorithmto p−rod−uceapproximationsforthe roots z ,...,z , andthenperformsa clustering
1 k
and certificationstep to verifythatthe candidatesare of highenoughquality. Forconcreteness, we pick
Pan’sfactorizationalgorithm[16]forthefactorizationstep,whichalsocurrentlyoffersthebestworstcase
bitcomplexity. Ifthecandidatesdonotpasstheverificationstep, we reapplythefactorizationalgorithm
withahigherprecision.Givenapolynomialpwith z 1for1 i k,andapositiveintegerbdenoting
i
| |≤ ≤ ≤
thedesiredprecision,thefactorizationalgorithmcomputesnrootapproximationszˆ ,...,zˆ . Thequalityof
1 n
approximationandthebitcomplexityareasfollows:
Theorem2(Pan[16]). Supposethat z 1for1 i k. Foranypositiveintegerb nlogn,complex
i
| |≤ ≤ ≤ ≥
numberszˆ ,...,zˆ canbecomputedsuchthattheysatisfy
1 n
p p (cid:213) n (x zˆ) 2 b p
− n i=1 − i ≤ − k k
using O˜(n) operations performed w(cid:13)ith the precision of(cid:13)O(b) bits (or O˜(bn) bit-operations). We write
(cid:13) (cid:13)
pˆ:= p (cid:213) n (x zˆ). Thealgorithmreturnstherealandimaginarypartofthezˆ’sasdyadicfractionsof
n i=1 − i i
theformA 2 BwithA Z,B NandB=O(b). Allfractionshavethesamedenominator.
−
· ∈ ∈
Theparameterbcontrolsthequalityoftheresultingapproximations.NotethatPan’salgorithmrequires
allrootsoftheinputpolynomialtoliewithintheunitdiskD (0,1).Hence,inordertoapplytheaboveresult
toourinputpolynomial,wefirstscale psuchthattherootscometolieintheunitdisk.Thatis,wecompute
a G as in Theorem 1, and then considerthe polynomial f(x):= p(s x)=(cid:229) n fxi with s:=2G . Then,
f(x)hasrootsx =z/s D (0,1), andthuswecanusePan’sAlgorith·mwithi=b0:=i nG +btocomputean
i i ′
approximatefactorization∈fˆ(x):=(cid:229) ni=0 fˆixi:= fn(cid:213) ni=1(x−xˆi)suchthat f−fˆ <2−b′kfk. Letzˆi:=s·xˆi
foralliand pˆ(x):=p (cid:213) n (x zˆ)= fˆ(x/s)=(cid:229) n fˆ/sixi. Then
n· i=1 − i i=0 i (cid:13) (cid:13)
(cid:13) (cid:13)
n n n
pˆ p =(cid:229) fi/si fˆi/si (cid:229) fi fˆi 2−b′(cid:229) fi
k − k | − |≤ | − |≤ | |
i=0 i=0 i=0
n
2−b′sn(cid:229) fi/si =2−b p .
≤ | | k k
i=0
Forthe factorizationof f, we need an approximationof precision b of f, and thus an approximationof
′
precision L of p with L=O(nS+b)=O˜(nG +b). The total cost is O˜(n2G +nb) bit operations. We
p
summarizein:
Corollary1. Foranarbitrarypolynomialpandanintegerb nlogn,complexnumberszˆ ,...,zˆ canbe
1 n
≥
computedsuchthat
p p (cid:213) n (x zˆ) 2 b p
− n i=1 − i ≤ − k k
iumsiangginOa˜(rny2pG a+rtobfn)thbeitzˆ-o’spaersadtiyoandsi.c(cid:13)(cid:13)fWraectwiornitseopˆft:h=epfonr(cid:213)mni=(cid:13)(cid:13)A1(2x−Bzˆwi)i.thTAhe aZl,goBrithNmarnedtuBrn=s tOhe(br+eanlGan)d.
i − p
· ∈ ∈
Allfractionshavethesamedenominator.
We now examine how far the approximations zˆ ,...,zˆ can deviate from the actual roots for a given
1 n
valueofb. LetD (z,r)bethediskwithcenterzandradiusrandletbdD (z,r)beitsboundary. Wefurther
definePi:=(cid:213) j6=i(zi−zj)mj. Then, p(mi)(zi)=mi!pnPi.
Lemma3. Ifr s /n,then
i
≤ p(x) >rmi p P /4
n i
| | | |
forallxontheboundaryofD (z,r).
i
5
Proof. Wehave
p(x) = p x z mi(cid:213) x z mj
n i j
| | | || − | | − |
j=i
6
p x z mi(cid:213) z z mj (1 x z /z z )mj
n i i j i i j
≥| || − | | − | · −| − | | − |
j=i
6
rmi(1 1/n)n mi p (cid:213) z z mj >rmi p P /4.
− n i j n i
≥ − | | | − | | |
j=i
6
BasedontheaboveLemma,wecannowuseRouche´’stheorem4,5 toshowthat,forsufficientlylarge
b,thediskD (zi,2−b/(2mi))containsexactlymirootapproximations.
Lemma4. Let pˆbesuchthat p pˆ 2 b p . If
−
k − k≤ k k
b max(8n,nlog(n)),andbisapoweroftwo (5)
≥
2−b/(2mi) 1/(2n2), (6)
≤
2 b/(2mi) s /(2n), and (7)
− i
≤
P
2−b/2≤ 16(n+1|)2i|tpM(zi)n (8)
for all i, the disk D (zi,2−b/(2mi)) contains exactly mi root approximations. For i= j, let zˆi and zˆj be
6
arbitraryapproximationsinthedisksD (zi,2−b/(2mi))andD (zj,2−b/(2mj)),respectively. Then,
1 1
(1 ) z z zˆ zˆ (1+ ) z z .
i j i j i j
−n − ≤ − ≤ n −
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
Proof. Let (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
d := 16 (n+1) 2 b2tp P 1M(z)n 1/mi.
i − i − i
· · | |
(cid:16) (cid:17)
It is easy to verify that d i 2−b/(2mi) min(1,s i)/(2n). The first inequality follows from (8) and the
secondinequalityfollowsfr≤om(6)and(≤7). WewillshowthatD (z,d )containsm approximations.Tothis
i i i
end, is suffices to show that (p pˆ)(x) < p(x) forall x on the boundaryof D (z,d ). Then, Rouche´’s
i i
theoremguaranteesthatD (z,|d )−contain|sthe| sam|enumberofrootsof pand pˆ countedwithmultiplicity.
i i
Sincez isofmultiplicitym andd <s /n,thediskcontainsexactlym rootsofpcountedwithmultiplicity.
i i i i i
Wehave(notethat x (1+1/(2n2)) M(z)forx bdD (z,d ))
i i i
| |≤ · ∈
(p pˆ)(x) p pˆ xn<2 b p xn
−
| − |≤k − k·| | k k| |
2−b p (1+1/(2n2))n M(zi)n
≤ k k· ·
4 2 b 2tp p (n+1) M(z)n
− n i
≤ · · | |· ·
d mi p P /4< p(x),
≤ i | n i| | |
t
wheretheinequalityinlinethreefollowsfrom p (n+1) pn 2 p,thefirstoneinlinefourfollowsfrom
thedefinitionofd i,andthelastinequalityfollokwskf≤romLemm| a|3. ItfollowsthatD (zi,2−b/(2mi))contains
exactly m approximations. Furthermore, since d s /(2n) for all i, the disks D (z,d ), 1 i k, are
i i i i i
≤ ≤ ≤
pairwisedisjoint.
Forthesecondclaim,weobservethat zˆℓ zℓ 2−b/(2mℓ) s ℓ/(2n) zi zj /(2n)forℓ=i,jand
| − |≤ ≤ ≤ −
hence zˆ z + zˆ z z z /n. Theclaimnowfollowsfromthetriangleinequality.
i i j j i j
| − | − ≤ − (cid:12) (cid:12)
(cid:12) (cid:12)
4Rouche´’stheore(cid:12)mstatest(cid:12)hati(cid:12)f f andg(cid:12)areholomorphicfunctionswith (f g)(x) < f(x) forallpointsxontheboundaryof
somediskD ,then f (cid:12)andghav(cid:12)ethe(cid:12)samenu(cid:12)mberofzeros(countedwithmult|ipli−city)in|D . | |
5ThereadermaywonderwhywearenotusingGershgorincircles toclusterandverifyrootapproximations. Theproblemisthat
verygoodapproximationsofamultiplerootgenerateverylargeGershgorincircleswhichmaythenfailtoisolatetheapproximations.
Indeed,iftwoapproximationsareequal,thecorrespondingdiskshaveinfiniteradius.
6
We have now established that the disks D (zi,2−b/(2mi)), 1 i k, are pairwise disjoint and that the
≤ ≤
i-th disk contains exactly m root approximationsprovided that b satisfies (5) to (8). Unfortunately, the
i
conditionson b are stated in terms of the quantities m, s and P which we do know. Also, we do not
i i i
| |
knowthecenterz.Intheremainderofthesection,wewillshowhowtoclusterrootapproximationsandto
i
certifythem. Wewillneedthefollowingmorestringentpropertiesfortheclusteringandcertificationstep.
s 8 s
2 b/(2mi)<min i , i (9)
− 4n 1024n2
(cid:18)(cid:16) (cid:17) (cid:19)
2 b/8<min(1/16, P /((n+1) 22nG p+8n)) (10)
− i
| | ·
Letb bethesmallestintegersatisfying(5)to(10)foralli. Then,
0
b0=O(nlogn+nG p+maxi(milogM(s i−1)+logM(Pi−1)).
We nextprovideahigh-leveldescriptionofouralgorithmto isolate therootsof p. Thedetailsofthe
clusteringstepandthecertificationsteparethengiveninSections2.2.2and2.2.3,respectively.
2.2.1 OverviewoftheAlgorithm
Oninput pandthenumberk ofdistinctroots,thealgorithmoutputsisolatingdisks D =D (z˜,R)forthe
i i i
rootsof paswellasthecorrespondingmultiplicitiesm. TheradiisatisfyR <s /(64n).
i i i
Thealgorithmusesthefactorizationstepwithanincreasingprecisionuntiltheresultcanbecertified.
If either the clustering step or the certification step fails, we simply double the precision. There are a
coupleoftechnicalsafeguardstoensurethatwedonotwastetimeoniterationswithaninsufficientlylarge
precision(Steps2,5,and6);alsorecallthatweneedtoscaleourinitialpolynomial.
1. Computethebound2G forthemodulusofallrootsof p,whereG fulfillsInequality(4). According
toTheorem1,thiscanbedonewithO˜(n2G )bitoperations.
p
2. Computea2-approximationl =2ll ,ll Z,of p / pn .AccordingtoLemma2andthesubsequent
remarks,wecancomputel withO˜(n2G ∈)bitokpekrati|ons|.
p
3. Scale p,thatis, f(x):=p(s x),withs:=2G ,toensurethattherootsx =z/S,i=1,...,k,of f are
i i
·
containedintheunitdisk.Letbbethesmallestintegersatisfying(5)
4. RunPan’salgorithmoninput f withparameterb :=b+nG toproduceapproximationsxˆ ,...,xˆ for
′ 1 n
therootsof f. Then,zˆ :=s xˆ areapproximationsoftherootsof p,and pˆ p <2 b p ,where
i i −
pˆ(x):=p (cid:213) n (x zˆ). · k − k k k
n i=1 − i
5. Ifthereisazˆ withzˆ 2G +1,returntoStep4withb:=2b.
i i
≥
6. If(cid:213) n M(zˆ)>8l ,returntoStep4withb:=2b.
i=1 i
7. Partition zˆ ,...,zˆ into k clustersC ,...,C . Compute (well separated) enclosing disks D ,...,D
1 n 1 k 1 k
fortheclusters.Iftheclusteringfailstofindkclustersandcorrespondingdisks,returntoStep4with
b:=2b.
8. Foreach i, letD denotethedisk withthe same centeras D butwith ann-timeslargerradius. We
i i
nowverifytheexistenceof C roots(countedwithmultiplicity)of pinD . Iftheverificationfails,
i i
| |
returntoStep4withb:=2b.
9. Iftheverificationsucceeds,outputthedisksD (inStep7,weguaranteethatthedisksD arepairwise
i i
disjoint)andreportthenumber C ofrootapproximationszˆ zˆ ,...,zˆ containedinthedisksas
i 1 n
| | ∈{ }
thecorrespondingmultiplicities.
NotethatSteps5and6ensurethatlogM(zˆ)=O(G +logn)foralli,andlog(cid:213) n M(zˆ)=O(log( p / p ))=
O(logn+t )=O˜(nG ). ThefollowingLemmi aguaranpteesthatthealgorithmpassi=es1thesiestepsifbk kb .| n|
p p 0
≥
7
Lemma5. Foranyb b ,itholdsthat zˆ <2G +1foralli,and(cid:213) n M(zˆ)<8l .
≥ 0 | i| i=1 i
Proof. IntheproofofLemma4,wehavealreadyshownthat zˆ (1+1/(2n2)) M(z)foralli. Hence,
i i
itfollowsthat zˆi (1+1/(2n2)) 2G p <2 2G p 2G p+1,and| |≤ ·
| |≤ · · ≤
(cid:213) n M(zˆi) 4 (cid:213) k M(zi)mi < 4Mea(p) 4kpk2 4kpk <8l .
i=1 ≤ ·i=1 |pn| ≤ |pn| ≤ |pn|
2.2.2 Clustering
Aftercandidateapproximationszˆ ,...,zˆ arecomputedusingafixedprecisionparameterb,weperforma
1 n
partitioningoftheseapproximationsintokclustersC ,...,C ,wherekisgivenasaninput. Theclustering
1 k
is described in detail below. It works in phases. At the beginningof a phase, it chooses an unclustered
approximation and uses it as the seed for the cluster formed in this phase. Ideally, each of the clusters
corresponds to a distinct root of p. The clustering algorithm satisfies the following properties: (1) For
b<b ,thealgorithmmayormaynotsucceedinfindingk clusters. (2)Forb b ,theclusteringalways
0 0
≥
succeeds.
Wheneverthe clustering succeeds, the clusterC with seed z˜ is contained in the disk D :=D (z˜,r),
i i i i i
wherer min(1, s˜i ),ands˜ =min z˜ z˜ .Furthermore,forb b ,D containstherootz (under
suitablein≈umberinn2g)2a56nnd2exactlyim manyj6=aippir−oxijmations. ≥ 0 i i
i
(cid:12) (cid:12)
Fortheclustering,weexploitthefactth(cid:12)atana(cid:12)pproximatefactorization pˆ(x)= pn(cid:213) i(x zˆi)of pwith
p pˆ 2−b p yieldsmiapproximationszˆoftherootziwithdistancelessthan2−b/(2mi−)tozi(atleast,
k − k≤ k k
forb b ). Thus,insimpleterms,weaimtodetermineclustersCofmaximalsizesuchthatthepairwise
0
distan≥cebetweentwoelementsinthesameclusterislessthan2 2 b/(2C). Wenextgivethedetails.
− | |
·
1. InitializeC totheemptyset(ofclusters).
2. InitializeCtothesetofallunclusteredapproximationsandchoosezˆ Carbitrarily.Leta:=2 logn +2
⌊ ⌋
andd :=2 b/4. ∈
−
3. UpdateCtothesetofpointsq Csatisfying zˆ q 2a/√2d .
∈ | − |≤
4. If C a/2,addCtoC. Otherwise,seta:=a/2andcontinuewithstep3.
| |≥
5. Iftherearestillunclusteredapproximations,continuewithstep2.
6. IfthenumberofclustersinC isdifferentfromk,reportfailure,doublebandgobacktothefactor-
izationstep.
Notethat,forb b0,thedisksD (zi,2−b/(2mi))aredisjoint.LetZidenotethesetofrootapproximations
in D (zi,2−b/(2mi)). ≥Then, Zi = mi according to Lemma 4. We show that, for b b0, the clustering
| | ≥
algorithmterminateswithC=Z ifcalledwithanapproximationzˆ Z.
i i
∈
Lemma6. Assumeb b ,zˆ Z,zˆ Z ,andi= j. Then,
0 i i j j
≥ ∈ ∈ 6
zˆ zˆ 2 (2 b/(16mi)+2 b/(16mj)).
i j − −
− ≥ ·
Proof. Sinceb b0,wehave2−b/(cid:12)(cid:12)(2mℓ) (cid:12)(cid:12)s ℓforℓ=i,jby(7)and2−b/(16mℓ)=(2−b/(2mℓ))1/8 s ℓ/(4n)
s /8by(9). Th≥us, ≤ ≤ ≤
ℓ
zˆ zˆ max(s ,s ) 2 b/(2mi) 2 b/(2mj)
i j i j − −
− ≥ − −
s /2+s /2 s /4 s /4
(cid:12) (cid:12) i j i j
(cid:12) (cid:12)≥ − −
2 (2 b/(16mi)+2 b/(16mj)).
− −
≥ ·
8
Lemma 7. If b b , the clustering algorithm computesthe correct clustering, that is, it producesclus-
0
≥
ters C to C such that C =Z for all i (under suitable numbering). Let z˜ be the seed of C and let
1 k i i i i
s˜ =min z˜ z˜ . Then, (1 1/n)s s˜ (1+1/n)s andC aswell asthe rootz is containedin
i j=i i j i i i i i
D (z˜,min(61, s−˜i )). − ≤ ≤
i n2(cid:12)256n2 (cid:12)
(cid:12) (cid:12)
Proof. AssumethatthealgorithmhasalreadyproducedZ toZ andisnowrunwithaseedzˆ Z. We
1 i 1 i
provethatit terminateswithC=Z. Let ℓ be a powerof two s−uch that ℓ m <2ℓ. The proof∈that the
i i
≤
algorithmterminateswithC=Z consistsoftwoparts. Wefirstassumethatsteps2and3areexecutedfor
i
a=2ℓ. We showthatthealgorithmwillthenterminatewithC=Z. Inthesecondpartoftheproof,we
i
showthatthealgorithmdoesnotterminateaslongasa>2ℓ.
Assumethealgorithmreachessteps2and3witha/2=ℓ,i.e.a/2 m <a.Foranyapproximationq
i
≤ ∈
Zi,wehave zˆ q 2 2−b/(2mi)=2mi/√2d 2a/√2d .Thus,Zi C.Conversely,consideranyapproximation
q /Zi. Then|,−zˆ |≤q · 2 2−b/(16mi)>24m√≤i d 22√ad ,andth⊆usnosuchapproximationiscontainedinC.
∈ | − |≥ · ≥
ThisshowsthatC=Z. Since C a/2,thealgorithmterminatesandreturnsZ.
i i
| |≥
It is left to argue that the algorithmdoes not terminate before a/2=ℓ. Since ℓ and a are powers of
two,assumeweterminatewitha/2 2ℓ,andletCbetheclusterreturned. Then,m <a/2 C <aand
i
≥ ≤| |
Z isapropersubsetofC. Consideranyapproximationq C Z,sayq Z with j=i. Sinceq /Z,we
i i j i
∈ \ ∈ 6 ∈
have q zˆ 2 (2−b/(16mi)+2−b/(16mj))>2 2−b/(16mi) >24m√i d . And since q C, we have q zˆ
| − |≥ · · ∈ | − |≤
2a/√2d . Thus,4m a/2and,hence,thereareatleast3a/8manyapproximationsinC Z. Furthermore,
i i
≤ \
zi zj zi zˆ + zˆ q + q zj 2−b/(2mi)+2a/√2d +2−b/(2mj) 2−b/(16mi)+2a/√2d +2−b/(16mj)
− ≤| − | | − | − ≤ ≤ ≤
3a/√2d . Consequently,thereareatleast3a/8rootsz =z countedwithmultiplicitywithindistance3a/√2d
(cid:12) (cid:12) (cid:12) (cid:12) j6 i
(cid:12)toz. T(cid:12)hisobservationallows(cid:12)ustou(cid:12)pperboundthevalueof P ,namely
i i
| |
P =(cid:213) z z mj (3a/√2d )3a/82(n mi 3a/8)G p
i i j − −
| | | − | ≤
j=i
6
<3nd 3/42nG p 3n2 3b/162nG p <3n2 b/8 2nG p,
− −
≤ ·
acontradictionto(10).
Wenowcometotheclaimsabouts˜ andthedisksdefinedintermsofit. Therelationbetweens and
i i
s˜i followsfromthesecondpartofLemma4. AllpointsinCi=Zi havedistanceatmost2 2−b/(2mi) from
·
z˜. Also,by(6)and(9),
i
2 2 b/(2mi)<min(1/n2,s /(512n2)) min(1/n2,s˜ /(256n2))
− i i
· ≤
Hence,C aswellasz iscontainedinD (z˜,min(1/n2,s˜ /(256n2))).
i i i i
Lemma8. Forafixedprecisionb,computingacompleteclusteringneedsO˜(nb+n2G )bitoperations.
p
Proof. For each approximation, we examine the number of distance computations we need to perform.
Recallthatb(property(5))andaarepowersoftwo,a 4nbydefinition,andb 8n 2abyporperty(5).
Then, a/√2d =2 b/(2a) 2 N. Thus,thenumber a/√2d≤hasaverysimpleforma≥tinbi≥narynotation. There
− −
∈
is a single one, and this one is b/(2a)positions afterthe binary point. In addition, all approximationszˆ
G
haveabsolutevaluelessthan2 2 duetoStep5intheoverallalgorithm.Thus,eachevaluationoftheform
zˆ q 2a/√2d canbedonew·ith
| − |≤
O(G +logd 2/a)=O((b/a)+G )=O((b/a)+G +logn)
− p
bitoperations.
Forafixedseedzˆ,inthei-thiterationofstep2,wehaveatmosta n/2i 2manyunclusteredapprox-
−
≤
imationsleft inC, since otherwise we would haveterminated in an earlieriteration. Hence, we perform
at most a evaluations of the form zˆ q 2a/√2d , resulting in an overall number of bit operations of
a O((b/a)+G )=O(b+aG ) for a| fi−xed|≤iteration. As we halve a in each iteration, we have at most
lo·gn+2iterationsforafixedzˆ,leadingtoabitcomplexityofO(blogn+nG )=O˜(b+nG )=O˜(b+nG ).
p
Intotal,performingacompleteclusteringhasabitcomplexityofatmostO˜(nb+n2G ).
p
9
Whentheclusteringsucceeds,wehavekclustersC toC andcorrespondingseedsz˜ ,...,z˜ zˆ ,...,zˆ .
1 k 1 k 1 n
Fori=1,...,k,wedefineD :=D (z˜,r),wherez˜ istheseedfortheclusterC and ⊆{ }
i i i i i
r :=min(2 2logn ,2 logs˜i/(256n2) ) min( 1 , s˜i ). (11)
i −⌈ ⌉ ⌈ ⌉ ≥ 2n2 256n2
In particular, r is a 2-approximationof min(1/n2,s˜ /(256n2)). Notice that the cost for computing the
i i
separationss˜ isboundedbyO˜(nb+n2G )bitoperationssincewecancomputethenearestneighborgraph
i p
ofthepointsz˜ (andthusthevaluess˜ )inO(nlogn)steps[10]withaprecisionofO(b+nG ).
i i
Now, suppose that b b , Then, according to Lemma 7, the cluster C is contained in the disk D.
0 i i
≥
Furthermore, D contains exactly one root z of p (under suitable numbering of the roots), and it holds
i i
thatm =mult(z,p)= C andmin(1/(2n2),s /(512n2)) r min(1/n2,s /(64n2)). Ifthe clustering
i i i i i i
| | ≤ ≤
succeeds for a b<b , we have no guarantees (actually, the termination condition in step 4 gives some
0
guarantee,however,wehavechosennottoexploitit). Hence,beforeweproceed,weverifythateachdisk
D actuallycontainstheclusterC. Ifthisisnotthecase,thenwereportafailure,returntothefactorization
i i
stepwithb=2b,andcomputeanewcorrespondingclustering.
Inthenextandfinalstep,weaimtoshowthateachoftheenlargeddisksD :=D (z˜,R):=D (z˜,nr),
i i i i i
i= 1,...,k, contains exactly one root z of p, and that the number of elements in C D equals the
i i i
multiplicity of z. Notice that, from the definition of r and D , it obviousthat the disks⊆D are pairwise
i i i i
disjointandthatC D D .
i i i
⊆ ⊆
2.2.3 Certification
InordertoshowthatD containsexactlyonerootof pwithmultiplicity C ,weshowthateachD contains
i i i
| |
thesamenumberofrootsof pand pˆcountedwithmultiplicity. Forthelatter,wecomputealowerbound
for pˆ(z) ontheboundarybdD ofD ,andcheckwhetherthisboundislargerthan (pˆ p)(z) forallpoints
i i
z |bdD |. Ifthisisthecase,thenwearedoneaccordingtoRouche´’stheorem.Oth|erw−ise,we|startoverthe
i
∈
factorizationalgorithmwithb=2b. Wenowcometothedetails:
1. Letl =2ll bethe2-approximationof p / p asdefinedinstep2oftheoverallalgorithm.
n
k k | |
2. Fori=1,...,k,letz :=z˜ +n r D . Notethat z (1+1/n) M(z˜)sincenr 1/n.
∗i i · i∈ i | ∗i|≤ · i i≤
3. Wetrytoestablishtheinequality
|pˆ(z∗i)/pn|>Ei:=32·2−bl M(z˜i)n (12)
forall i. We willsee inthe proofofLemma10thatthisimpliesthateachdiskD containsexactly
i
onerootz of pandthatitsmultiplicityequalsthenumber C ofapproximationswithinD . Inorder
i i i
toestablishtheinequality,weconsiderr =1,2,4,8,...and| c|ompute pˆ(z )/p toanabsoluteerror
less than 2 r . If, for all r b, we fail to show that p(z )/p >E| , w∗i e renp|ort a failure and go
backtothe−factorizationalgo≤rithmwithb=2b. Otherw|ise,∗iletr n|bethei smallestr forwhichweare
i
successful.
4. If,atanystageofthealgorithm,(cid:229) r >b,wealsoreportafailureandgobacktothefactorization
i i
algorithmwithb=2b. Lemma9thenshowsthat,forfixedb,thenumberofbitoperationsthatare
usedforallevaluationsisboundedbyO˜(nb+n2t +n3).
p
5. If we can verify that pˆ(z˜ +nr)/p >E for all i, we return the disks D and the multiplicities
i i n i i
| |
m = C .
i i
| |
r
Lemma9. Foranyi,wecancompute pˆ(z )/p toanabsoluteerrorlessthan2 withanumberofbit
| ∗i n| −
operationslessthan
O˜(n(n+r +nlogM(z˜)+t )).
i p
Forafixedb,thetotalcostforallevaluationsintheabovecertificationstepisboundedbyO˜(nb+n2t +
p
n3).
10