Structured Matrix Based Methods for Approximate Polynomial GCD

207 Pages·2011·1.15 MB·English

by  Paola Boito (auth.)

15
TESI
THESES
tesidiperfezionamentoinMatematicasostenutail5ottobre2007
COMMISSIONEGIUDICATRICE
MarianoGiaquinta,Presidente
BernhardBeckermann
DarioBini
LucaGemignani
PatriziaGianni
StefanoMarmi
AlessandroProfeti
PaolaBoito
Universite´deLimoges/CNRS
123avenueAlbertThomas
87060LimogesCedex,France
StructuredMatrixBasedMethodsforApproximatePolynomialGCD
Paola Boito
Structured Matrix Based
Methods for Approximate
Polynomial GCD
(cid:2)c 2011ScuolaNormaleSuperiorePisa
ISBN:978-88-7642-380-2
e-ISBN:978-88-7642-381-9
“Hefoundhimunderapinetree,sittingontheground,
arrangingfallenpineconesinaregulardesign: an
isoscelestriangle. AtthathourofdawnAgilulfalways
neededtoapplyhimselftosomepreciseexercise:
countingobjects,arrangingthemingeometricpatterns,
resolvingproblemsofarithmetic. Itwasthehourinwhich
objectslosetheconsistencyofshadowthataccompanies
themduringthenightandgraduallyreacquirecolors,but
seemtocrossmeanwhileanuncertainlimbo,faintly
touched,justbreathedonbylight;thehourinwhichone
isleastcertainoftheworld’sexistence. He,Agilulf,
alwaysneededtofeelhimselffacingthingsasiftheywere
amassivewallagainstwhichhecouldpitthetensionof
hiswill,foronlyinthiswaydidhemanagetokeepasure
consciousnessofhimself. Butiftheworldaroundwas
insteadmeltingintothevagueandambiguous,hewould
feelhimselfdrowninginthatmorbidhalflight,incapable
ofallowinganyclearthoughtordecisiontoflowerinthat
void. Insuchmomentshefeltsick,faint;sometimesonly
atthecostofextremeeffortdidhefeelhimselfableto
avoidmeltingawaycompletely. Itwasthenhebeganto
count: trees,leaves,stones,lances,pinecones,anything
infrontofhim. Orheputtheminrowsandarranged
theminsquaresandpyramids.”
ItaloCalvino,TheNonexistentKnight
Contents
Introduction xi
Notation xv
1 ApproximatepolynomialGCD 1
1.1. Coefficient-baseddefinitions . . . . . . . . . . . . . . . 2
1.1.1. Quasi-GCD . . . . . . . . . . . . . . . . . . . . 2
1.1.2. (cid:2)−GCD . . . . . . . . . . . . . . . . . . . . . . 4
1.1.3. AGCD . . . . . . . . . . . . . . . . . . . . . . 5
1.2. Ageometricalinterpretation . . . . . . . . . . . . . . . 6
1.3. Pseudozerosandrootneighborhoods . . . . . . . . . . . 8
1.4. Aroot-baseddefinition . . . . . . . . . . . . . . . . . . 13
1.5. Graph-theoreticaltechniques . . . . . . . . . . . . . . . 16
1.6. FormulationsoftheapproximateGCDproblem . . . . . 19
2 Structuredandresultantmatrices 21
2.1. ToeplitzandHankelmatrices . . . . . . . . . . . . . . . 21
2.2. Displacementstructure . . . . . . . . . . . . . . . . . . 22
2.2.1. Toeplitz-likematrices . . . . . . . . . . . . . . . 25
2.2.2. Cauchy-likematrices . . . . . . . . . . . . . . . 25
2.3. Computationofdisplacementgenerators . . . . . . . . . 26
2.3.1. Sumofdisplacementstructuredmatrices . . . . 26
2.3.2. Productofdisplacementstructuredmatrices . . . 26
2.3.3. Inverseofadisplacementstructuredmatrix . . . 27
2.4. FastGEPPforstructuredmatrices . . . . . . . . . . . . 27
2.4.1. TheGKOalgorithm . . . . . . . . . . . . . . . 28
2.4.2. TheToeplitz-likecase . . . . . . . . . . . . . . 30
2.5. TheSylvestermatrix . . . . . . . . . . . . . . . . . . . 31
2.6. TheBe´zoutmatrix . . . . . . . . . . . . . . . . . . . . 34
2.7. Moreresultsonresultantmatrices . . . . . . . . . . . . 39
viii PaolaBoito
3 TheEuclideanalgorithm 45
3.1. Theclassicalalgorithm . . . . . . . . . . . . . . . . . . 45
3.1.1. Matrixrepresentation . . . . . . . . . . . . . . . 46
3.1.2. Generalizations . . . . . . . . . . . . . . . . . . 47
3.2. Stabilizednumericalversions . . . . . . . . . . . . . . . 47
3.3. Relativeprimality . . . . . . . . . . . . . . . . . . . . . 49
3.4. Alook-aheadmethod . . . . . . . . . . . . . . . . . . . 53
3.5. Pade´ approximation . . . . . . . . . . . . . . . . . . . . 56
3.6. EuclideanalgorithmandfactorizationoftheBezoutian . 57
4 MatrixfactorizationandapproximateGCDs 59
4.1. ApproximaterankandtheSVD . . . . . . . . . . . . . 59
4.1.1. TheSVDofSylvesterandBe´zoutmatrices . . . 61
4.1.2. TheSVDofsubresultants . . . . . . . . . . . . 64
4.1.3. Statisticalapproaches . . . . . . . . . . . . . . . 65
4.2. AnSVD-basedalgorithm . . . . . . . . . . . . . . . . . 66
4.3. QRfactorizationoftheSylvestermatrix . . . . . . . . . 67
5 Optimizationapproach 71
5.1. Thecaseofdegreeone . . . . . . . . . . . . . . . . . . 72
5.2. Structuredlow-rankapproximation . . . . . . . . . . . . 73
5.2.1. Lift-and-projectalgorithm . . . . . . . . . . . . 74
5.2.2. Structuredtotalleastnorm . . . . . . . . . . . . 76
5.3. Thedivisor-quotientiteration . . . . . . . . . . . . . . 79
5.4. Moreoptimizationstrategies . . . . . . . . . . . . . . . 81
5.4.1. AnotherSTLNapproach . . . . . . . . . . . . . 81
5.4.2. Sumsofsquaresrelaxation . . . . . . . . . . . . 82
5.4.3. Gradient-projectionmethods . . . . . . . . . . . 82
6 Newfactorization-basedmethods 83
6.1. Realvs. complexcoefficients . . . . . . . . . . . . . . . 84
6.2. Sylvestervs. Be´zoutmatrices . . . . . . . . . . . . . . . 85
6.3. QRFactorization . . . . . . . . . . . . . . . . . . . . . 85
6.3.1. StabilityoftheQRfactorization . . . . . . . . . 86
6.3.2. Degreeofan(cid:2)-GCD . . . . . . . . . . . . . . . 91
6.3.3. Coefficientsofan(cid:2)-GCD . . . . . . . . . . . . 93
6.3.4. Iterativerefinement . . . . . . . . . . . . . . . . 94
6.3.5. Rank-deficientJacobian . . . . . . . . . . . . . 98
6.3.6. AnalgorithmbasedonQRdecomposition . . . . 102
6.4. Conditioning . . . . . . . . . . . . . . . . . . . . . . . 103
6.4.1. Degree . . . . . . . . . . . . . . . . . . . . . . 103
6.4.2. Coefficients . . . . . . . . . . . . . . . . . . . . 103
ix StructuredMatrixBasedMethodsforApproximatePolynomialGCD
6.5. Tridiagonalization . . . . . . . . . . . . . . . . . . . . . 105
6.5.1. Theexactcase . . . . . . . . . . . . . . . . . . 105
6.5.2. Theapproximatecase: rankdetermination . . . . 111
6.5.3. Thealgorithm . . . . . . . . . . . . . . . . . . . 112
6.6. Morefactorization-basedalgorithms . . . . . . . . . . . 112
6.6.1. SVDoftheBe´zoutmatrix . . . . . . . . . . . . 112
6.6.2. QRdecompositionoftheBe´zoutmatrix . . . . . 113
6.6.3. QRdecompositionofthecompanionmatrixres-
ultant . . . . . . . . . . . . . . . . . . . . . . . 114
7 AfastGCDalgorithm 117
7.1. Stabilityissues . . . . . . . . . . . . . . . . . . . . . . 117
7.2. Fastandstablefactorizationofrectangularmatrices . . . 120
7.3. ComputingatentativeGCD . . . . . . . . . . . . . . . 124
7.4. Fastiterativerefinement . . . . . . . . . . . . . . . . . . 124
7.4.1. Iterativerefinementwithlinesearch . . . . . . . 127
7.5. Choiceofatentativedegree . . . . . . . . . . . . . . . . 128
7.5.1. IsLUfactorizationrank-revealing? . . . . . . . 129
7.6. TheFastgcdalgorithm . . . . . . . . . . . . . . . . . . 131
8 Numericaltests 133
8.1. Tolerance-sensitivedegree . . . . . . . . . . . . . . . . 134
8.2. Mignotte-likepolynomials . . . . . . . . . . . . . . . . 137
8.3. HighGCDdegree . . . . . . . . . . . . . . . . . . . . . 138
8.4. Anill-conditionedcase . . . . . . . . . . . . . . . . . . 139
8.5. Unbalancedcoefficients . . . . . . . . . . . . . . . . . . 140
8.6. Multipleroots . . . . . . . . . . . . . . . . . . . . . . . 141
8.7. Smallleadingcoefficient . . . . . . . . . . . . . . . . . 143
8.8. Effectivenessoftheestimateonthe(cid:2)-GCDdegree . . . 143
8.9. Computationtime . . . . . . . . . . . . . . . . . . . . . 145
8.10. Acomparisonwithδ-GCD . . . . . . . . . . . . . . . . 150
9 Generalizationsandfurtherwork 157
9.1. ApproximateGCDofmanypolynomials . . . . . . . . . 158
9.1.1. Generalizedresultants . . . . . . . . . . . . . . 159
9.2. ApproximateGCDofmultivariatepolynomials . . . . . 161
9.3. PolynomialGCDinotherbases . . . . . . . . . . . . . 162
9.4. FastQRdecomposition . . . . . . . . . . . . . . . . . . 163
9.5. Astructuredapproachtothecompanionmatrixresultant 164
x PaolaBoito
A Distancesandnorms 167
A.1. Vectornorms . . . . . . . . . . . . . . . . . . . . . . . 167
A.2. Matrixnorms . . . . . . . . . . . . . . . . . . . . . . . 168
A.3. Polynomialmetrics . . . . . . . . . . . . . . . . . . . . 169
B Specialmatrices 171
B.1. Convolutionmatrices . . . . . . . . . . . . . . . . . . . 171
B.1.1. Univariatecase . . . . . . . . . . . . . . . . . . 171
B.1.2. Multivariatecase . . . . . . . . . . . . . . . . . 172
B.2. TheFouriermatrixandpolynomialcomputations . . . . 173
B.3. Circulantmatrices . . . . . . . . . . . . . . . . . . . . . 175
References 179
Index 191
Introduction
ThecomputationofpolynomialGCDisabasicalgebraictask,whichhas
many applications in several fields such as in polynomial root-finding,
controltheory,imagedeblurring,CAGD.
Theproblemisusuallystatedasfollows: giventhecoefficientsoftwo
polynomials u(x) and v(x), compute the coefficients of their greatest
commondivisorg(x). Wefocushereontheunivariatecase.
The usual notion of polynomial GCD, however, is ill-suited to deal
with many applications where input data are affected by errors (due for
example to roundoff, or to the fact that the data come from physical
experiments or previous numerical computations). Indeed, if the given
polynomialsu(x)andv(x)haveanontrivialGCD,thenarbitrarilysmall
perturbationsinthecoefficientsofu(x)andv(x)maytransformu(x)and
v(x)intorelativelyprimepolynomials. Thereforetheproblemoffinding
anexactGCDisill-posedinanapproximatesetting.
ThisiswhythenotionofapproximateGCDhasbeenintroduced. Sev-
eraldefinitionsofapproximateGCDarefoundintheliterature; herewe
will mostly use the so-called (cid:2)-GCD. Roughly speaking, a polynomial
g(x) is an (cid:2)-GCD of u(x) and v(x) if there exist polynomials uˆ(x) and
vˆ(x)suchthat
(i) uˆ(x)andvˆ(x)are“close”tou(x)andv(x),thatis,d (u,uˆ) < (cid:2) and
d(v,vˆ) < (cid:2) forsomefixedpolynomialmetricd andtolerance(cid:2);
(ii) g(x)isanexactGCDofuˆ(x)andvˆ(x),and
(iii) g(x) has maximumdegree among the exact GCDs of pairs of poly-
nomialsthatsatisfy(i).
The first analysis of the approximate GCD problem dates back to 1985
([118]); several approaches to the problem have been proposed since
then. We seek here to give a comprehensive overview of the existing
literatureonthesubject,withafocusonmatrix-basedmethods. Wenext
exploreindetailtherelationshipbetweenapproximateGCDandresultant