Rigorous Polynomial Approximations and Applications

195 Pages·2017·2.46 MB·English

by  Mioara Maria Joldes

Rigorous Polynomial Approximations and Applications
Mioara Maria Joldes
To cite this version:
Mioara Maria Joldes. Rigorous Polynomial Approximations and Applications. Other [cs.OH]. Ecole
normale supérieure de lyon - ENS LYON, 2011. English. ￿NNT: 2011ENSL0655￿. ￿tel-00657843v2￿
HAL Id: tel-00657843
https://theses.hal.science/tel-00657843v2
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N◦d’ordre:
N◦attribuéparlabibliothèque:
ÉCOLE NORMALE SUPÉRIEURE DE LYON
Laboratoire de l’Informatique du Parallélisme
THÈSE
présentéeetsoutenuepubliquementle26Septembre2011par
Mioara JOLDES
,
pourl’obtentiondugradede
Docteur de l’École Normale Supérieure de Lyon
spécialité : Informatique
autitredel’ÉcoleDoctoraledeMathématiquesetd’InformatiqueFondamentaledeLyon
Approximations polynomiales rigoureuses et
applications
Directeursdethèse: Nicolas BRISEBARRE
Jean-Michel MULLER
Aprèsavisde: Didier HENRION
Warwick TUCKER
Devantlacommissiond’examenforméede:
Frédéric BENHAMOU
Nicolas BRISEBARRE
Didier HENRION
Jean-Michel MULLER
Warwick TUCKER
ForSoho,whereIbelong.
Acknowledgements
I cite G. Cantor, for saying that the art of proposing a question must be held of higher value than
solvingitandexpressmydeepestgratitudeandappreciationformyadvisors. Nicolasfoundme∗
duringanexaminaMaster2courseatENS,andproposedtherightquestionforme. Althoughthe
answerwasnotgivenduringmymaster,asinitiallythought,althoughthereweremomentswhen
I wanted different questions, he encouraged me every day, he believed in me, he was always
available when I needed his advice in scientific and non-scientific matters. His altruism when
sharing his knowledge with me, his desire for perfection when carefully reviewing all my work,
werealwaysastonishingandmotivatingforme. Thisthesis,andthenicetimeIhad,owesalotto
his devotion, ideas, suggestions† and so good sense of humor (not the Romanians’ related jokes,
though!). Jean-Michel always kept an experienced and kind eye over my work. I felt honored to
haveassupervisorthebestexpertinComputerArithmetic: heansweredallmyrelatedquestions
and he always made so good and often mediating suggestions‡. It was a great chance for me
to have had advisors who guided my path, but let me make my own choices and find my own
valuesandinterestsinresearch. Ithankthemforallthat,andforhavingmadeunedocteuroutof
unechipie. Mercibeaucoup,NicolasetJean-Michel.
IwanttothankFredericBenhamou,DidierHenrionandWarwickTuckerforhavingaccepted
to participate in my jury, for their pertinent questions and useful suggestions. I would like to
express my gratitude to the reviewers of this manuscript. I thank them for their hard work, for
their understanding and availability regarding the schedule of this work. It seems that in what
follows I will have the chance and the honor to work more with them. I think that they brought
tremendousopportunitiesinmyyoungresearcherlifeandIthankthemforthatalso.
Moreover,Iwouldliketothankmycollaborators: supnormandSollya(andbigbrothers)Sylvain
and Christoph, for having helped me with their hard work, bright ideas and (endless, since I
don’tgiveineither)debates;BogdanandFlorent,wholetmekeepintouchwithmyoldlove,the
FPGAs;AlexandreandMarcforhavingsharedwithmethemagicofD-finitefunctions;Érik,Ioana
and Micaela for their formalization efforts in our Coqapprox meetings; Serge for his unbelievable
kindness.
IwouldalsoliketothankthemembersoftheArenaireteamforhavingmademystaytherean
unforgetable experience: my office mates Andy, Guillaume, Ivan and Xavier for having listened
to all my usual bla-bla and for having accepted my bike in the office; Claude-Pierre (thanks a lot
foryourworkfortheProjetRégionthatfinancedmyscholarship!),Damien(thanksforthebeauti-
fulEuclideanLatticescoursethatledmetoArenaire!), Gilles(thanksforourshortbutextremely
meaningfuldiscussions!), Guillaume(manythanksforyouradvicesandhelp!), Nathalie(thanks
for often pointing me interesting conferences!), Nicolas L. (thanks for the fun I had teaching and
joking with you!), Vincent (thanks for the 17 obsession!), and our assistants Damien and Séver-
ine for their great help with les ordres de mission. I also thank Adrien, Christophe, Diep, Fabien,
Eleonora, Nicolas Brunie, Jingyan, Adeline, Philippe, David and Laurent-Stéphane for their help
∗. unepioupiou
†. Pasdecalendriersetderépétitions,quandmême!
‡. Someofuswouldbestilldebatingtoday,otherwise.
4 Acknowledgements
inmyArenaireday-by-daylife.
Then, I am very grateful to my external collaborators who helped and inspired me: Marius
Cornea,JohnHarrison,AlexanderGoldsztejn,MarcusNeher,andmynewteamCAPAatUppsala.
I would like to say a "pure" Multam fain! to my ENS Romanian mafia for being there for me,
,
for our coffees, lunches (sorry I always had to leave earlier!) and iesit la sca˘ri. I adored our non-
,
scientificandnotpoliticallycorrectdiscussionsandjokes.
My thesis would not exist without the basis of knowledge I accumulated during my studies
in Romania. I will always be grateful to my teachers there. Especially, I would like to thank
OctavianCretnotonlyforhavingsentthee-mailannouncingthescholarshipsatENS,butalsofor
,
allhishardandfairworkindifficultandunfairtimes. IthankalsomyprofessorsAlinSuciu,Ioan
GavreaandDumitruMirceaIvanwhoimpressedmystudentmindwiththeiruniqueintelligence,
modestyandhumor.
IwouldliketogivemyspecialthankstoBogdan(wemadeit,we’redoctors!) andmyfamily
(parents,grand-parents,dearsisterOanaandDia). Theirtotalsupport,careandloveenabledme
tocompletethiswork. IthankalsomycousinsCa˘linandClaudiafortheirhelpandsupportand
longtriptoFrance.
Last but not least I thank the ENS janitor who brought me the hot chocolate the night before
mydefenseandmyfatherforhavingtaughtmethebeautyofmathematics. Theirsimplekindness
keepsmegoingon.
4
Contents
1 Introduction 13
1.1 Introductiontorigorouspolynomialapproximations-outlineofthethesis . . . . . . 15
1.2 Computerarithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3 IntervalArithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.4 Fromintervalarithmetictorigorouspolynomialapproximations . . . . . . . . . . . 36
1.4.1 Computingtheapproximationpolynomialbeforeboundingtheerror . . . . 37
1.4.2 Simultaneouslycomputingtheapproximationpolynomialandtheerror . . . 41
1.4.3 Practicalcomparisonofthedifferentmethods . . . . . . . . . . . . . . . . . . 42
1.5 Datastructuresforrigorouspolynomialapproximations . . . . . . . . . . . . . . . . 43
2 TaylorModels 45
2.1 BasicprinciplesofTaylorModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.1.1 Definitionsandtheirambiguities . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.1.2 Boundingpolynomialswithintervalcoefficients . . . . . . . . . . . . . . . . . 48
2.2 TaylorModelswithAbsoluteRemainder . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.2.1 TaylorModelsforbasicfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.2.2 OperationswithTaylorModels . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.3 Theproblemofremovablediscontinuities–theneedforTaylorModelswithrelative
remainder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.3.1 TaylorModelswithrelativeremaindersforbasicfunctions . . . . . . . . . . . 65
2.3.2 OperationswithTaylorModelswithrelativeremainders . . . . . . . . . . . . 69
2.3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3 EfficientandAccurateComputationofUpperBoundsofApproximationErrors 83
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.2 Previouswork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.2.1 Numericalmethodsforsupremumnorms . . . . . . . . . . . . . . . . . . . . 86
3.2.2 Rigorousglobaloptimizationmethodsusingintervalarithmetic . . . . . . . . 86
3.2.3 Methodsthatevadethedependencyphenomenon . . . . . . . . . . . . . . . . 87
3.3 Computingasafeandguaranteedsupremumnorm . . . . . . . . . . . . . . . . . . . 88
3.3.1 Computing a validated supremum norm vs. validating a computed supre-
mumnorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.3.2 Schemeofthealgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.3.3 Validatinganupperboundon(cid:107)ε(cid:107) forabsoluteerrorproblemsε = p−f . . 89
∞
3.3.4 Caseoffailureofthealgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.3.5 Relativeerrorproblemsε = p/f −1withoutremovablediscontinuities . . . 91
3.3.6 Handlingremovablediscontinuities . . . . . . . . . . . . . . . . . . . . . . . . 92
6 Contents
3.4 ObtainingtheintermediatepolynomialT anditsremainder . . . . . . . . . . . . . . 93
3.5 Certificationandformalproof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.5.1 FormalizingTaylormodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.5.2 Formalizingpolynomialnonnegativity . . . . . . . . . . . . . . . . . . . . . . 95
3.6 Experimentalresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4 ChebyshevModels 105
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.1.1 Previousworksforusingtighterpolynomialapproximationsinthecontext
ofrigorouscomputing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.2 Preliminary theoretical statements about Chebyshev series and Chebyshev inter-
polants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.2.1 SomebasicfactsaboutChebyshevpolynomials . . . . . . . . . . . . . . . . . 107
4.2.2 ChebyshevSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.2.3 DomainsofconvergenceofTaylorversusChebyshevseries . . . . . . . . . . 113
4.3 ChebyshevInterpolants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.3.1 Interpolationpolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.4 Summaryofformulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.5 ChebyshevModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.5.1 ChebyshevModelsforbasicfunctions . . . . . . . . . . . . . . . . . . . . . . . 129
4.5.2 OperationswithChebyshevmodels . . . . . . . . . . . . . . . . . . . . . . . . 131
4.5.3 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.5.4 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.5.5 Composition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.6 Experimentalresultsanddiscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.7 Conclusionandfuturework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5 RigorousUniformApproximationofD-finiteFunctions 147
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.1.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.2 ChebyshevExpansionsofD-finiteFunctions . . . . . . . . . . . . . . . . . . . . . . . 149
5.2.1 ChebyshevSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.2.2 TheChebyshevRecurrenceRelation . . . . . . . . . . . . . . . . . . . . . . . . 150
5.2.3 SolutionsoftheChebyshevRecurrence . . . . . . . . . . . . . . . . . . . . . . 152
5.2.4 ConvergentandDivergentSolutions . . . . . . . . . . . . . . . . . . . . . . . . 154
5.3 ComputingtheCoefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.3.1 Clenshaw’sAlgorithmRevisited . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.3.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.3.3 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.4 ChebyshevExpansionsofRationalFunctions . . . . . . . . . . . . . . . . . . . . . . . 162
5.4.1 RecurrenceandExplicitExpression . . . . . . . . . . . . . . . . . . . . . . . . 163
5.4.2 Boundingthetruncationerror . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.4.3 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.5 ErrorBounds/Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5.6 Discussionandfuturework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
5.7 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6
Contents 7
6 Automatic Generation of Polynomial-based Hardware Architectures for Function Eval-
uation 175
6.1 Introductionandmotivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.1.1 Relatedworkandcontributions . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.1.2 RelevantfeaturesofrecentFPGAs . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.2 Functionevaluationbypolynomialapproximation . . . . . . . . . . . . . . . . . . . . 177
6.2.1 Rangereduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.2.2 Polynomialapproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.2.3 Polynomialevaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.2.4 Accuracyanderroranalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.2.5 ParameterspaceexplorationfortheFPGAtarget . . . . . . . . . . . . . . . . 182
6.3 Examplesandcomparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7

List of Figures
1.1 Approximationerrorε = f −pforeachRPAgiveninExample1.1.3. . . . . . . . . . 18
1.2 Approximationerrorε = f −pforeachRPAgiveninExample1.1.4. . . . . . . . . . 18
1.3 Approximationerrorε = f −pforeachRPAgiveninExample1.1.5. . . . . . . . . . 19
1.4 Approximationerrorε = f −pfortheRPAgiveninExample1.1.6. . . . . . . . . . . 20
1.5 Valuesofulp(x)around1,assumingradix2andprecisionp. . . . . . . . . . . . . . . . . 25
2.1 ATM(P,[−d,d])oforder2forexp(x),overI = [−1, 1]. P(x) = 1+x+0.5x2 and
2 2
d = 0.035. We can view a TM as a a tube around the function in (a). The actual error
R (x) = exp(x)−(1+x+0.5x2)isplottedin(b). . . . . . . . . . . . . . . . . . . . . 46
2
3.1 Approximationerrorinacasetypicalforalibm . . . . . . . . . . . . . . . . . . . . . 84
4.1 DomainofconvergenceofTaylorandChebyshevseriesforf. . . . . . . . . . . . . . 113
z+z−1
4.2 Joukowskytransformw(z) = mapsC(0,ρ)andC(0,ρ−1)respectivelytoε . 115
ρ
2
1
4.3 The dots are the errors log (cid:107)f −f (cid:107) in function of n, where f(x) = is
10 n ∞ 1+25x2
the Runge function and f is the truncated Chebyshev series of degree n. The line
n √
1+ 26
hasslopelog ρ∗,ρ∗ = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
10 5
4.4 Certifiedplotoff(x) = 2π−2xasin((cos0.797)sin(π/x))+0.0331x−2.097forx ∈
I = [3,64]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.1 NewtonpolygonforChebyshevrecurrence. . . . . . . . . . . . . . . . . . . . . . . . . 155
5.2 ApproximationerrorsforExample5.7.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.3 ApproximationerrorsforExample5.7.2. . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.4 ApproximationerrorsforExample5.7.3. . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.5 CertifiedplotsforExample5.7.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.1 Automatedimplementationflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.2 Alignmentofthemonomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.3 Thefunctionevaluationarchitecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180