Table Of ContentLecture Notes in Computer Science 3218
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DavidHutchison
LancasterUniversity,UK
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CarnegieMellonUniversity,Pittsburgh,PA,USA
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UniversityofSurrey,Guildford,UK
JonM.Kleinberg
CornellUniversity,Ithaca,NY,USA
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GerhardWeikum
Max-PlanckInstituteofComputerScience,Saarbruecken,Germany
Jürgen Gerhard
Modular Algorithms
in Symbolic Summation
and Symbolic Integration
1 3
Author
JürgenGerhard
Maplesoft
615KumpfDrive,Waterloo,ON,N2V1K8,Canada
E-mail:[email protected]
ThisworkwasacceptedasPhDthesisonJuly13,2001,at
FachbereichMathematikundInformatik
UniversitätPaderborn
33095Paderborn,Germany
LibraryofCongressControlNumber:2004115730
CRSubjectClassification(1998):F.2.1,G.1,I.1
ISSN0302-9743
ISBN3-540-24061-6SpringerBerlinHeidelbergNewYork
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ToHerbertGerhard(1921–1999)
Foreword
This work brings together two streams in computer algebra: symbolic integration
andsummationontheonehand,andfastalgorithmicsontheotherhand.
Inmanyalgorithmicallyorientedareasofcomputerscience,theanalysisofal-
gorithms–placedintothelimelightbyDonKnuth’stalkatthe1970ICM–provides
acrystal-clearcriterionforsuccess.Theresearcherwhodesignsanalgorithmthatis
faster(asymptotically,intheworstcase)thananypreviousmethodreceivesinstant
gratification: her result will be recognized as valuable. Alas, the downside is that
suchresultscomealongquiteinfrequently,despiteourbestefforts.
An alternative evaluation method is to run a new algorithm on examples; this
hasitsobviousproblems,butissometimesthebestwecando.GeorgeCollins,one
ofthefathersofcomputeralgebraandagreatexperimenter,wrotein1969:“Ithink
this demonstratesagainthata simple analysisisoftenmorerevealingthana ream
ofempiricaldata(althoughbothareimportant).”
Within computer algebra, some areas have traditionally followed the former
methodology,notably some parts of polynomialalgebra and linear algebra. Other
areas, such as polynomialsystem solving, have not yet been amenable to this ap-
proach. The usual “input size” parameters of computer science seem inadequate,
and although some natural “geometric” parameters have been identified (solution
dimension, regularity), not all (potential) major progress can be expressed in this
framework.
Symbolic integration and summation have been in a similar state. There are
somealgorithmswithanalyzedruntime,butbasicallythemathematicallyoriented
world of integrationand summation and the computerscience world of algorithm
analysisdidnothavemuchtosaytoeachother.
Gerhard’sprogress,presentedinthiswork,isthreefold:
• aclearframeworkforalgorithmanalysiswiththeappropriateparameters,
• theintroductionofmodulartechniquesintothisarea,
• almostoptimalalgorithmsforthebasicproblems.
Onemightsaythatthefirsttwostepsarenotnew.Indeed,thebasicalgorithms
and their parameters – in particular, the one called dispersion in Gerhard’s work
– have been aroundfor a while, and modularalgorithmsare a staple of computer
algebra. But their combinationis novel and leads to new perspectives, the almost
optimalmethodsamongthem.
VIII Foreword
A fundamental requirement in modular algorithms is that the (solution mod-
ulop)oftheproblemequalthesolutionofthe(problemmodulop).Thisisgener-
ally notvalidforall p, anda firsttask isto finda nonzerointeger“resultant”r so
thattherequirementissatisfiedforallprimespnotdividingr.Furthermore,r has
to be “small”, and one needs a bound on potential solutions, in order to limit the
sizeandnumberoftheprimesprequired.Thesetaskstendtobethemajortechnical
obstacles;thedevelopmentofamodularalgorithmisthenusuallystraightforward.
However,inordertoachievethetrulyefficientresultsofthiswork,oneneedsathor-
oughunderstandingoftherelevantalgorithmics,plusalotoftricksandshortcuts.
Theintegrationtaskisnaturallydefinedviaalimitingprocess,buttheOldMas-
terslikeLeibniz,Bernoulli,Hermite,andLiouvillealreadyknewwhentotreatitasa
symbolicproblem.However,itsformalization–mainlybyRisch–inapurelyalge-
braicsettingsuccessfullyopenedupperspectivesforfurtherprogress.Now,modular
differentialcalculusisusefulinsomecontexts,andcomputeralgebraresearchersare
awareofmodularalgorithms.Butmaybethesystematicapproachasdevelopedby
Gerhardwillalsoresultinaparadigmshiftinthisfield.Ifatall,thiseffectwillnot
bevisibleatthe“highend”,wherenewproblemareasarebeingtamedbyalgorith-
micapproaches,butratheratthe“lowend”ofreasonablydomesticatedquestions,
wherenewefficientmethodswillbringlargerandlargerproblemstotheirknees.
It was a pleasure to supervise Ju¨rgen’s Ph.D. thesis, presented here, and I am
lookingforwardtotheinfluenceitmayhaveonourscience.
Paderborn,9thJune2004 JoachimvonzurGathen
Preface
Whatfascinatedmemostaboutmyresearchinsymbolicintegrationandsymbolic
summation were not only the strong parallels between the two areas, but also the
differences. The most notable non-analogy is the existence of a polynomial-time
algorithm for rational integration, but not for rational summation, manifested by
suchsimpleexamplesas1/(x2+mx),whoseindefinitesumwithrespecttoxhas
the denominator x(x + 1)(x + 2)···(x + m − 1) of exponential degree m, for
allpositiveintegersm. ThefactthatMoenck’s(1977)straightforwardadaptionof
Hermite’s integration algorithm to rational summation is flawed, as discussed by
Paule(1995),illustratesthatthedifferencesareintricate.
TheideaforthisresearchwasbornwhenJoachimvonzurGathenandIstarted
theworkonourtextbookModernComputerAlgebrain1997.Ourgoalwastogive
rigorousproofsandcostanalysesforthefundamentalalgorithmsincomputeralge-
bra.WhenwecametoChaps.22and23,aboutsymbolicintegrationandsymbolic
summation,we realizedthatalthoughthereisnoshortageofalgorithms,onlyfew
authorshadgivencostanalysesfortheirmethodsortriedtotunethemusingstan-
dardtechniquessuchasmodularcomputationorasymptoticallyfastarithmetic.The
pioneersinthisrespectareHorowitz(1971),whoanalyzedamodularHermitein-
tegration algorithm in terms of word operations, and Yun (1977a), who gave an
asymtotically fast algorithm in terms of arithmetic operations for the same prob-
lem.Chap.6inthisbookunitesHorowitz’sandYun’sapproaches,resultingintwo
asymptoticallyfastandoptimalmodularHermiteintegrationalgorithms.Formodu-
larhyperexponentialintegrationandmodularhypergeometricsummation,thiswork
givesthefirstcompletecostanalysisintermsofwordoperations.
Acknowledgements. Iwouldliketothank:
Mythesisadvisor,JoachimvonzurGathen.
KatjaDaubert,MichaelaHuhn,VolkerStrehl,andLuiseUngerfortheir
encouragement,withoutwhichthisworkprobablywouldnothavebeen
finished.
MyparentsJohannaandHerbert,mysisterGisela,mybrotherThomas,
andtheirfamiliesfortheirloveandtheirsupport.
MycolleaguesatPaderborn:theResearchGroupAlgorithmic
Mathematics,inparticularMarianneWehry,theMUPADgroup,
X Preface
inparticularBennoFuchssteiner,andSciFaceSoftware,inparticular
OliverKluge.
Myscientificcolleaguesallovertheworldforadviceandinspiring
discussions:PeterBu¨rgisser,Fre´de´ricChyzak,WinfriedFakler,
MarkGiesbrecht,Karl-HeinzKiyek,DirkKussin,UweNagel,
ChristianNelius,MichaelNu¨sken,WalterOevel,PeterPaule,
ArneStorjohann,andEugeneZima.
Waterloo,23rdJuly2004 Ju¨rgenGerhard
Table of Contents
1. Introduction................................................ 1
2. Overview .................................................. 7
2.1 Outline ................................................... 12
2.2 StatementofMainResults ................................... 13
2.3 ReferencesandRelatedWorks................................ 21
2.4 OpenProblems ............................................ 24
3. TechnicalPrerequisites....................................... 27
3.1 SubresultantsandtheEuclideanAlgorithm ..................... 28
3.2 TheCostofArithmetic...................................... 33
4. ChangeofBasis............................................. 41
4.1 ComputingTaylorShifts .................................... 42
4.2 ConversiontoFallingFactorials .............................. 49
4.3 FastMultiplicationintheFallingFactorialBasis ................ 57
5. ModularSquarefreeandGreatestFactorialFactorization ......... 61
5.1 SquarefreeFactorization..................................... 61
5.2 GreatestFactorialFactorization............................... 68
6. ModularHermiteIntegration ................................. 79
6.1 SmallPrimesModularAlgorithm ............................. 80
6.2 PrimePowerModularAlgorithm ............................. 85
6.3 Implementation ............................................ 87
7. ComputingAllIntegralRootsoftheResultant ................... 97
7.1 ApplicationtoHypergeometricSummation.....................103
7.2 ComputingAllIntegralRootsViaFactoring ....................109
7.3 ApplicationtoHyperexponentialIntegration....................112
7.4 ModularLRTAlgorithm ....................................116
8. ModularAlgorithmsfortheGosper-PetkovsˇekForm ............. 121
8.1 ModularGP(cid:1)-FormComputation..............................134
XII TableofContents
9. PolynomialSolutionsofLinearFirstOrderEquations ............ 149
9.1 TheMethodofUndeterminedCoefficients .....................155
9.2 BrentandKung’sAlgorithmforLinearDifferentialEquations.....158
9.3 Rothstein’sSPDEAlgorithm.................................161
9.4 TheABPAlgorithm ........................................165
9.5 ADivide-and-ConquerAlgorithm:GenericCase ................169
9.6 ADivide-and-ConquerAlgorithm:GeneralCase ................174
9.7 Barkatou’sAlgorithmforLinearDifferenceEquations ...........179
9.8 ModularAlgorithms ........................................180
10. ModularGosperandAlmkvist&ZeilbergerAlgorithms .......... 195
10.1 HighDegreeExamples......................................198
References ..................................................... 207
Index.......................................................... 217
