lng.ThornasPARISINI UniversitadiG';T'Y,"i,::)"o","r;',otIiInformatica $.:,.:'..; Viaall'Cpt";, 13.161 .. 5G..:'DO"JS"!talia Tel.+jl)lO35327.99- fax+3::110353 29.41 thomas@rust.unige.it Harald Niederreiter AustrianAcademyofSciences nNer rtion an nte Carlo ts SOCIETYFORINDUSTRIAL ANDAPPLIEDMATHEMATICS PHILADELPHIA,PENt--ISYLVANIA1992 CBMSNSF REGIONAL CONFERENCE SERIES INAPPLIED MATHEI\J1ATICS A series of lectures on topics of current research interest in applied mathematics under the direction of the Conference Board of the Mathematical Sciences, supported by the National ScienceFoundationandpublishedbySIAM. GARRETI'BIRKHOFF,TheNumericalSolutionof EllipticEquations D.V.LINDLEY,BayesianStatistics,~Review R.S.VARGA,Functional Analysis and Approximation TheoryinNumerical Analysis R.R.BAHADUR,SomeLimit TheoremsinStatistics PATRICKBILLINGSLEY,WeakConvergenceof Measures:ApplicationsinProbability J.L.LIONS,SomeAspectsof theOptimal Control of Distributed Parameter Systems ROOERPENROSE,Techniquesof Differential TopologyinRelativity HERMANCHERNOFF,Sequential Analysis and OptimalDesign 1.DURBIN,DistributionTheory for TestsBased ontheSampleDistribution Function SOL1.RUBINOW,MathematicalProblemsintheBiological Sciences P.D.LAX,HyperbolicSystemsof ConservationLawsand theMathematicalTheoryof Shock Waves 1.J.SCHOENBERG,CardinalSplineinterpolation IVANSINGER,TheTheoryof Best Approximation mid Functional Analysis WERNERC.RHElNBOLDT,Methodsof SolvingSystemsof Nonlinear Equations HANSF.WEINBERGER,VariationalMethods forEigenvalueApproximation R.TYRRELLROCKAFELLAR,Conjugate Duality and Optimization SIRJAMESLIGHTHILL,MathematicalBiofluiddynamics GERARDSALTON,Theoryof indexing CATHLEENS.MORAWETZ,NotesonTime Decay and Scattering for SomeHyperbolicProblems F.HOPPENSTEADT,MathematicalTheoriesof Populations:Demographics,Geneticsand Epidemics RICHARDASKEY,OrthogonalPolynomials and Special Functions L.E.PAYNE,improperly Posed Problems inPartialDifferential Equations S.ROSEN, LecturesontheMeasurement and Evaluationof thePeiformance of Computing Systems HERBERTB.KELLER,Numerical SolutionafTwo Point Boundary ValueProblems J.P.LASALLE,TheStabilityof Dynamical Systems- Z.ARTSTEIN,Appendix A: Limiting Equationsand Stabilityof NonautonomousOrdinaryDifferentialEquations D.GOTTLIEBANDS.A.ORSZAG,Numerical Analysisof SpectralMethods:Theoryand Applications PETER1.HUBER,RobustStatisticalProcedures HERBERTSOLOMON,GeometricProbability FREDS.ROBERTS,GraphTheoryand its Applications toProblemsof Society (continuedoninside backcover) Contents PREFACEv CHAPTER1.MonteCarloMethodsandQuasi-MonteCarlo Methods L 1Introduction........ 1.2MonteCarlomethods... 1.3Quasi-MonteCarlo methods. Notes............ . CHAPTER 2.Quasi-Monte Carlo Methods for Numerical Integra-1 1 3 9 11 t ~ n13 2.1Discrepancy..13 2.2Errorbounds.18 Notes....21 CHAPTER 3.Low-Discrepancy PointSets andSequences23 3.1Classicalconstructions.....23 3.2General discrepancybounds.33 Notes.............44 CHAPTER 4.Nets and (t,s)-Sequences47 4.1Definitionsand discrepancy bounds.47 4.2Combinatorialconnections...60 4.3General construction principles....63 4.4A special construction of nets.....74 4.5A specialconstructionof (t,s)-sequences.90 Notes....................98 CHAPTER 5.Lattice RulesforNumerical Integration101 5.1The method of goodlattice points......102 5.2Existencetheorems forgoodlattice points.109 5.3Generallattice rulesandtheirclassification.125 lVCONTENTS 5.4Existence theorems forefficientlattice rules.. Notes..................... . 138 145 CHAPTER 6.Quasi-Monte Carlo Methods forOptimization147 6.1General theory of quasirandom search methods..147 6.2Low-dispersionpoint sets and sequences..152 Notes........................158 CHAPTER 7.Random Numbers and Pseu.dorandomNumbers161 7.1Randomnumber generation..161 7.2Pseudorandom numbers.164 7.3Classicalgenerators.168 Notes..........175 CHAPTER 8.NonlinearCongruential Pseudorandom Numbers177 8.1The generalnonlinearcongruential method.177 8.2The inversivecongruential method.182 Notes.....................189 CHAPTER 9.Shift-Register Pseudorandom Numbers191 9.1The digital multistep method...............191 9.2The generalizedfeedbackshift-register(GFSR)method.198 Notes........................204 CHAPTER 10.Pseudorandom VectorGeneration205 10.1The matrix method.205 10.2Nonlinear methods..213 Notes........215 APPENDIX A.Finite Fields and Linear Recurring Sequences217 APPENDIX B.Continued Fractions219 BIBLIOGRAPHY223 Preface The NSF-CBMS Regional Research Conference on Random Number Generation andQuasi-MonteCarloMethodswasheldattheUniversityof AlaskaatFair-banksfromAugust13-17,1990.Thepresentlecturenotesareanexpanded writtenrecordof aseriesof ten talkspresentedbythe authorastheprincipal speakerat thatconference.It wasthe aimof this seriesof lecturestofamiliar-izeaselectedgroupof researcherswithimportantrecentdevelopmentsinthe related areas of quasi-Monte Carlo methods and uniform pseudorandom number generation.Accordingly,theexpositionconcentratesonrecentworkinthese areasandstressestheinterplaybetweenuniformpseudorandomnumbersand quasi-MonteCarlomethods.Tomaketheselecturenotesmoreaccessibleto nonspecialists,some background material wasadded. Quasi-Monte Carlo methods can be succinctly described as deterministic ver-sions of Monte Carlo methods.Determinism enters in two ways,namely, by work-ing with deterministicpoints rather than random samplesandby the availabil-ity of deterministicerrorboundsinsteadof the probabilisticMonteCarloerror bounds.It could be argued that most practical implementations of Monte Carlo methods are,in fact,quasi-MonteCarlo methods since the purportedly random samplesthatareusedinaMonteCarlocalculationareoftengeneratedinthe computerbyadeterministicalgorithm.Thisisonegoodreasonforaserious study of quasi-Monte Carlo methods,and another reason isprovided by the fact thataquasi-MonteCarlomethodwith judiciouslychosendeterministicpoints usuallyleadsto afasterrateof convergencethan acorresponding MonteCarlo method.Theconnectionsbetweenquasi-MonteCarlomethodsanduniform pseudorandomnumbersariseinthetheoreticalanalysisof variousmethodsfor the generation of uniformpseudorandom numbers. Thelastfiveyearshaveseentremendousprogressinthesubjectareasof theselecturenotes.Thefieldof quasi-MonteCarlomethodswasenrichedby thesystematicdevelopmentof thetheoryof latticerulesandof thetheoryof netsand(t, s )-sequences,andnewandbetterlow-discrepancypointsetsand sequences wereconstructed.Important advancesin the area of uniform pseudo-random number generationincludethe introduction and the analysisof nonlin-earcongruentialmethodsandthemuchdeeperunderstandingwehavegained v VIPREFACE ofshift-registerpseudorandomnumbers.Furthermore,asystematicstudyof methods forpseudorandom vectorgeneration wasinitiated. The main aim of these lecture notes is to give an account of the recent develop-mentsmentioned above.The material inChapter 4on nets and(t,s)-sequences andinChapter5onlatticerulesisof centralimportance,sincemanyof the newadvancesinquasi-MonteCarlomethods,andeveninuniformpseudoran-domnumbergeneration,revolvearoundtheconceptsandresultsinthesetwo chapters.Indeed,netsandlatticesappearinsomanydifferentcontextsthat they mustbeviewedas basic structures.Another fundamental notion isthat of discrepancy,forwhich the essential facts are presented in Chapters 2 and 3.The conceptof dispersionplaysaroleinChapter6onquasi-MonteCarlomethods foroptimization.In our discussion of uniform pseudorandom number generation inChapters7,8,and9,weemphasizethosealgorithmsforwhichadetailed theoretical analysis has been performed. For some results, especially for classical ones, the proof has been omitted, but areferenceisalwaysprovided.Thenotestoeachchaptercontainsupplemen-tary information andhistoricalcomments.The bibliography isnotmeant to be comprehensive,but listsonlythosereferencesthatarecited in the text.Since adetailed bibliography up to 1978isavailable in[225],the presentbibliography concentrates on the more recentliterature. Theconferencewouldnothaveoccurredwithouttheinitiativeandtheen-thusiasm of ProfessorJohn P.Lambert of the University of Alaska at Fairbanks, whooriginatedtheidea,didallthenecessarypaperwork,andorganizedthe conferenceinaperfectmanner.Many thanks,Pat,forthe generoushospitality you extended to all participants and formaking sure that everybody survived the white-water rafting on the Nenana River.I also want to express my gratitude to NSF-CBMS forthe financial support of the conference.The actual production of these lecture notes relied heavily on the word-processing skills of Rainer Gottfert and Irene Hosch.Special words of appreciation go to Rainer Gottfert for his dedi-cated help in proofreading the manuscript and to Dipl.-Ing. Leonid Dimitrov and Dipl.-Ing.Reinhard Thaller forexpert technical advice.The cooperation of the SIAM staff isalso noted gratefully.Last, but by nomeans least,I wish to thank Gerlindeforputtingup withahusbandwhowassomewherebetweenthereal worldand mathematical elysium during the last15months. Vienna,August 1991H.NIEDERREITER CHAPTER MonteCarloMethodsand Quasi-MonteCarloethods

Inthischapterweset forthemoredetaileddiscu.ssi,otofquasi- Carloinc4 the o(quaai-Vente Car19 metpods,is,without anrudi,wentary undemtan..),then,forany N;?:1,wehave ProofWrite 9= f- E(f)i thenfAgd..\= 0and 1N.1N 'N Lf(an )-E(f) = NLg(an). n=ln=l MONTE CARLO METHODS ANDQUASI-MONTE CARLOMETHODS5 Thus = L'" , .1 Theorem 1.1may be interpreted to mean that the aboolutev,alue of the error in(1.4)is,on the a v e r ~ ju(f)N-1/2, where a(f) =(u2(fV2 the standard deviationofI,Furtherprobabilisticinformationabouttheerrorisobtained fromthe central limit theorem, which states that, if 0< u(f) < 00,then foranyconstantsCl'$(12(1). j=1NjJAjA(Aj)JAjN p.roof,'Nit h the sTIBc1alfo:rm.theNiit sufficesto show that t {(I ____I1,1dA)2 d)'$f(f _E(J2 dA. j=l JAj>'(Aj)AjiA By expanding the square on both sides, it is seen that this is equivalent to proving 2k1(1\2 E(f)$L >'(A .) II d>.), j=1J\Aj andthelatter isobtainedfromtheCauchy-Schwarzinequalityby noting that Another for variance reduction is the method of antithetic variates. .-We explain this method in the simple case of an integral-E(f) = 11 I(u)du. We introduce the auxiliary function (1.7) y(u) =! (f(u) + - ufor0 $u$1 and use the estimate :1N1N E(f) =E(g) NL 9(Xn} =2N L (f(Xn) + 1(1- xn \n=ln=l MONTE CARLOMETHODS ANDQUASI. MONTE CARLOMETHODS9 withNindependentanduniformlydistributedrandomsamplesXi, E [0,1].AccordingtoTheorem1.1,theerrorof thisestimateis rr (g) IN.Since there are 2N functionvalues of /involved in this estimate,the mean-eq1W'e er1'Ofu2(g)/N should beoompared. with the quantity (72(J)/(2N). The fOllowing result showsacue in which avariance reduction isachieved. PROPOSITION1.3.II/is acontinuousmonotonefunction00[0, 1]and g istkfinetlby(L7) I Proof.We have o"(g) = J.' (g(,,) - E(g))' 11"= J.' G/(tt) +u) - E(f), du = 11 /2(u) du + 11 I(u)!(l - u) du - E(f)2. The de&red inequality iEltherefore eqluvalent to .11 I(u)f(l- u)du '5E(f)2. Byreplacing,if necessary,fby-I, wecanassumethatfisnondecreasing. Then the function F(u) = lu J(1- t) dt - E(f)u on [0,1] has the nonincroosing derivative F'(u) =J(1-u) -E(J). Since F(O)= F(l) =0,it followsthat F(u)2::0forall u E[0,1].This implies that 11 F{u) dJ(u)2::o. Integration byparts yields 11 f(u) dF(u)= 11 J(u)F'(u)$O. By inserting the formula forF'(u), wearriveat(1.8).0 1.S.Quasi-Monte Carlo methods. We:recallthatinMonteCarlointegrationwithNrandomnodestheabsolute value of the error has the average order of magnitude N-l/2.Clearly, there exist sets of Nnodesforwhichthe absolute valueof the error isnotlargerthan the average.IT we could construct such sets of nodes explicitly, this would already be auseful methodological advance.Thequasi-MonteCarlomethodfornumerical integration aims much higher!as it seeks to construct sets of nodes that perform significantly better than average.The integration rules for the quasi-Monte Carlo method are taken fromthe appropriate MonteCarloestimate.For instance,for 10CHAPTER 1 the normalizedintegration domainill, wehavethequasi-MonteCarloapproxi-mation (1.9) f1N }f8 feu) du NL f(Xn), 1n=l whichformallylooksliketheMonteCarloestimatebutisnowusedwithde-terministicnodesXll".,XNEis.Thesenodesshouldbechosenjudiciously soastoguaranteeasmallerrorin(1.9).Inanalogywiththemoregeneral Monte Carlo estimate (1.6),wehave thequasi-MonteCarloapproximation (1.10) f1N }1feu) au NLf(x.n), Bn=l xnEB where Bisasubset of /sand Xl, , XNEIsare deterministic points. The error analysis forthe approximations(1.9)and(1.10)will be performed in 2.2and foraspecial situation in Chapter 5;Explicit constructions of sets of deterministic nodes guaranteeing small errors will be the subject of ChapterS 3, 4,and 5.Wedonot wishto preview theseresultsat length,but the fonowing veryconcisesummary maybe givenat this early stage.If weassume(1.9)to be the standard case,then the benchmark is the probabilistic Monte Carlo error boundO(N-l/2).The quasi-MonteCarlomethodyieldsamuchbetter result, givingusthe deterministicerrorbound D(N-'-l(logN)S-l)forsuitably chosen sets of nodes and forintegrands with a relatively low degree of regularity.In the specialcircumstandfornumericalintegration,whichshouldbecomparedwiththelistof deficienciesof theMonteCarlomethod in1.2.The verynature of the quasi-Monte Carlo method, with its completely deterministic procedures, implies that wegetdeterministic-andtl1usguanmteed-:-er.rorbounds.Inprinciple,itis therefore always possible to determine in advance an integration r-ulethat yields a prescribed level of accuracy.Moreover, with thesame computational effort, Le., with the same number of functionevaluations(which a.rethe costly operations in numerical integration), the quasi-Monte Carlo method achieves a significantly higher accuracy than the Monte Carlo method.Thus, on two crucial accounts--determinismandprecision-the Carlomethodissuperiortothe MonteCarlo method.For aspecial type of quasi-MonteCarlo method, the lat-ticerulesto be discussediuChapter 5,wehavethedesirablepr9perty thata of regularity of theleadstoprecision in the inte-gration rule.,The one problem with the Monte Carlo method that attains almost philosophiFal dim,ensions,namely, the difficulty of generating truly random sam-ples, whenweconsideraquagi-MonteCarlomethod,sinceherewe just implement ready-roMe constructions to obtain the required nodes. , MONTE CARLOMETHODS ANDQUASI-MONTE CARLO METHODS11 Therearequasi-MonteCarlomethodsnotonlyfornumericalintegra.tion, butalsoforvariousother numericalproblems.Infact,formanyMonteCarlo methods,it is possible to developcorresponding quasi-MonteCarlo methods as their. deterministicversions.Invariably,the basicidea is toreplacethe random samples in the Monte Carlo method by deterministic points that are well sUited for the problem at hand.A further illustra.tion of this principle will be provided Chapter 6, where we discuss quasi-Monte Carlo methods for global optimization. Quasi-Monte Carlo methods have firstbeen proposed in the 19508,and their theoryhassincedevelopedvigorously.However,forara.therlongtime,these methodsremainedtheprovinceof specialists.Thewideracceptanceof quasi-Monte Carlo methods W8Bpioneered in computational physics where large-scale MonteCarlocalculationsareverycommon' andwhereitwasrecognizedthat quasi-MonteCarlomethodscanleadto significantgainsin' efficiency.There-finementsof quasi-MonteCarlomethodsandthe expanding scopeof theirap-plicationshaverecentlymadethesemethodsknownto 'largersegmentsof the scientific computing community. oh";ioUJlmethodsDfoblere.B of numericalanalysisthat canbe reducedto numericalintegration.Anexam-ple is the numerical solution of integral equations forwhichquaai-MonteCarlo methodsarediscussedinthebooksof Korobov[160,Chap.4]andHuaand Wang[145,Chap.10]and in the morerecentworkof Sarkarand Prasad[301] and Tichy [349].For applications to initial value and boundary value problems, we refer to HUB. and \Vang [145,Chap. 10]and Taschner [34OJ.Important applica-tions of quasi-Monte Carlo methods to the numerical solution of difficult integro-differentialequationssuchastheBoltzmann equationhavebeendevelopedby Babovsky etal.[12},Lecot[181],[182],[184J,and Motta etal.[214].Applica.-tions to approximation theory can already be found in the books of Korobov [160, Chap.4]andHuaandWang[145,Chap.9].Quasi-MonteCarlomethodsfor thenumericalsolutionof systems of equationsare studiedinHlawk..a[140]and Taschner[341 J.A survey of quasi-Monte Carlo methods in computational statis-ticsispresentedinShaw[309].Dealandsatisfyinglime-+D+ bee)=o. Then weletMb/bethe familyof all Lebesgue-measurableB18 forwhich A8(Be "B) :5bee)andAs(B" B-e) :5bee)foralle > O. EveryBEMbisactuallyJordanmeasurable,and,conversely,everyJordan-measurablesubsetof i8 belongstoMbforasuitablefunctionb(seeNieder-reiter[219,pp.168-169]). ForafamilyMb,wenowconsiderthediscrepancyDN(Mbi P)definedac-cordingto(2.4).NiederreiterandWills[278]havegivenaboundforthisdis-crepancyintermsof DN(P).If thefunctionb satisfiesbeE:)2:Eforallc> 0, then the bound canbe simplified to yield DN(Mbi P):54b{2.jSDN(P)1/B). In many cases of interest,the functionb Mll havethe formb(e)=Cforsome constantC> 0,and then the bound in[278]reduces to 18CHAPTER 2 Sinceeveryconvexsubsetofi8 belongstoMbawithbo beingthefunction bo(e)= 2se,wehave IN(P) $;DN(Mb;P) whenever the function b satisfies b( e)~2Se for all e> 0.IT the sequence S is uni-formly distributed in ja, then, for any function b,we have limN__ CDDN(Mbi S) = 0,whereDN(Mbi S)isthe appropriate discrepancyof thefirstNterms of S. Under suitable conditions on the functionh,e.g.,if b(e)~2seforall e> 0,we a.lso have the converse result;namely, limN-+CDDN(Mb; S)=0 implies that S is uniformly distributed in ia 2.2.Error bowids. Wediscuss the most important error bounds forthe quasi-Monte Carlo approx-imation(2.1)and analogousboundsformoregeneralintegration domains.All theseboundsinvolveasuitablenotionof discrepancy.Westartwiththe one-dimensionalcasein whichtheproofsarequiteeasy.Aclassicalresultisthe followinginequality of Koksma[155]. THEOREM2.9.If fhasboundedvariationV (f)on[0, 1],then,forany Xl! .. ,XN E[0,1 j,wehave Proof.Wecan assume that Xl::5X2::5. . . ::5x N.Put Xo=0andx N +1= 1. Using summation by parts and integration by parts,weobtain 1N11. N.11 NL f(Xn)- f(u) du = - L ; (f(Xn+d - f(xn}) +udf(u) n=1'()n=O() N/Zn+l (n) = ~ l z : "u- Ndf(u). For fixednwith ::511,::5N, wehave lu- ;1 ::5l)N(Xl!'",XN)forxn::5u::5Xn+l by Theorem 2.6,and the desired inequality followsimmediately.0 In Theorem2.12,below,wewillprovearesultthatimpliesthatKoksma's inequalityis,ingeneral,thebestpossible,evenforCoofunctions.Werecall that,foracontinuousfunctionfon[0,1], itsmOOulmof contin/uityis defined by w(fj t) =supIf(u) - f(v)1fort~O. u,vE[ 0,1] lu-vl:9 The followingerror bound forcontinuous integrands was established by Nieder-reiter[218]. QUASI-MONTE CARLOMETHODS FOR NUMERICALINTEGRATION19 2.10.II Iiscontinuowon[0,1],then,foranyXli'" jXNE 1],wehave Proof.WecanagainassumethatXlX2::;X N.Bythemean-value theorem forintegrals,wehave r1N(n.IN1N JoI(u) du = L J(,f(u) du =NL J(tn } on=l(n-l)/Nn=l with(n -l)IN < tn< niN.Therefore NowIx", - tnl:::;D:N(Xl,'") XN)for1 n::;Nby Theoremfollows.0 the result To extend Koksma's inequa.1..ityto the multidimeooional. COOiEi,an appropriate conceptoftotalvariationforfunctionsofseveral'!2\Tiablesillneeded.Fora functionJ on i8 and asubintervalJof 11\letl},,(f; J)be an alternating sum of the values of Iat the vertices of J(Le.,function -v'aluesat adjacent vertices have opposite signs).The variationoff on isin these'f!.JJeisdefinedby V(S)(f)= sup L J)I, 'PJE'P wherethesupremumisextendedoverallpartitionsPof isintosubintervals. The more convenientformula (2.5) 1111oaJ V(8) (f) =...I----ldu1dus o0871,1'"8us holds whenever the indicated partial derivative is continuous on is.For 1 ::;k::;s and1::;il(V(f) + 1/(1, . .. ,1 )i)JN(P). x",EB The most general situation that weconsideriswhereBbelongsto a.family Mbof Jordan-measurable sets introduced in2.1.The followingtheoremisan improvementby de Clerck[54]on aresult of Niederreiter[219]. THEOREM2.15.If BE Mband fhasbounded variationV(f)on is inthe senseHardy and Krause,then,for any point setPconsisting of Xl, IXNE we ha've It. I(x..) - 1. f( u) dul!>(V (f) + 1/(1, ... ,1 )I)DN(M,; Pl x"EB The term 11(1, ... ,1)1is needed in the inequalities in Theorems 2.14 and 2.15 because without this term the bounds would failto hold even forconstant func-tions f. Notes. The notion of discrepancywasfirststudiedinitsownrightbyBergstrom[24J, but the applications to numericalanalysisbecame apparentonly after the later workofKoksma[155].Theuseoftheisotropicdiscrepancywassuggested byHlawka[136].ThediscrepanciesDN(Mb; P)wereintroducedbyNieder-reiter[219].If,insteadof themaximumdeviationbetweentheempiricaldis-tributionandthe uniformdistribution,weconsidel'themean-squaredeviation, then -1;heL2discrepancyisobtained,and,moregenerally,anIJ>discrepancy 22CHAPTER 2 maybedefinedfor1~P 0dependsonly ons.This wouldmean that the star discrepancy of an N -elementHammersley pointsetin pairwise relatively prime basesattainstheleastpossibleorderof magnitude.Asmentionedpreviously inthis section,(3.8)isobviousinthecasewheres=1.Fors= 2,(3.8)was establishedby Schmidt[303],but(3.8)isstill open fors3.The best general result in this direction isdue to Roth [296],who showed.that, forany N-element point setPin dimensions,wehave For s= 3,a slight improvement was recently obtained by Beck [19],namely, that Div(P)BsN-1 (log N) (log log N)CforN2:3, wherec > 0 isanabsolute constant. LOW-DISCREPANCY POINT SETSANDSEQUENCES33 If (3.8)holds,then, with the help of Lemma 3.7,wewould easily obtain that any s-dimemlional sequence S satisfies (3.9)DN(S)forinfinitely manyN, wheretheoomltant >0dependsonlyon8.Thiswouldmeanthatthe star discrepancy of aHalton sequence in pairwise relatively primebasesattains the leastpossible order of magnitude.By aresultof Schmidt[303],mentioned previouslyinthissection,(3.9)holdsfor8=1,butthisistheonlycasein which(3.9)isknown.Theresultsof Roth[296]andBeck[19],statedabove, haveanaloguesforsequences.Forarbitrary8andanys-dimensionalsequence S,wehave DN(S)forinfinitely manyN, while any two-dimensional sequence S satisfies > manyN, where d> 0isan absolute constant.These resultsonirregularities of distribu-tion arequitedifficultto prove.Expositoryaccountsof thistopicaregivenin the books of Beck and Chen[20]and Kuipersand Niederreiter[163J. 3.2.Gen.eral discrepancy bounds. Becauseof thefinite-precisionarithmeticof computers,allpointsetsandse-quences of practical interest consist only of points with rational coordinates.For thiscase,thereareseveralimportantprinciplesforobtainingupperandlower discrepancy bounds. For an integer M2,let G(M)= (-M/2,M/2] nz and G*(M) = G(M) "-{O}.Furthermore, let Gs(M)be the Cartesian product of scopies of GeM)and G:(M) = Gs(M) "- {o}.Put r(h, M)= {Ml sin forhEG*(M), forh= O. 8 r(h, M) = II r(hi' M). i=l Wewritee(u)= e2'1rv'-Tu foruEJ.Rand X Y forthe standard innerproduct of x,yEJ.RB. LEMMA3.9.Letti, UiE[0,1]for1 :::;i:::;sandletvE[0,1 Jbesuchthat Iti - uil:::;vfor1 :::;i:::;s.Then. I Bti - 8Ui I :::;1 - (1- v) 8. i=li=l 34CHAPTER 3 Proof.Weproceedbyinductionons,withthecase8=1beingtrivial.If the inequality isshown forsome82::1 and weassume without lossof generality that ts+!2::Ua+!, then lIt ti-It Uil~(t'+1- U'+1) gti +u,+lg ti - i ~ 'Uil :::;ts+l- Us+l+ us+! (1- (1- v)S) = ts+! (1- (1- v)S) + (ts+l- us+d (1- v)S :::;1- (1- v)S+ v(l - v)S= 1 - (1- v)s+l. o THEOREM3.10.Foraninteger M2::2andYo, ... ,YN-lEZS,letPbethe pointsetconsistingof thefractionalparts{M-lyO}"",{M-1YN_d.Then ( 1)s11N-l(1) DN(P)::;1- 1- lvI+LrehM)N2: e'Mh' Yn. hEC:(M)'n=O ProofFork=(k1, ... ,ks)EZS,letA(k)bethenumberof nwith0:::; n:::;N- 1andYn= kmod M,whereacongruencebetweenvectorsismeant componentwise.Then sincetheinnersumhasthevalueMSif Yn= kmodMandthevaluezero otherwise.Therefore N1(1) N-l(1) (3.10)A(k) - Ms= MsLe- Mh. kL eMh. Yn. hEC;(M)n=O Now let J= rr:=l [Ui, Vi)be a subinterval of IS.For each i,1 :::;i:::;8,we choose the largestclosedsubinterval of [Ui' Vi)of the form[ai/M, bi/M]with integers ai:::;bi,whichweagaindenoteby[ai/M,bi/M].The casewhereforsome ina suchsubintervalof[Ui' Vi)existscanbe easilydealtwith,sincewethenhave A(J; P) =0and Vi- Ui< ljM; hence (3.11) A(J; P)_A(J)= A(J)< ~< 1 - l1 - ~ I)8 Ns8M- !M \ Intheremaining Fase,theintegersai, bi,1:::;i:::;8,arewelldefined,andwe LOW-DISCREPANCYPOINT SETSANDSEQUENCES35 obtain, by(3.10), _A.(J)(At) - + -a; + 1) - A.(J)

Now Ibi-ai+1 I1 M- (Vi- Ui) 1_ (1_ ~ .)\ 8 - \- \.M \ by Theorem 3.14.On the other hand, if the points ofare denoted by M-1y.n,) :::;11,:::;N- 1,then,foranyh= (hI, .. ,,hs) EC;(JlIl),wehave 42CHAPTER 3 and soTheorem 3.10implies that DN(P)~1 - (1- M-I r. Therefore This shows,in particular,thatthe term1 - (1- AI-I) sin Theorems 3.10and 3.14is,ingeneral,thebestpossible.Theexpression1 - (1- M-Ir maybe viewedasthediscretizationerror,sinceitarisesfromthespecificdiscreteness property of the point sets in Theorem 3.14.Note that in firstapproximation the discretization error iss / Mforlarge M. It is often easier to obtain formulas or lowerbounds forcosine or exponential sumsthanfordiscrepancies.Thefollowingresultscan thenbeusedtoderive lowerbounds fordiscrepancies.For h= (hI, . 0 ,hs) E ZS,wewrite s r(h) = II max(l, Ihil). i=l Note that Theorem 3.16and Corollary 3.17 can be viewedas special versions of Theorem 2.11. THEOREM3.16.Forarbitraryto, tlJ ... ,tN-l ElIts letPbethepointset consistingof the fractional parts{to}, {tl}, ... ,{tN-I}.Then,forany nonzero hE ZSandany real6,wehave 11N-lI2 N~cos 21r(b . tn - 6)~;((11" + l)m - l)r(h)DN(P), where misthenumber of nonzerocoordinatesofh. ProofWemstconsiderthecasewhereh= (hI, ... ,ha)withIhil= 1for 1 ~i~s.Then the functionf(u!, ... ,us) = feu)=coo 211"(hu- 6)satisfies f(ul,'",Uk-ll Uk+ !, Uk+h 0 ,'Us)= - f(ul, ... ,Uk-I, Uk, Uk+lI. 0, Us) foreach k with 1 ~k:5s and all UI, 0,Uk-lJ Uk+l,"0,UsE [0,1], UkE(0,~]. Thus wecan apply[234,Lemma.5.3].With suitable 6i ElR,this yields I N-II ~~coo 21r(h . tn - 6) Now .- rl/2 rl/'},/(j)I1 Jo... Jocos 21rt; hiui - 6idUl ... dUj= 2i-I1r' \ LOW-DISCREPANCY POINT SETSANDSEQUENCES43 and wearriveat the inequality of the theorem in the caseunder consideration. Next,wetreatthecasewhereh=(hI,.,.,hs)withallhii=O.Lettn= ... ,t}:))andputWn=... ,lh.'llt!:for0::;n::;N- 1.Then b tn=I W n ,whereallcoordinatesof Ihaveabsolutevalue1.Accordingto wehavealready established, 11N-lI11N-lI Ncos 211"(b.tn - 0)=Ncos 211"(1Wn- 0) :::; + 1)8-l)DN({wo}, ... ,{WN-l})' 11" To bound the last discrepancy,wenote that, forapositiveintegerh and t ER, wehave{ht}E[u, v)[0, 1)if and only if {t}belongs to one of the h intervals [(q + u)jh, (q + v)jh), q = 0,1, ... ,h - 1.Using this property,weobtain this againyieldsthe inequality of the theorem, Finally,weconsideranarbitraryh=(hI, ... ,hs)i- Wecanassume loss of generality that hi=f.0 for1 i:::;mand hi=0m + 1 :::;i8. T. bi - Ihh . )d'f t- (t(l)t(8)"",- ft(l) .- \1, .. "m, an1n- n, .,n.,weput[,n- \n)... ,n o Sn:::;N- 1.According to whatwehavealready established,weobtain 11N-lII1N-l! NL c08211"(b tn - 0)= I NL cos 2'li(h' .- 8)1 n=On=OI :::; + 1)7n- l)r(h')DN( ...)}). 11"-Nowr(h') = r(h),and,fromthe definitionof the discrepancy,weseethat Thus wehaveprovedthe theorem in the general case.0 COROLLARY3.17.For arbitmryto,, ... , tN-l ER87 letPbethepoint set consistingof thefractionalparts {to}, {td, ... ,{tN-d.Then,foranynonzero hEZ 8,wehave whe'i'emisthenumber of nonzerocoordinatesofh. Pmoj.Wewrite N-lIN-1 I 2: e(h . tn)=e(8)e(h . tn) n=On=O 44CHAPTER 3 with some real 0,a representation that is possible forany complex number.Then and,taking real parts,weobtain IN-lIN-l e(h.tn)1=cos27r(htn -0). The desiredbound nowfollowsfromTheorem 3.16.0 Notes, ForthevanCorputsequence82 inbnse'2,a proofofDN(S,),)= O(N-llog(N + 1)canbefoundin[163,p.127].Haber[122]showedthat NDj.,(S2)(logN)j(log8) + 0(1)andthattheconstantIj(log8)is thebest possible.The vandel'Corputsequencein basebcanbe obtainedby iteration of an ergodictransformationon[0,1),aswasdemonstratedby Lambert[165]; see also Lapeyre and Pages[169].This factwas usedby Lambert[166]foranal-ogousconstructionsin dimensions8=2,3,4.Furtherworkon the(star)dis-crepancy of generalized van del' Corput sequences was carried out by Bejian [21], Faure[94],[96],[97],[99],[100],andThomas[347].The study of the discrep-ancy of the sequences S(z)with irrational z is aclassical area of number theory; see[163,p.128]forworkpriorto1974.Morerecentpapersonthistopicare Dupain[72],Dupainand80s[73],Ramshaw[291J,SchoiBengeier[305],[306], [307],and 80s [331].In particular, SchoiBengeier[305]has shown the converse of Corollary3.4;i.e.,if DN(8(z))=O(N-llogN) forallN;::::2,thennecessarily E::l ai = Oem). Distribution properties of the multidimensional sequences 8(z) were recently studiedbyLarcher[172],[176],[178]andLiardet[191];see[163,p.129]for references to earlier work.The sequences 8(z) play an important role in a quasi-MonteCarlomethod forthe numericalintegration of periodicfunctions,which isbased on the theory of diophantine approximations(see[225,5]). HaltonsequenceswereintroducedinHalton[125],andHammersleypoint setsinHammersley[129].Discrepancyboundsoftheordersof magnitudein Theorems 3.6 and 3.8, respectively,were established in Halton [125],and the im-plied constants were firstimprovedby Meijer[209]and then by Faure[93].The bounds in Theorems 3.6and 3.8 are essentially those of the latter paper, but we haveused asomewhat differentmethod of proof.The principle in Lemma 3.7 is attributed to Roth [296];for a converse of this principle, see [163,p.106, Ex.2.2]. The paperof Roth[296]alreadycontainstheconstructionof two-dimensional Hammersleypointsetsin thebase 2.Forthiscase,andforthe numberNof beingapowerof 2,an exactformulaforthe star discrepancywases-tablishedbyHaltonandZaremba[128J.Moregenerally,anexactformulafor the star discrepancy of two-dimensional N -element Hammersley point sets in an \ LOW-DISCREPANCY POINT SETS ANDSEQUENCES45 arbitrarybaseb,withNbeingapowerof b,wasgivenbydeClerck[55],[56]. Computer implementations of Halton sequences and Hammersley point sets are described in Fox [107],Halton and Smith [127], and Ucot [183].Generalized Hal-ton sequences,whicharethe obviousanaloguesof generalizedvanderCorput sequences,werefirstconsidered by Braaten and Weller[32],and furtherresults on these sequencescanbe foundinHellekalek[132].ForgeneralizedHammer-sleypointsetsinthetwo-dimensionalcase,seeFaure[98].Othervariantsof Hammersley point sets are discussed in[225,pp.977-978]. Theorem3.10isduetoNiederreiter[224].Theorem3.12isaspecialcase of aresultof Niederreiter[241]in whichthepoints(3.14)maybe suchthat in each coordinate the digitexpansions can haveadifferentlength and adifferent base.Asomewhatweakerformof Theorem 3.12wasalready stated inNieder-reiter[229],andtheone-dimensionalcasewasprovedinNiederreiter[232].A discrepancyformulathat ismoregeneralthan thatinRemark3.15wasestab-lishedinNiederreiter[241];notethat thediscrepancyformulainRemark3.15 can also be derived by elementary counting arguments.Theorem 3.16 was shown dimensional case, the problem of obtaining a result like Corollary 3.17 with opti-mal constants was studied by Horbowicz and Niederreiter[143]and Niederreiter and Horbowicz[265]. Low-discrepancypointsetsandsequencesandirregularitiesof distribution havealsobeenconsideredfordomainsotherthanintervals.Ausefulsurvey canbe foundinBeckandChen[20].Thecaseof sphereshasreceivedspecial attention;seeHlawka[138]andLubotzky,Phillips,andSarnak[195],[196]for important work on thiBcase,and Tichy[351]forapplications.Adetailed study of various other special domains wasrecently carried outby Hlawka[139]. CHAPTER4 N'etsandces Wehaveseenin3.1that,forans-dimensionalHaltonsequenceinpairwise ,relativelyprimebases,wehaveDiv(S)= O(N"71(logN)S)forallN2.By ,optimizingthe choiceof bases,wearrivedat the discrepancybound(3.6).Let mi' now take acloser look at tHe coefficient 'Asof the leading term in thiSbound. "VehaveAs=A(Pl,'",Ps),wherePb'"JP&afethemstsBythe forA(pl, ... ,Ps)and by prime r.nllnbertheorem,weobtain limIog'A.'!= L s-+ooslogs ThusAsincreasessuperexponentiallyas' s'00.Similarly,forHammersley point sets with an optimal choice of bases,we have the discrepancy bound (3.7), wherethecoefficientof theleadingterm againincreasessuperexponentially as s00.Thisfastgrowthof ,As(comparealsowithTable4.4)makesthe bounds(3.6)and(3.7)practicallyuselessforallbutverysmalldimensionss. Formostapplications,weneedpointsetsandsequencessatisfying discrepancy boundswith' muchsmallerimpliedconstants.Constructions of suchpointsets and sequenceswill be describedin this chapter. In4.1 wedefinepointsetsandsequenceswithaveryregulardistribution behavior.Thesewillbecalled(t, m, s)-netsand(t, s)-sequences,respectively. On the basis of the strong properties enjoyedby these nets and(t, s)-sequences, wecanderiveverygooddiscrepancybounds.Thedefinitionsof(t, m, s }-nets and(t, s)-sequences have acertain combinatorial flavor,and some concrete con-nectionswithclassicalcombinatorialproblemsareexploredin4.2.General principles forthe construction of nets and(t, s}-sequences are presented in 4.3. Theseprinciplesare usedin4.4and4.5forthe construction of specialfami-liesof netsand(t, s)-sequences,respectively.Byoptimizingtheparametersin the construction of (t, s )-sequencesin4.5,weobtain sequencesforanydimen-sions2,whichasymptotically have the smallest discrepancy that iscurrently known. 4.1.Definitionsand discrepancy bounds. To motivate the following definitions,wefocuson the followingspecial property ofthevanderCorputsequenceXO,Xl, .inanarbitrarybaseb2(this 47 48CHAPTER 4 sequence wasintroduced in Definition 3.2).For fixedintegersk2::0and m2::1, considerthebm pointsXnwithkbm ~n t,then they can be written in the form :,btb; -t)1I+ O(bt(k - t)8-1), --aqq-P. In the secondcase,leta~(q-1)/3.Then,fromA(JjP) =0,weobtain Thus inboth cases wehave the firstinequality in the theorem. If,inaddition,wehavel1ij(O)=0for1::;i::;sand1::;j::;m,then from(4.34)wegetc=OJhencea=O.Thus,.fromthefirstcaseabove,we obtain D* (P)>q - a- 1q_p(0)=q -1q_ p(0} N- 2q2q' whichisthe secondinequalityin the theorem.0 REMARK4.31.Since the proof of Theorem 4.30 is based on the construction of an intervalJcontainingnoneof thepointsXnin(4.25),itfollowsthatthe lowerboundsinTheorem 4.30alsoholdforanypoint setconsistingof theXn with nrunning through an arbitrary nonempty subset of {O, 1, ... ,qm -1}. If qisaprime,thenFqandZqcan beidentified.Thereisalsoacanonical way of identifying elements of Fqand C( q) .Let C = {C)i.)EF;: : 1 ::;i::;s,1 ::; j::;m}be the system of vectors in(4.26).Usingthe quantities Wq(H)defined in(3.18),weset (4.35) Rq(C)= 2: Wq{H). H Herethe sum runs overaUnonzeroH= (hij)EC(q)8Xmwith where the hij are viewedaselements of Fq LEMMA4.32.If qisaprime,R=Fq;andeveTY"Iijistheidentitymap, thent h e . ~ t a rdiscrepancyof thepoint set Pin (4.25)satisfies NETS AND(t,S)-SEQUENCES ProofBy the assumption on the "Iii' wehave m-l (i)_""'(i) Ynj- L...JCj'1''r(ar(n for0:5 n< qm,1:5 i:58,1 :5j:5m. '1'=0 A:na.pplication of Theorem 3.12 yields (4.36) ( 1)81N-l(18m) DN(P) $1- ,1- N+ L Wq(H)NL e-' \HOn=Oqi=l j=l 69 where the outer sum isoverallnonzeroH=(hij)EC(q)8xm.ForfixedH, we have = II2:e -. ffl-1(q-l(b8m)) 1"=0b=Oqi=l j=1 ;;The lastexpressionisequal to qm= Nif for0 :5r:5m- 1, i=1 ;=1 andequalto zerootherwise,wheretheh ijareviewedaselementsofFq.The lemma now followsfrom(4.36)and thedefinitionof Rq(C)in(4.35).0 On the basis of Lemma 4.32,we can show that the construction of the point sets(4.25)yieldson the averagealow-discrepancypoint set. THEOREM4.33.Fora prime qand forintegers m1and s1,let 1 Mq(m, 8)= card(C)Rq(C) bethemeanvalueof Rq (C)extendedover theset C of allchoicesforasystem C= {cji)EF;":1:::; i:::;S,1 $j :::;m}.Then,withN= qffl,wehave 1(logN)81 Mq (m, s)=Nlog 4+ 1Nif q =2, 1(m""'nlhlm - 1) S1 Mq(m,s) =N.- L..tcsc-+m--qhEC*(q)qqN 2. Nn5 log qq log qq 70CHAPTER 4 Proof.Inserting the definition of Rq(G)into the expression forMq(m, s)and interchanging the order of summation,weobtain 1 Mq(m, s)=card(C)~Wq(H) ~1, where the outer sum is over all nonzero H= (hij )EC(q)sxmand the inner sum is overallC= (c)iEC forwhich (4.37) sm '" ""' h(i)0Fm L- L.,ijCj =Eq. i=l j=l ForafixednonzeroHEG(q)BXm,theinnersuminthelastexpressionfor iI.Jq(m,3)representsthenumbersolutions(c)tJ)ECofthevectorequa-tion(4.37).SinceatleastonehijisanonzeroelementofFq,wecanchoose ms - 1 vectorsc)i)EF::arbitrarily,andthe remaining vectoristhen uniquely determinedby(4.37).Thereforethe numberof solutions of (4.37)isq(ms-l)m. Since card (C)=qm2 s,it followsthat Mq(m,8)= q-m28 q(ms-l)mL Wq{H)= ~L Wq(H). H#OH,eO H q= 2,then the second part of Lemma 3.13yields ~(m)S(lOgN)8 f::o Wq (H)=2" + 1- 1 =log 4+ 1- 1, andthe formulaforMq (m, 8)follows.For q> 2,the desiredresultisobtained fromthe firstpart of Lemma 3.13.0c, If wecombineLemma4.32andTheorem4.33,thenweseethat,if qisa prime,R= Fq ,andeveryrli';isthe identity map,andif m2:fands2:1are fixed,then the construction of the point sets (4.25)yields on the average apoint set Pwith Div(P)=O(N-l(logN)s).We now show that the quantities p(G)in Definition4.27and R,.;( G)in(4.35)are connectedbythe followinginequalities. THEOREM4.34.For s2:2andany prime q,wehave q-p(C)-l:$Rq(C):$(1- t)k(q)8 ((m + lr _( p ( C ~+ s) )q-P(C\ wherek(q)=1if q =2and k(q)=csc(-n-jq)+ 1if q > 2. Proof.If C= (c)i))is given and anonzero H= (hij)EG(q)sxm is such that Bm (4.38)L :E hijc;i)= 0EF::, i=l j=l NETS AND(t,S)-SEQUENCES71 thenthesystem{C)i):1$j$di(H),1$i$s}islinearlydependentover Fq, wheredi(H)=d(hi1, ,him)for1$i$swiththenotationintroduced in (3.15).From the definitionof p(G),itfollowsthatE:=lp(G) + 1. Weobtain' thelowerboundinthetheoremfromthefactthatthereisanH .sAtisfying(4.38)and E:=l di(H)=p(G) + HI:." ,'il Toprovetheupperbound,weputD(H)' =E:=1 di,(H),andnotethat from(3.16),(3.17),and(3.18)weobtain Wq(H)$k(q)8q-D(H). Therefore (4.39)Rq(G)$k(q)8 L q-D(H)=: k(q)8 Sq(G), H wherethesumisextendedoverallnonzeroH=(hij )EG(q)8Xmsatisfy-ing(4.38).Wehave (4.40) Sq( G)=L A(d)q-d1--ds , d wherewesum overall d=(d1,,ds)EZ8with 0 $di $mfor1 $i$sand E:=l dip(G)+ 1,andwhereA(d)isthenumberof H=(hij )EG(q)8Xm 'satisfying (4.38)and di(H)= di for1 $i$s.If dis fixedand Hiscounted by A(d), then(4.38)attains the form i=l j=1 Choose integersIiwith 0 $Ii $difor1 $i$sand E:=l Ii= p(G).Then the last vector identity can be written in the form (4.41 ) Suppose that weallow arbitrary choices forthe coefficients hij on the right-hand sideof (4.41),the only stipulationbeing that hid.=1=0 wheneverdi> Ii.Since the vectorsC;i)on the left-hand side of (4.41)arelinearly independentoverFq , therecanbeatmostonechoiceforthecoefficientshi';ontheleft-handside. Therefore A(d)$ir (q- 1)qdi -f.-1 =8q - 1 . ir qdi-f.$(1- ! )qd1 +... +ds - P(C). i=li=lqi=lq Together with(4.40),this yields Sq(C)$(1- =(1- Dq-P(C)(m+ 1)' _+ 8)), and,by invoking(4.39),wecomplete the proof.0 72CHAPTER 4 It followsfromtheresultsalreadyshowninthissectionthatapointset constructedby(4.25)isalow-discrepancypointsetpreciselyif p( C)islarge. Theproblemofmaximizingthevalueofp( C)foragivenfinitefieldFqand forgivenintegersm2::1and82::1 isan interesting combinatorial questionfor vector spaces over finite fields,which is also connected with a classical problem in algebraic coding theory; see Niederreiter [244,7],[260].An explicit construction of systemsCwithalarge value of p( C)willbe givenin4.5. Wenowgivethedescriptionof ageneralprinciplefortheconstructionof (t, s)-sequencesinbaseb.Letthe integerss2::1andb ;:::2be given.Then we choosethe following: (81)Acommutative ring Rwith identity andcardeR)= b; (82)Bijections 'l/Jr: ZbRforr2::0, with 1p,JO)= 0{(iTtl..U 8ufficipn.tlvlarQ"Br: ;.J'Y) (83)Bijections 17ij: RZbfor1 isand j2::1; (84)ElementsERfor1 is,j2::1,and r 2::O. For n= 0, 1, ... , let 00 n= Lar(n)br r=O be the digitexpansion of nin base 0,wherear(n)EZbforr2::0and ar(n)= 0 forallsufficiently large r.We put 00 =forn2::0and1 is, ;=1 with (i) (i)) Ynj= 17ijejr 'l/Jr(ar(n))EZb forn2::0,1 is,andj2::1. Notethatthe sum overrisafinitesum,since'l/Jr(O)=0and ar(n)= 0forall sufficiently large r.Wenowdefinethe sequence (4.42) _((1)(8)) Xn- Xn, ... ,Xn forn= 0,1, .... Toguarantee that the points Xnbelong to [8(andnot just to ]8),andalsofor the analysisof the sequence(4.42),weneedthe follovlingcondition: (85)For each n0 and1 $is,wehave < b -1 forinfinitely many j. This conditionisalwaystacitly assumed when weconsiderthe sequence(4.42). A sufficient. condition for(85)is: (86) 1]ij(O)= 0 for1 i8and all sufficiently large j, and for each 1 $i$s and r2::0,wehave= 0 forall sufficiently large j. NETS AND(t,S)-SEQUENCES73 Condition(86)guaranteesthat,foreachn0and1::;i::;s,wehave = 0forallsufficientlylargej, and00eachisgivenbyafinitedigit expansion in base b. THEOREM4.35.Supposethattheinteger t0satisfiesthefollowingprop-erty:Foranyintegers m> tand dl, ... ,dB0 withE:=l di =m- tandany !fi) ER,1 ::;j::;d;,1 ::;i::;s,thesystemof m- tlinearequations m-l z,.= fP)for1 ::;jdi ,1::; i::;s ,.=0 l,ntheunknownsZo, ... ,Zm-loverRhasexactlybt solutions.Thenthese-quence(4.42)isa (t, s)-sequencein baseb. Proof.Forintegersk0andm>t,considertheXnwithkbffl ::;n< (1: + l)bm.In this range,the digits ar(n)of nare prescribed forrm, whereas the ar(n)with 0 ::;l' ::;m- 1 can vary freelyoverZb.Let an elementary interval inbaseb withAIJ{E)= bt-mjhenceL::=1 di = m- t. Using condition (S5)and proceeding asin the proof of Theorem 4.26,we obtain that XnEEif and only if 00 = l1ijl(aij)for1::; j::;di !1 $;i ::;8. ,.=0 Since the ?/I,.(a,.(nwith rmare given,this reducesto m-l I: =fj')for1 ::; j::;di ,1::; i::;8, ,.=0 withsuitablef?)ER.Byhypothesis,thissystemof linearequationsinthe unknowns?/I,.(a,.(n)) ,0::;r::;m- 1,overRhasexactlybt solutions.Each solutioncorrespondsto auniquem-tupleCao(n), ... ,am-l (n)),andeachsuch m-tuple corresponds to aunique integer nwith kbm ::;n< (k+ l)bm.Therefore thepointsXnwithkbm ::;n t,thesystem c(m)consistingof thevectors (i)_((i)(i)pm Cj- CjO "",Cj,m-lEq for1 i5,1 ::;jm, satisfies p(c(m)m-t, then thesequence(4.42)isa(t,s)-sequenceinbaseq. 74CHAPTER 4 Proof.WeverifytheconditioninTheorem4.35forthegivenvalueoft. Considerintegersm>tandd1, ... ,dB0with2::=1 di=m- t.From p(c(mm - tand Definition 4.27,it followsthat the coefficientmatrix of the systemof m- tlinearequationsinTheorem4.35hasrankm- t,andsothe system alwayshas exactly qtsolutions.0 There isalsoan analogue of Theorem 4.29for(t, s)-sequences.If the ringR ischosen as inthat theorem,then the elements in(84)can again be written in the form(4.27). THEOREM4.37.Let b = n:=l qvbea productof primepowersqv,letR= Fq",andlet t0bean-integer.Supposethat,foreachinteger m> tand eachvwith1 vh,thesystemconsistingof thevectors satisfiesm- t.Thenthesequence(4.42)isa(t, s)-sequencein baseb. Proof.Proceedin analogywith the proofs of Theorems 4.29and 4.36.0 REMARK4.38.ThederCorputsequenceinbaseb(seeDefinition3.2) arises as a special case from the general construction based on (81) - (85).Choose s=1andlettheringRbethe residueclassringZ/bZ.ThenRandZbcan beidentified,andsoallbijections'l/Jrand'fJijin(S2)and(S3),respectively, canbetakentobeidentitymaps.Putc);)=1ifr = j- 1,andcj;)=0 otherwise.Thencondition(S6)issatisfied,andthe sequence(4.42)isthevan derCorputsequenceinbaseb.It is atrivial verificationthatthe conditionin Theorem 4.35is satisfied with t = O.Thl.'IBthe vander Corput sequence in base b isa(0, I}-sequencein base b(compare aloowith thebeginning of 4.1), 4.4.Aspecial construction of nets. 'Wediscussaspecial familyof nets arisingfromthe generalconstructionbased on(Nl) - (N4)in 4.3.We choose the base to be aprime power q,and the ring Rto be the finitefieldFqwith qelements.LetFq((x-1)be the fieldof formal LaurentseriesoverFqinthe variable x-I,Thus the elementsof Fq((x-1))are formalLaurentseries 00 L= Ltkx-k, k=w wherewisan arbitraryinteger andall EFq .The discreteexponentialval-uationvonFq ( x-I))isdefinedbyv( L)=-w ifL# 0andwistheleast index with i,w# 0,andby v(O)=-06.vVehavev(J)= deg(f)forall nonzero polynomials fEFq[xJ.Wealso note that Fqx-1)) contains the fieldof rational functionsoverFqasasubfield. Foragivens2,choosefEFq [xlwithdeg(f)= m1 and let ',', F. NETS AND(t,S)-SEQUENCES75 9}' . .. ,98EFq [X].Consider the expansions 9i(X)= t E Fq(x-1for1 i8, f(x)k=Wi .where Wi1 for1 ::; i8.Define the elements in(N4) by (4.43)for1 is,1 jffi)0 rSm- 1. Choosethe bijections in(N2)and(N3)arbitrarily.Then the generalconstruc-tionprinciple' fornetsdescribedin4.3yieldsthepointset(4.25)consisting ofqmpointsin[8.Wedenotethispointsetbypeg, f))wherewewrite g=(91, ... ,9s)EFq [XJ8forthes-tupleofpolynomialsg}, ... ,98'Foran arbitrary h= (hI!"., hs)E Fq [x] s ,wedefine the"inner product" s h'g= Lhigi . i=l .!:.Susual,wewriteJig if fdivides9EFq[xJ.Inthefollowing,weusethe conventiondeg(O)= -1. ,DEFINITION4.39.For fand gas above,wedefine 8 p(g,f) =s -1 + minLdeg(hi ), i=1 wherethe minimum is extended over all nonzero h= (hI, ... ,hs) EFq [XJ8with deg(hi )< mfor1 isandfI h g. LEMMA4.40.Let (i)-' ((i)(i) cFm Cj- CjO "'"Cj,m-l .Cq fDr1 is,1 jSm, wherethe aregivenby(4.43).Then,for hij EFq,1is,1jm,we have (4.44) 81n "" "\:""'"h(i)- 0Fm ijCj- Eq i=1j=l if andonlyif fI h g,whereh=(hI!'",hs) EFq[x]Swith m (4.45)hi(x) =L hijxi-1 EFq[x]for1 SiS s. j=l ProofBy comparing components,weseethat(4.44)isequivalentto (4.46)for0rm- 1. i=l j=l 76CHAPTER 4 For 1 i:::;:s,wehave hi(x)gi(X)_h- __ i-1) (i)-k)_h- _ (i)-1I:+j-1 ()- LtJ;L- L.,Ukx- L.tL.,2JUkx fXj=lk=Wij=1k=Wi m00 "\:"""'h"\:"""'(i)- r-1 = L.,ijL.,Ur+jX. j=1r=wi-j Thus,forr0,thecoefficientof x-r-1 inhigdfisE.i=lTherefore, condition(4.46)isequivalentto the following:For 0rm-1 the coefficient of x-r-1 in2.::=1 higd fisO.This meansthat 1 ,h.g=p+L, wherepEandLEFq((x-1)with1/(L) t,the vectors (1)_((1)(1m Cj- CjO "'"Cj,m_lEFq, 1 $j 5:m-t, arBindependent overFirSuppose that) forsomeTn>t, wehad m-t 2: hje)1)= 0EF:;, j=1 where not all hj EFqarezero.Then m-t L= 0for05: r$m-l. j=l With h(x) =E;:-;.t hjxj-1 EFq[x]weobtain. hLl =(f ui1)x-k)=hjfui1)x-k+j-1 3=1)k=Wl3=1k=Wl m-t00 =L hjL j=1r=Wl-j and so the coefficient of x-r-1 in hLl is zero for 05: r5:m-l. Thus v(hL1-p) < -m forasuitable P EFq[x].Since deg(h)5:m- t - 1,it followsthat (4.65)deg(h) + v(hL1 - p)< -t - 1 = -K(Lt). Ontheotherhand,withnotationandresultsfromAppendixB,weseethat there existsad1 with deg(Qd-d5:deg(h)< deg( Qd)and that deg(h) + v(hLl- p)=2deg(h) + v(L1- *)2deg(Qd-l) + v(L1-= deg(Qd-l) - deg(Qd)=- deg(Aa)-K(Ll)' This isacontradiction to(4.65).0 90CHAPTER 4 4.5.Aspecial construction of (t,s)-sequences. WefirstconsideraprimepowerbaseqandwechoosetheringRtobethe finitefieldFqin the general construction based on (81) - (84)in 4.3.To obtain suitableelementsEFqin(84),weagainusethe method of formalLaurent series employed in 4.4.In the following,we describe the most interesting special case of aconstruction introduced in Niederreiter[247]. Foragivendimensions2::1,letPI,'",PsEFq[x]bedistinctmonicirre-duciblepolynomials overFq Put ei= deg(Pi)for1 ::;i::;s.For1 ::;i::;sand integers j2::1 and 0::;k< ei,consider the expansions (4.66) Ie00 X_I: (i) (.k)-r-l c:::F.((-1 -(-)-. - aJ,,r x'- qx. PXJ 1r=O ( i) the elements cjr (841by \/" (4.67)= a(i)(Q + 1, k, r)EFqfor1 ::;i::;8,j2::1,r2::0, wherej- 1=Qei+kwithintegersQ=Q( i, j)andk=k( i, j)satisfying 0::; k< ei.Weassumethat the bijections 'TJijin(83)are chosenin suchaway that 'TJij (0)=0for1::;i::;sand allsufficientlylargej.Thebijectionsin(82) are selectedarbitrarily,subject only to the condition imposed in(82).For each 1::;i::;8andr2::0,theelementsin(4.67)satisPJ =0forallsufficiently largej.Thuscondition(86)holds,andso(4.42)yieldsasequenceof points in IS. THEOREl'44.49.Theaboveconstruction,withthedefinedby(4.67), yieldsa(t, s)-sequence in baseqwith t= L::=l (ei- 1). Proof.By Theorem 4.36 and Definition 4.27,it suffices to verify the following property:Foranyintegerm >L::=l (ei- 1)andanyintegersdll ... ,ds2::0 with1::; L::=1 di::;m- L::=l(ei -1), the vectors (i)(i)(i))Fm Cj =CjO j,Cj,m-lEq are linearlyindependentoverFq 8uppose that wehave Sdi LLf?)C;i) = 0EF:; i=l j=l forsomefP)EFq ,wherewecanassumewithoutlossof generalitythatall di 2::1.By comparing components,weobtain 8d. (4.68)= 0for0::; r m-l. i=l j=l Consider the rational function NETS AND(t,8)-SEQUENCES91 whereweused(4.66)and(4.67)inthesecondidentity.Inviewof(4.68),we haveveL) Tq,,(s)andeachvwith 1 $v$h.Thus the condition in Theorem 4.37 holds with t= Tq,,(s); hence the result.0 REMARK4.52.If qisaprimepowerand8isanarbitrarydimension$q, thenwecanchooseforPll'",PsthelinearpolynomialsPi(X)=x- bifor 1 $i$s,wherebb'",bs aredistinct elementsof Pq.Thus wehaveTq(s)=0 fors:5q.Furthermore,(4.67)reduces to = a(i)(j,O,r)for1 $i$s,j;;::1,r;;:: 0, and these elements are obtained fromthe expansion = xi (1_ = x-; t. (r;l)biX-' =(r-1'-1 L,;'-1Ix. r=j-1J Thus,for1 $i$sand j;;::1,wehave C\i)=0 31' for$r< j- 1, \i)=(r)C]r.1' )-forr;;::j-1, where weuse the convention 0= 1 E Pq This choice of the yields the (0,8)-sequences in base q constructed in Niederreiter [244],and, if we specialize further to q being a prime, then this yields the sequences introduced by Faure [95].Note that,asfarastheconstructionof(0, s )-sequencesinbaseqisconcerned,the conditions:5qonthe dimensionsisbestpossible,sinceCorollary4.24shows thats$qisanecessary conditionforthe existence of a(0, s )-sequencein base q.Weobtain the choice of theleading to the van del'Corputsequenceina prime baseq (seeRemark 4.38)if weput s =1 and PI (x)=x. WenowconsiderinmoredetailthequantityTq(s)defined(4.69).Let Iq(n)be the numberof monic irreduciblepolynomials overFqof degreen,and letJq (n)be the number of monicirreducible polynomials overFqof degree $n, withJq(O)=O.Forgivens;;::1,letn=nq(s)bethelargestintegerwith Jq(n)$s.Then itfollowsfromthe definition of Tq(s)that nq(s) (4.70)Tq(s)=L (h - l)Iq(h) + nq(s)(s - Jq(nq(s))). h=l Thevaluesof Tq(s)forq=2,3,5and1$s$30aregiveninthefonowing tables.AgeneraluJ?perbound forTq{s)isshownin Theorem4.54. \; NETSAND(t,S)-SEQUENCES TABLE4.1 ValuesofT2(s)jar 1 s30. s1234561891011 T2(S)00135811141822,26 s1611181920212223242526 T2(S)48535863681318838995101 TABLE4.2 ValuesoITg(s)for1 s30, I 1234567891011 0001235791113 17181920212223242526 __' __

28313431404346495255 TABLE4.3' Valuesof Ts(s)lor 1 s30. S1234561891011 Ts(s)00000123456 s1617181920212223242526 Ts(s)1214161820222426283032 LEMMA4.53.Forany prime power q,wehave 1 Jq(n)2:_qnlor alln2:1. n ProofBy Appendix A,wehave Iq(n)= L JL(S)qdforalln2:1, dIn 93 ._ .. 12131415 30343843 21282930 107113119125 12131415 15111922 27282930 58616467 12131415 78910 27282930 34363840 whereJ-ListheMobiusfunction.Thisformulaimmediatelyyieldsthelower boundinthe lemma forn=1,2,3.Forn2:3 weuseinduction to obtain o 94CHAPTER 4 THEOREM4.54.Let qbeanyprimepower.Then,for1::;s~q,wehave Tq(s)= 0,and,for s> q,wehave Tq(s)< 8(logq 8+ logq logq 8 + 1), wherelogqdenotesthelogarithmtothebaseq. Proof.The Jirstpart of the theorem was already noted inRemark 4.52.For S> q,wehave (4.71) by (4.70).Put k=llogq S+ logq logq sJ+ 2. If either q=2,Y ~4,or q~3,y> 1,then Withy= logq s,we obtain q logq 8:2::logq S+ logq logq S+ 2 :2::k if either q=2,8:2::16,or q:2::3,S> q.In these cases,it followsthat and so Lemma 4.53yields 1 Jq(k)~kll > s. By the definition of nq(s), wethen see that nq (s)~k- 1~logq S+ logq logq S+ 1, and the bound forTq(s)followsfrom(4.71).In the remaining case where q= 2, 3~S::;15,the bound forTq(s)ischecked directly by using Table4.1.0 WenowdiscllilBthe important question of fi:ru:HngasequenceS of pointsin IS,whosestar discrepancy satisfies where the constant Cs isas small as possible.The case where 8 =1 has already beenconsidered in3.1,andsowetake8:2::2.Foranyprimepowerq,there exists a(Tq(s), s)-sequence in base qby Corollary 4.50,and,forsuch asequence S,wehave byTheorem4.17,wheretheimpliedconstantdependsonlyonqand8.If we optimize inthis family of sequences,then wearrive at the value (4.72) NETS AND (t,S)-SEQUENCES95 wheretheminimumisextended overallprime powersq.Thus,forany8?::2, there exists asequenceSwith (4.73) whereGs isgivenby(4.72)and where the impliedconstantdependsonly ons. For 8=2,wehave 1 G2 = G(2,2)= 8(log 2)2 . For s> 3,letql(S)be the least even prime power?::s,let Q2(S)be the least odd prime power~s,and put G ~=mID Foranintegerk;::::0,thekthhyperderivaiiveistheFq-linear operator H(le)on the polynomial ring Fq[x]defined by H(k) (xT)= forr2:k and H(k)(x'f')=for0::; r< k.Weagain use the general principle for the construction of (t, s)-sequences based on(81) - (84)in4.3,with R = Fq THEOREM4.56.Fors::;q,let bI, . ..,bs besdistinctelementsof Fq For 1 is,let 9iE Fq[x]with 9i(bi )=1=O.Define =[H(j-1}(Xr 9i(X)](bi)for1 ::;is,j;::::1,r2:0, 98CHAPTER 4 andsupposethat thebijections 'fJijaresuchthat 'fJij (0)= 0for1 ::;i::;sandall sufficiently largej.Thenthesequence(4.42)isa (0, s)-sequenceinbaseq. Proof.Itisclearthatcondition(S6)in4.3issatisfied.Toprovethatthe given sequence is a(0, s )-sequence in base q,it suffices to show by Theorem 4.36 that, foranyintegersm~1 and d1, ... ,dB~0withL::=l di,=m,the vectors (i)(i)F m CjO "..,Cj,m-lEq are linearly independent over FqSuppose that the columns of the m x mmatrix formed by these row vectors satisfy a linear dependence relation.Then there exist 1o, ... , fm-lEFqsuchthat m-l Lfr[H(j-l)(x1"gi(X))]Cbi)=0for1 :-:;j::; di1 1 ~i::;s. 1"=0 Withf(x)= L:::o1 l1"x1" EFq[x],this yields [H(k) (fYi)](bi)= 0for0::; k::;di - 1,1::; i~s. For1::;i::;5,thisimpliesby[192,Lemma6.51]thatbi isarootoffgiof multiplicityatleastdiSincegi(bi)i- 0,itfollowsthatbi.isarootofIof multiplicityat leastdi.However,deg(J)< m= 1::=1 di"and so wemusthave f= 0,I.e.,f1"= 0for0 ::;r::;m- 1.0 Notes. Thebasicpaperforthetheoryof (t, m, B)-netsand(t, B)-sequencesisNieder-reiter[244].Theresultsof4.1and4.2andmostoftheresultsin4.3 stemfromthispaper.Forbase2,earlierresultsareduetoSobol'[323J. SurveysoftheworkofSobol'canbefoundinNiederreiter[225,3]and Sobol'[324].Aconstruction of(0,2)-sequencesinbase2isalsogiven in Srini-vasan[333].Forageneralbackgroundonlatinsquares,werefertothebook of Denesand Keedwell[59].Conversesof Theorems 4.28and 4.36are shown in [244,6].Note that om definition of p(C) is slightly different fromthat in [244]. Theorem4.30,Lemma4.32,andTheorem4.33areduetoNiederreiter[261 J. Theorem 4.34 is a special case of aresult in Niederreiter [241],and Theorem 4.37 wasshowninNiederreiter[247]. The construction of nets in 4.4 was introduced in Niederreiter [261],whereas theideaof usingformalLaurentseriesfortheconstructionof netsand(t, s)-sequenceswaspreviouslyestablishedinNiederreiter[247J.Theorem4.43isa newresult.Theorem4.46isshowninthe samewayasaspecialcasethat was consideredinNiederreiter[241].The quantityK(g/f)wasstudied in detailin Niederreiter[243].Calculations of the figureof meritpeg, f) havebeen carried out in certain special cases, e.g.,for gof the formg= (1, xm, x2m, ... jx(s-l)m); seeAndre,Mullen,and Niederreiter[8]and Mullenand Niederreiter[215].This figureofmeritalsooccursinthecontextofpseudorandomnumbergenera-tion(comparewith9.1).Theanalysisofthesequencesconstructedin4.4 NETSAND(t,S)-SEQUENCES99 leads to interesting connections with problems of diophantineapproximationin Fqx-1 )), which are further explored by Larcher and Niederreiter [180].For the construction in 4.5,weneedtables of irreducible polynomials overfinitefields; wereferto[192,Chap.10]forsuchtables.An interesting applicationof (t,8)-sequencestothenumericalsolutionofintegro-differentialequationsoccursin Lecot[184].Fox[107]describedacomputer implementation of Faure sequences and compared their efficiency with that of Halton sequences and pseudorandom numbers;seealsoSarkarandPrasad[301]foranothercomparativestudy.A computer implementationof Sobol'sequenceswascarriedoutbyAntonovand Saleev[9],and an improved scheme wasdevelopedby Bratley and Fox[33J. 5 [Y :()j'I,_";\' ,'.. ':,;\V, .. Rules,forNumerical Integration quasi-Monte Carlo methods for numerical integration that we have discussed .'ibasedonlow-discrepancypointsetsandsequences.Aninspectionof the ml'll'lI"uU:Perrorboundsin 2.2revealsafeaturethatmaybeconstruedas 'Ati7nA4"1i 1and C> 0,forany gEZSand anyinteger N1,wehave maxNtL fNn g- ff(u)du= CPa(g,N). N-1() n=O.ljB ProofFor fE!(C), the bound followsimmediatelyfrom(5.4)andDefinitions5.1and5.2.Nowletfobe the special function lo(u) = CL r(h)-Oe(h u)foruE R8. hez Then /0E and fo (;g) -1. fo(u} du = CPa(g, N) by(5.4)and Definition 5.2.0 Theorem5.3Showsthat,forgivenaandN,thelatticepointgshouldbe chosen in such away that Pa(g, N) is small.We now introduce a related quantity that doesnot depend on the regularity parameter a.The idea forits definition stemsfromtheobservationthatthemaincontributionstothesumdefining Pa(g, N)come fromthe lattice points hclose to O.In the followingdefinition, weusethe set C: (N)introduced in3.2. DEFINITION5.4.For gEZS,82,and an integer N2,weput R(g, N) =I: r(h)-l, h wherethe sum isoveraUhEC:(N) with h g==0 mod N. Next,we establish abound forPo(g, N) in terms of R(g, N) in an important specialcase.Amoregeneral,butsomewhatweaker,bound",linbeprovedin Theorem 5.26.Let (01.)=E:=l m-O forCIt> 1 be the lliemann zeta-function. THEOREM5.5.Let N2bean integer and let g=(911".jUs)EZS,s2) withgcd(gi,N) =1 lor18.Then,for anyreal0> 1,'we have Pafg,N)< + (1 + 2(a)N-a)S-1 -It+ 2(01.)+ 2((a)N1-o)8- + 2(0)S, LATTICE RULES FOR NUMERICAL INTEGRATION105 Proof.EveryhEcan be uniquely written in the formh=k + N mwith kE ell(N), mE 1,;8.Thus wecan write (5.5) Pa(g,N) =Lr(Nm)-a + Now mEZ" m#O,I' kEC:(N)mEZ", k:g=:OmodN (5.6)81=Lr(Nm)-a -1 = (L:: r(N'Tn)-a)8 -1 mEZ8mEZ = (1+ 2 f, (Nm)-a)' ~1 = (1 + 2a)N-O)' ~1. Furthermore, (5.7) wherek= (k1 ; ... jks ).Fork =0,wehave + Nm)-a=L:: r(Nm)-a =1 + 2({a)N-0I. mEZmEZ For 0< Ikl:::;N /2,wehave mEZmEZ 0000 =Ikl-a+ L{k+Nm)-a+ I)-k+Nm)-a m=lm=l m=lm=l Thus,in both cases, Lr{k + Nm)-Ot:::;r(k)-a + 2a((a)N-a, mEZ 106CHAPTER 5 and so from(5.7)weobtain s II(r(ki)-O + 2(a)N-a) kEC:(N)i=1 kg=OmodN < Ra(g, N) + 20S(a)B N-OS#{k E Cs(N) : k. g= 0mod N} s-1 + L 201(8-j)(0:)8-j N-o(s-j) j=ll$;il 1.The quantity R(g, N) '" be used to provide an upper bound forthediscrepancyof thepointset(5.1.For82andN2,weusethe notation in 3.2to define (5.9)R1(g,N) = 2: r (h,N)-I, It! where wesum overall hEC:(N) with II g.....O'modN. THEOREM5.6.For gEZS,82,af'd2,letPbethepoint set (5.1).Then".,",'!("nil)''.)')' DN(P) S1- (1-d+ R1(g,N)"5;+ Proof.ByTheorem 3.10withM=Nand Yn= ng for0SnSN- 1,we obtain ( 1 "s11N-l() DNfp) < 1- 1- -)+'""- '"" e\- ...NL- r(hN)NL- N bEe; (N)'n=O =1- (1- ! r+R,(g,N). Thesecondinequalityofthetheoremisobtainedfromr(h, N)2r(h)for hE C;(N), which follows,in turn,from sin(-Jrt)2t for0 ::;t0 Onthebasisof the discrepancyboundinTheorem5.6,thepointset(5,1) canalsobeusedforthenumericalintegrationofnonperiodicfunctionsof low regularity,e.g.,of boundedvariationonjsinthe senseof HardyandKrause. However,thefullpowerof thepointset(5.1)isachievedonlyforperiodicin-tegrandsbelongingto afunctionclass There aremethods forperiodization, i.e.,fortransformingasufficientlyregularnonperiodicintegrandintoaninte-grand belongingto asuitablewithoutchanging the value of theintegral.A simple periodization technique isthe replacementof agivenfiJ.nctionfonjsby the function 11 !(UI, ... ,us) =2-s L: ... L f(cl+ (-1)e1Ub .,Cs+ (-lyBus) el=Oes=O for(Ull ... ,us)EIs.Thistechniquemaybeviewedasananalogueofthe methodof antitheticvariatesdescribedin1.2,butitisoflimitedusefulness 108CHAPTER 5 becauseof apossiblelackof regularityoftheperiodicextensionof !to RSat the boundary of Is.More satisfactory techniques are based on suitable changes of variablesoronthedeviceof addingfunctionstofthatareobtainedfrom Bernoulli polynomials and certain partial.derivatives of f.Wereferto Hna and Wang[145,Chap.6],Korobov[160,Chap.1],andZaremba[365]fordetailed discussions of periodization methods. Anotherquantitymeasuringthequalityof latticepointsgisthepositive integer introduced in Definition 5.7 below.The idea is that Pa (g, N) and R(g, N) will be small if the nonzero lattice p8filts' b.withh. g=0 mod Nare rather far ,l fromO. DEFINITION5.7.ForgE7/,s2,andanintegerN2)thefigureof merit peg, N) isdefined by (,..,i\T\ Pvt.,., 'i; ,.".., - - ."\ where the minimum isextended overall nonzero hEzawith h g= 0 modN. LEMMA5.8.Wealwayshave1 :s;peg, N):s;N/2. " Proof.If g= (9b .. ',98)withgCd(gl,N)= 1,thenhgl+ g2- 0modN forsomehEZwith-N/2 1,then there existsaproper divisordof Nwithdgl=0 modN, and 80 peg, N)::::;red)::::;N/2.0 REMARK5.9.It followsfromLemma 5.8that it sufficesto extend the min-imum inDefinition5.7over all hEC:(N) withh g= 0mddN.If anonzero bE'Ls issuchtha.th. g= 0 mod Nandpeg, N)=reb),thenLemma5.8 impliesthat notallcoordinates of haredivisible by N.Thus,if wereduceall coordinates of hmodulo Nto obtain apofut hoE C: (N), then ho . g= 0 mod N and r(ho)::::;r(h)ihence peg, N) = r(ho) .The qua.ntities:Pa(g, l'{)irdm below a.nda.bove in terms of the figureof merit peg, Nt ill detiill,forany gEZS,82,and any integer N'? 2;wehave'.I.,'"; (5.10) 2< P.(N) < c(s,a) (1 + logp(g, N))8-lforalla> 1, peg, N)a- ag,.peg, N)a where the constant c(s, a) depends only on sand a.The lowerbound in (5.10) istrivial,and the upper bound isaspecial.caseof Theorem5.34.Similarly,we have (5.11) 1"< R(N" (N) - g,} - (' N), pg,..,pg," wherethe constantc(s)depends only on s.T4elowerboundin(5.11)follows fromRemark5.9, the upper bound isaspecialcaseof Theorem5.35.For LATTICE RULESFOR NUMERICAL INTEGRATION 109 the discrepancy of the point setPin (5.1),wehave the bounds (5.12) Cl(S)< D(P)< c2(s)(logN)8 peg, N)-..N- peg, N), wheretheconstantsCl(S)> 0andC2(S)dependonlyons.Thelowerbound in(5.12)isaspecialcaseof Theorem5.37andtheupperboundfollowsfrom Theorem 5.6,Lemma 5.8,'and(5.11).Weinferfrom(5.10),(5.11),and(5.12) that,foragivenN~2,thelatticepointsgE1.s,s~2,whicharesuitable fornumerical integration-in the sense of yielding asmall integration error-are those for which peg, N) is large.Such agis informally called agoodlattice point mod N, hence the name"method of good lattice points." . Goodlatticepointsinlargedimensionsautomaticallyyieldgoodlattice points in smaller dimensions.Concretely,foragiven dimensions~3, let g(s)= (91,'",9s) E1.s, and,forany dimension twith 2 :::;t< 5,put (t)_() g- 91,,9tE Then, fromDefinition 5.7,weimmediately obtain (5.13) p(g(t) , N) ~p(g(s) , N)forany integerN~2. Thus,if g{ s)isana-dimensionalgoodlatticepointmod N,theng(t)isat-dimensionalgoodlatticepointmodN.However,thereisnoobviouswayof .. constructing higher-dimensional good lattice points from lower-dimensional good lattice points.In analogy with(5.13),wehave Pa(g(t) , N):::;Pa(g(s) , N)foranyN~2anda> 1 and R(g(t) , N) :::;R(g(s) , N)foranyN~2. 5.2.Existence theorems forgood lattice points. Toguaranteethatthemethodofgoodlatticepoints ispracticable,weneed resultsthatdemonstratethat,forgiven8~2andN~2,thereexistpoints gEzs such that peg, N) is large and Pa(g, N) and R(g, N) are small.We start with the last quantity and show that the averagevalue of R(g, N), taken over a wen-chosen set of lattice points g,issmall.Wefirstnote that gisonly relevant modulo N,so that it sufficesto take gECs(N).Weput Recall that the Euler-Mascheroni constant isgivenby 'Y=lim( ~~-lOgn)= 0.577 .... n---+ 00L....tm m=l 110CHAPTER 5 THEOREM5.10.Foranyintegerss~2and N~2,put Thenwehave 1 M8(N)=card(G8(N))LR(g, N). gEGs(N) M(N)= ~ ( 2 1N)8_2s1ogNo((loglogN)2) 8Nog+CN+N' where c = 2-y -log 4 + 1 =0.768 . ..and wheretheimplied constant dependsonly on s. Proof.Note that card(Gs(N))= (NY\ where isEuler's totient function. Bythe definitionof R(g, N),wethen aerhre 1 r(h)-l =;(N)SLA(h)r(h)-l, hEC;(N) whereA(h)isthenumber of gEGB(N)withh g= 0modN.SinceA(O)= ;(N)Band 1'(0)=1,weobtain (5.14) 1 M8(N)=;(N)8A(h)r(h)-l - 1. hEC,,(N) For anyh, wehave N-l(k) A(h) =~eNh. g. gEG.(N)11:=0 In this proof,wewrite (m, n)for gcd(m, n).Then 1N-I = NI: T(k, N)8 k=O with T(k,N)=e ( ~ h g)r(h)-l. \hEC(N) gEC(N) (g,N)=l LATTICERULESFOR NUMERICAL INTEGRATION111 If weput L(N) =Er(h)-l =LIhl-I, hEq*(N)hEC-(N) . :'," .\.I .then T(O, N) = (N)(l + L(N}), and so ..N-l (5.15)LA(b)r(b)-l= + L(N8 + L T(k, N)8. hECs(N)k=l Forfixed1 kN- 1,wehave T(k,N) =:Er(h)-l:E LJL(d) , hEC(N)gEC(N)dl(g,N) '.,":'_A(;0(,;'.'','., where JLis the Mobius function and dis restricted to positive divisors;if wealso fixh EC(N), then Therefore T(k, N) = LJL( Lr(h)-l, diNhEC(N) dlkh Now d divideskhif and only if d/{d, k)dividesh;hence (5.16) where,forapositive divisorb of N,weput If b < N,then L(b, N) =Lr(h)-l. hEC(N) blh L(b,N) = 1 +LIhl-1 = 1 +I:labl-1 = 1 +hEC"(N)aEC"(N/b) blh 112CHAPTER 5 If weput L(1)= 0,then this formula forL(b, N)also holds forb =N.Together with(5.16),this yields (5.17)T(k,N) = LJL( + L( diN = (N) + LJL( diN By[226,Lemmas1,2Jwehave,forany integer m1, (5.18)L(m) =210gm+c-1 +c(m)withIc(m)1< 4m-2 Consequently, = (21ogN +c--l)B(k,N) - 2H(k,N) + V(k,N) diN with B(k,N) =diN H(k,N) =diN. V(k,N) =diN Foraprimepowerpmwith1,wehaveB(k,pm)= (pm,k)-(pm-l,k)j . hence B(k,pm) =pm - pm-l =(Pffl)if pmI k and B(k,pm) =0 otherwise.For fixedk,wenote that(d,k)isamultiplicative function of d,and 80B(k, N)isa multiplicative function of N.Thus, for any positive integers kand N, we obtain (5.19) B(k, N) ={cf>(N)if NI ootherwlse. In particular,for1 kN- 1,wehaveB(k, N) =0,and so (5.20)T(k,N) =(N) - 2H(k,N)+ V(k,N)for1 kN-1. In the rest of the proof,pwillalways denote aprime number.For apositive integern,letep(n)be the largestnonnegativeintegersuchthat pep(n)divides n.WenowconsiderH(k, N)forafixedk.Since(d, k)isamultiplicativeand log(d/{d, k)an additive function of d,it followsbyinduction on the number of distinct J:l!imefactors. of Nthat (5.21) H(k, N) =L H(k,pep(N))B(k, N/pep(N)). piN LATTICERULESFOR NUMERICALINTEGRATION113 For an integer m1,wehave ep(k),'then.(pm,k)...::.pmand(pm-l,k)=pm-l jhenceH(k,pm)=o. H m> ep(k),then(pm,k)=(pm-1,k)=pep(k);henceH(k,pm)=pep(k)logp. Together with(5.19)and(5.21),this yields (5.22) H(k, N)=Lpep(k)>{N/pep(N) logpl p where the sum runs over all p satisfying the fonowing two conditions:(i) ep(N)> ep(k)j(li)N/pep(N)dividesk.Note that (li)means that epl (N)ep1 (k)forall primes PI=f:.p.Therefore(i)and(li)holdsimultaneouslyifandonlyif there exists aunique pwithep(JV)> ep(k).Hence it followsfrom(5.22)that if there exists aunique prime pwith ep(N)> ep(k),then H(k, N) =pep(k)4>(N/pep(Nlogp, and H(k, JV)= 0 otherwise. Totreat V(k,N), we use the bound forc{m)in(5.18)to obtain IV(k, N)I4 I:IJL(I2= 4 I:= 0(1) diNdiN withanabsoluteimpliedconstant.Combiningthiswith(5.14),(5.15), and(5.20),weobtain 11N-l(( 1))8 MsCN)=N(1 + L(NS + Nk=l1 - 2J(k, N)+ 04>(N)- 1, whereJ(k, N) =H(k, N)/>(N).The formulaforH(k, N),givenabove,shows thatJ(k, N)= 0(1);hence (5.23) M.(N) =L(N))" +(:) (-2)' '%;' J(k,N)i + 0 with an implied constant depending only on s. Wenowconsiderthe sum overkin(5.23)for1is.Fromtheformula forH(k, N),weobtain 114CHAPTER 5 if there exists aunique prime pwith ep(k)< ep(N),andJ(k, N)=0 otherwise. We put N-l Qi(N) = LJ(k,N)ifor1 ~i::;s, and we claim that Qi is an additive function.Let NIand N2be positive integers with(NI,N2 )= 1.H1~k~NIN2- 1issuchthatthereexistsauniquep withep(k)0,thento2::2andBeforallsufficientlylargeN; henceto(logtO)s-1sBforsuchN.If c(s)ischosensuitably,thenitfollows from(5.30)and Lemma 5.11that K(s,N,to) cl(s)(N)S foracO'nf(tantCl(S)>0"TIdalllargeInparticular,wehave K(s, iV, to)>0forallsufficientlylargeN,andso(5.29)isapplicablewith t =to.:ForagE Gs(N) satisfying(5.29),wethen obtain R,,(g,N) =o( N-"(logN)a(B-')+1 (0-')(0-1). Since by(5.8)wehave the result of the theorem follows.0 If wecarefully keeptrack of the constantsin the aboveproof,asisdone in the original paper [2641,then we obtain the bound forPOt(g, N) in Theorem 5.13 with the coefficientof the leading term being 2Q (s-1)+10:a (.)Q-l (s-1)!(a-1), which decreases superexponentially as s increases and 0: is fixed.Using (N)-l = (N-1 log log ( N + 1) ), we may also write the bound in Theorem 5.13 in the form PQ(g,N) = O(N-Q(logN)Q(s-I)+1 (loglog(N + 1{0I-l)(S-1). THEOREM5.14.For any integers s2and N2and any real 0:> 1,there existsagE Es(N)with PQ(g, N) =(N-Q(log N)0I(s-1) (1 +'T(N)))if 83, (logN)s-l N) =0( N-Ot (log N)Q+ ))if s = 2, wheretheimplied constantsdependonly on 8and Ctand 'where'T(N)denotesthe number of positiveH.ivisorsof N. LATTICE RULES FOR NUMERlCAL INTEGRATION119 ProofForfixeds,N,and aand an integer t2::2,cOIlBider Het)=H(s,N,a,t):=LPa(g,N). geE.(N) p(g,Nt ForhE2;6,letD(h,N) be thenumberof g,EE.(N)withb g=0mod A!. Then,usingthe definitionof Pa(g, N)andthe factthat, if peg, N)> t,then a nonzerohEZ can satisfy h . g==0 mod Nonly if reb) > t,weseethat (5.31)H(t}~Lr(h)-a D(h, N}. heZ" r(ht TodetermineDCh, N)foranyh= (hb'",h.)E,wenotethat,ifg= (1,g21'",g.) EE6(N)isasolution of h. g=hI+ h2g2+ ... + h.ga=0mod lV, then d =gcd( N, ha, . .. ,ha)must dividehI'In this case,wedividethe congru-enceby dand obtain (5.32) withm=N/d andgcd(m,ka,.oo,ka)=1.If m>1,thenlet pBbeaprime powerinthecanonicalfactorizationofm.Sinceatleastoneki!2~i~s, isnotdivisiblebyp,thenumberof solutionsmodpBof(5.32)ispa(a-2).By combiningth'8solutioIlBbytheChineseremaindertheorem,thenumberof s0-lutionsmodrnof(5.32)isma-a,andthisholdsform=1aswelLThus D(h, N)= (N/m)8-1mB-2= Na-2d if d divides hI, and D(h, N)= 0 otherwise. Using this formula(5.31),weobtain N(t)::;Ns-2 2: gcd(N, ha, .. Iha)r(h)-Ot, h wherethesumisoverallh=(hI. ... ,hs)Ezsforwhichr(h)> tandd= gcd(N, h2l ..,hs) divideshI'By splitting up the sum according to the value of d,weobtain (5.33) with H(t)~Ns-2 L dTd(t) diN Td(t)=Lr(dh)-a. neZ r(dht Wesplit up the lastsum according to the number iof nonzerocoordinates of h and the position of these nonzero coordinates.This yields (5.34) 120 where Vi(u)= (hI,'",hi)EZ' Ih1 .. h.l>u CHAPTER 5 Forintegers1i8andk1,let be the number of (hI, ... ,hi) EZi with 0 < Ihl ... hil:::s;k,andput Ai(O) = O.Then k>uk>u < 0L k-Ot-1 Ai(k). k>u Intheremainderoftheproof,theimpliedconstantsintheLandausymbols a.lwaysde'Dendonly on 8and o.By induction on i,weseethat and so Vi(u)= 0(2: k-a(log(k + li-l)= O(ul-(log(u + li-l). k>u Thus,from(5.34), and so,by(5.33), (5.35) H(t) =0(N8-2tl-otOOgt)i-1 Ld1-i). i=ldiN Fori=1,wehaveEdlN d1-i =T(N),and,fori=2,wehaveEdlN d1-i = Ed Nd-I < N 14>( N),wheretheinequalityisobtainedbycomparingthetwo functionsatprimepowersandusingmllitiplicativity.Fori2::3,we haveLdlN d1-i =0(1).For 82::3,wethus obtain from(5.35), Let (8, N, t) be the numbers of gEEs.(N) with: p(g, N) > t.Then L(8, N, t) = Ns-l - Fs(t, N)ihence if weput l c(8)NJ to=(logC(S)N)B-l LATTICE RULESFOR NUMERlCALINTEGRATION121 with asuitableconstantc(s)> 0,then to;:::2andL(8, N, to);:::cl(s)NB-lfor allsufficientlylargeNand some constantCl (8)> 0,by Lemma 5.11.Itfollows then fromthe definitionof H(t)and from(5.36)that,fors;:::3,there existsa gE ElI(N)with peg, N) > toand For8=2,wesee,from(5.35),that H(t)=logt + r(N))). By choosingtoasaboveandusingasimilarargument,weobtainthe resultof the theorem for8=2.0 Themoredetailed performedin(2641yieldstheboundsTheo.-rem 6.14 wlth explicit coefficients of the main terms.These coefficients decrease superexponentiallyassincreasesandaisfixed:Since'T( N)hastheaverage order of magnitude log Nby [188,Thm. 6.30],the result of Theorem 5.14 is usu-ally better than that of TheOrem 5.13.However,there exist sequences of values of Nthrough which 'T(N)growsfaster than any givenpower of logN(see[188, p.164]),and, for such values of N, Theorem5.13yieldsthe better result. The lattice pointsg satisfying the bounds in Theorems 5.13 and 5.14 depend, inparticular,on Ot,so that theSetheorems can guarantee only thatPc:lg, N) is small forthe chosen value of a.However,the proofs of these theorems yieldthe additional property that the figureof meritpeg, N) islarge,and on the basis of this information it can be shown that P,a(g, N)is smallforallf3> 1 (see[264]). Fors=2,thereisaninterestingconnectionbetweengoodlatticepoints and continuedfractionsforrational numbers,which can be used forthe explicit constructionof goodlatticepoints.Foran integerN;:::2,letg=(1, g)EZ2 withgcd(g, N)=1.Letthe rationalnumbergiN havethecontinuedfraction expansion (5.37) wheretheajareintegerswithaj1for1::;j::;Iandwhereaz=1 forthe sakeof uniqueness(comparewithAppendix B).Recallthattheconvergentsto giN are definedby p. _L=[oo;al,a2,'",aj]forO::;j::; l. qj TheintegersPjandqjareuniquelydeterminedif weimposetheconditions qj;:::1 and gcd(pj, qj)=1.Then wehavethe followingexplicitformulaforthe figureof meritdue toBorosh and Niederreiter[30]. 122CHAPTER5 THEOREM5.15.If N~2 is an integer and g= (1, g)E Z2with gcd(g, N) = 1,then p(g, N)=minqjlqjg - pjNI. O ~ j < l Proof.BecauseofLemma5.8,itsufficestoextendtheminimuminDefi-nition5.7overall1ionzeroh=(hI, h2)EZ2withIhll 1,wehave + ~G) 2"('-;) a}'-; (1+ 2(aW N(l-a)(,-j) . In particular,if either s=20.1'Ot~2,then with animpliedconstantdependingonlyon sand a. Proof.EveryhEZBcan be uniquely representedinthe formh= k+ Nm with kECB(N),mEZ".Since L1..;?(NZ)8(see Remark 5.25), we have hEL1.. if and only if kEL1...Separating the cases where k=()and kf=0, weobtain (5.41)Pa{L) =Lr(Nm)-a +LLr(k + Nm)-a =: 81 + 82. mEZ"kEE(L) mEZ2 .mO Asin the proof of Theorem 5.5,wesee that 82 < Ra(L) + 2as(a)" N-as card(C,,(N) n L1..) ,,-1 + L2a("-;)((a)"-;N-a(s-j)LLr(kiJ-a .. r(ki;)-a, ;=11:5h 2s(a)nIatforall a> 1; (ii)R(L)c(s)nlllog(N/nl)'wherec(s)> 0dependsonlyon s; (iii)Div(P)1- (1- nil)" nIl,wherePisthenodeset of L. Proof.(i)SinceLl..;?(nIZ)/lby the proof of Lemma5.32,weseethat,for a> 1, Pat(L)Lr(h)-at- 1 =(.E r(n1h)-at) /l- 1 hE(nlZ)8hEZ = (1 + 2t,(n1h)-')' -1 = (1 + 2( 2s(a)n,'. (ii)FromLl..2('11IZ)B,weobtainE(L);?C;(N) n ('11lZ)Binthe notation of Definition5.24.The elementsof thisintersectionareexactlyallpoints'T11Jl with hEC;(N/'11l).Therefore R(L).Er(n1h)-1 - 1 =(.Lr(nIh)-I) 8-1 hECII(N/nt}hEC(N/nd =(1 + '1111LIhl-I)8- 1c(s)'11111og(N/'11I)' \hEC*(N/nt} (iii)BytheproofofLemma5.32,thecoordinatesofallpointsofPare rationalswithdenominator'111.Thus thedesiredresultfollowsfromTheorem 3.14.0 Existencetheoremsforefficientlatticerulesofrank1havealreadybeen shown in 5.2.These theorems can be used to obtain results forhigher ranks by amethod of extending lattice rules. LEMMA5.39.Fors2,letarankrwith1rsandinvariants nl, ... ,n".2with niHdividing nifor1 :::;i:::;r- 1begiven,and let Ll bean a-dimensional nl-point latticeroleof rank1 generated bygl = (g11),... ,gis)E zswithgcd(g?), nl) = 1.Thenthereexistsan s-dimensional latticeruleLwith rank randinvariantsnl, ...jnrsuchthatthenodesetof Lcontainsthenode setof L1. Proof.Wemay assumethatr2.WeconstructLby putting gi= ... EZSfor2ir with = 0for1ji-I andni)=1andlettingthenodesof L be allfractionalparts { ".k} .E i=l with0:::;ki < 'T1ifor1 ir. 140CHAPTER 5 Consider the mapping 'I/Jon the finite abelian group A= (Z/n1Z)EB'"EB(Z/nrZ) definedby Then 'I/Jis asurjective grouphomomorphism.Weclaim that thekernelof 'I/Jis trivial,i.e.,that (5.49) impliesthatki - 0 mod nifor1$i$r.Bycomparingthefirstcoordi-natesin(5.49),wederive=0mod 1;k1gi1)==0mod nl, andthuskl= 0mod nlsincegcd(gil) jnl)=1.Thus(5.49)reducesto = 0modZs,and,comparingsecondcoordinates,weobtain k2= 0mod n2'By continuing in thismanner,weestablishtheclaim.Conse-quently,L/,L,8is isomorphic to A, and so L has rank rand invariants nl, ... ,nr Clearly,the node set of Lcontains the node set of L1.0 LEMMA5.40.Fors;:::2,letL1L2betwos-dimensionalintegrationlat-tices,and let N1and N2bethe number of nodes of thecorrespondinglatticeroles Ll and L2,respectively,with NI;:::2.Then if L1is alatticeroleof rank1generatedby apointgEGs(N1). Proof.(i)From L1L2,weobtain Lf Lt, and so the desiredinequality followsimmediately from the definition of the figureof merit. (ii) UseLt ;2Lt and the definitions of Pa(LI)and Pa(L2). (ill)N-otethatNIdivides'N2l'sinceLd'ZlisasubgroupoforderNIof thegroupL2/ZB of orderN2EveryhECs(N.},)canbewrittenintheform .h=k+ N1mwithkEeB(Nl ),mEMil:=[-N2 /(2N1), N2/(2Nt}]11n ZII. LATTICE RULESFOR NUMERlCALINTEGRATION SinceN1mELt by Remark 5.25,wehavehELt if and onlyif kELt.Also using Lt ;?wesee that (5.50)R(L2)$ r(h)-1 - 1 hEC8 (N:a) nLt $L:r(Nlm)-l- 1 +L:Lr(k + N1m)-1 mEM.keE(Ll)meMs with the notation inDefinition 5.24.Now (5.51) Furthermore) T:=. LLr(k+N1m)-1 =LiI(Lr(ki+ N1m)-1), kEE(Lt}mEM.kEE(Lt} ;=1wherek=(kI, ... ,ka).Byproceedingasintheargumentfollowing(5.7),we obtain ()1)1,2eN'}. L..Jrk+N1m- $r(k- +N1ogN .11 forkEC(Nd, and so For fixed1 :::;j$s- 1and1 $il < ... < ij $sand anykEE(Ll),wehave and so 142CHAPTER 5 Since card(E(Ld) =N:-l - 1 by Remark 5.25,it followsfrom(5.52)that 8() (N)8-j28(N)8 T< R(Ld ~ ;1 +log N:+Nl1 +log N:' and,together with(5.50)and(5.51),this yields(iii). (iv)If Ll is as in (iv),then, forfixed1 ::;j::;s -1 and 1 ::;i1 < ... < ij ::;s, weobtain,asinthe proof of Theorem 5.5, Lr(kiJ-1 ... r(kij)-1 ::;N:-j-lLr(h)-l < N:-j-l(2 log Nl +l)j. kEE(L1 )hECj(N1 ) Using this in(5.52),wederive 28 (N)818-1()(N)s-j T< R(Ll )+N1 +log N2 +NL~2 log N2 +2(2 log Nl +1)j 111j=1J1 1 = R(L1 )+Nl ((2 log N2+3)8 - (210gNl +1)8), and,together with(5.50)and (5.51),this yields(iv).0 THEOREM5.41.Fors;::::2,letarankrwith1::;r 1,thereexistsan s-dimensionallattice rule Lwith thisrankand these invariantssuchthat wheretheimpliedconstantdependsonlyon sand a. Proof.ByTheorem5.13,thereexistsaglEG8(nl)suchthatPa(gl, nd satisfies the bound in Theorem 5.13 with N=nl.LetLI be the corresponding s-dimensionalnl-pointlatticeruleof rank1andletLbeasinLemma5.39. Then Ll ~L,and soPa(L)::;Pa(L1)=Pa(gI, nl)by part(ii)of Lemma 5.40. By the construction of gl, wegetthe desired result.0 THEOREM5.42.Fors;::::2,letarank randinvariantsnl, ... ,nr begiven asinTheorem 5.41.Then,for every a> 1,thereexistsan s-dimensionallattice ruleLwiththisrankand theseinvariantssuchthat wherer(nl)isthenumber of positivedivisorsof nland wheretheimpliedcon-stantsdependonlywn sand a. LATTICE RULESFOR NUMERICALINTEGFLl\TION143 Proof.Proceed as in the proof of Theorem. 5.41, but UB Theorem 5.14 :instead of Theorem5.13.0 The remarksfollowingTheorem 5.14apply,with the necessary changes hav-ing been made, to Theorems 5.41and 5.42 as well.An.lilogml existence theorems for large values of p( L) and small values of R( L) fOTlattice ,["lllesL ",lith prescribed rank and invariants can be deduced fromparts(i), (iii),and(iv)of Lemma 5.40 and fromthe corresponding existence theorems concerningpeg, N) andR(g, N) in5.2. Theorems5.41and 5.42havebeen obtainedbyamethodthatisinasense indirect,since,intheproof,wefirstconsiderefficientlatticerulesof rankl. Thereisalsoadirectmethodofestablishingexistencetheoremsforefficient latticerulesinwhichweimmediatelylookatlatticerulesof thedesiredrank. For adimension 8;::::2,let arank 'I'with 1 S;'I'S;sinvariants nl! ... , nr ;::::2 with niHdividing ni for1 S;iS;'I'- 1 beWe put Zi={gE Z: 0 S;9< niandgcd(g, ni) = 1}for1 S;iS;1'. LetC= (8; nl, ... ,nr). bethefamily of alla-dimensionallatticerulesLfor the node set COlilSlits with integers0 S;ki < nifor1 S;iS;'1', where the gihave the special form gi =(gil), .. . ,gis))with gii) =0 for1 j i-I andEZiforiS;js.It followsfromthe proof of Lemma 5.39that eachLEhas rank 'I'and invariants nl, ... ,n,..Let M(C) =L R(L) jLEe betheavera.gevalueofR(L)asLnmsthroughC.Notethatcard(C)= (ni)B-Hl.Inserting the definition of R(L) and interchanging the order of summation,weobtain 1 M(C)=I:A(h)r(h)-l - I, bECa(N) whereN=nl'"nrandA(h)isthenumberof LECwithhELi..-Wenow invokeLemma5.21toexpressA(h)intermsof exponentialsums,andweuse the special formof the node setof eachLECto obtain forh= (hI, ... ,hs)Ees(N).Therefore A(h)1nl-lnr-lS(1min(j,r) ( -=-I:'''I:III:- e bEC.(N)r(h)Nk1=Okr=O j=1hEC(N)r(h)i=lgEZi )) 144CHAPTER 5 The innermost sum iscalculatedby the method following(5.15),which yields The resulting expressionforM(.)israther complica.ted.Evenin the simplest higher-rankcasewherer= 2,veryela.boratenumber-theoreticargumentsare neededto deal with this expression.The followingresultisobtained in Nieder-reiter[259]forthis case. THEOREM5.43.Forevery822andany prescribedinvariantsnt, n222 with n2dividing n1,wehave M(l,) < Cll ((lOg N)lI+ log N) \Nfl'l/ withaconstantClI dependingonlyon8.Inparticular,thereexistsans-dimensionallatticerule Lof rank 2withinvariants n1and n2such that R(L) < + By combining this result with Theorem 5.27,wesee that there always exists alattice ruleof rank2with prescribed invariantsfQrwhichthe node set has a discrepancy of an order of magnitude close to that of the lower bound inpart(ili) of Theorem 5.38.Similarly,by virtue of ,Theorem5.26 or(5.42),we getavalue of Po:(L)of an order of magnitude close to that of the lowerbound in part(i)of Theorem 5.38. Efficientlatticerulescanalsobeobtainedbyconsideringsuitables-dimensionallatticerulesof maximalrankr=8,withcopyrolesofferingthe most promising option.H Lisan s-dimensional lattice ruleandk22 is an in-teger, then the k8-copy of Lis the lattice rule with integration lattice ,,-lL, Le., the lattice Lscaled by afactork-1The kll_ropy of Lmay also be described as the lattice rule obtained by partitioning f8into k8 cubes of, side k-1 and apply-ing aproperly scaled version of the lattice rule Lto each of these smaller cubes. Thus the useof copyrules maybe viewedasaspecialdeterministic versionof theMonteCarlotechniqueof stratifiedsampling(comparewith1.2).In the followingtheorem,wecollectthe basicinformation on copy rules. THEOREM5.44.If Lisans-dimensionalN -pointlatticeruleandk22 isaninteger,thenk-1 LisakilN-pointlatticeroleof rank8withduallattice (k-1 L)..L= kL..L. Proof.Thenumber of nodesof k-1 Lisequaltothenumberof pointsof the integrationlatticeLin(O,k)lJ,whichiskIJN.Furthermore,k-1Lcontains (k-1Z)3,and the group(k-1Z)8/ZJJis the direCt sum of 3cyclic groups of order k.Hence(k-1Z)8hasrank8,andsohask-1 Lby[317,Thm.3,3].Finally, fromthedefinitionoftheduallattice,weimmediatelyobtainthat(k-1L)..L containskL.l...NpwZB/(k-1L).l..hasorderkilNbyRemark5.25,and,from LATTICE RULESFOR NUMERICALINTEGRATION145 zsLJ..kLJ..andRemark5.25,weseethatZS/(kL1.)alsohasorderkS N. Therefore(k-1L)1.= kLJ...0 Onthebasisof the identity(k-1L)J..= kLJ..,correspondingquantitiesfor k-1 Land Lcan be related.FOl"instance,wehave Pa(k-1L) = Lr(kh)-aforanya> 1, h where the sum is over all nonzero hEL1. Iand this expression is closely connected withPa(L).ThisconnectioncanbeeffectivelyexploitedinthecasewhereL isalatticeruleof rank1.DisneyandSloan[68]calculatedthemeanvalueof Pa(k-1 L)whenLrangesoverall s-dimensionalN-pointlattice rulesof rank 1 generated by apoint gEG8(N),where Nisaprime number,and showedthat .fork=2 this mean value is asymptotically s.ma.ller than the corresponding mean vjUueof Pa(L)foracomparable This suggests that there are lattice rules. of' fw;Uc yiEtld. smaller bounds. than lattice rulesof rank1 with nodes..,. ofpr1:ndpa!probleT'Gsthe?xeaistofindgener?,lcon.structionsof efficientlattice rrues.Exceptforthe casewhere8= 2,in whichthe connection withcontinuedfractionscanbeused(see5.2),noexplicitconstructionsof infinitefamiliesof efficientlatticerulesareknown.Someattemptshavebeen madebyHuaandWang[144]andZinterhof[370]forlatticerulesof rank1, but the results fall farshort of the levelsthat are set by the existence theorems. Thus, at present, we still must resort to computer searches to filld efficient lattice rulesfordimensionss3.Thesearchforgoodlatticepointsmod Nmay proceedbytwostrategies,namely,an eliminationmethodinwhichthepoints gmod Nwithasmallvalueofpeg, N)aresystematicallyeliminated,anda randomsearchmethodinwhichpointsgmod Narechosenatrandomanda goodlattice pointmod Nisobtainedaftersufficientlymanytrials.Sometimes the search is restricted to points of the special form(1, g, 92,.,g8-1) proposed byKorobov[159].Extensivetablesof goodlatticepointswerecompiledby Maisonneuve [203J,and these were complemented more recently by the tables in the book of Hua and Wang [145]and in the papers of Bourdeau and Pitre [31]and Haber [123].The search for efficient lattice rules of higher rank is based on similar strategies.The most ambitious search project involving higher-rank lattice rules wasundertakenbySloanandWalsh[321]andyieldedmanypracticallyuseful lattice rules of rank2. Notes. The method of good lattice points was introduced by Korobov[158],and further pioneering papers on the subject include Bakhvalov[13]'Hlawka[135],and Ko-robov (159].Expository accounts ofthis method can be found in the books of Ko-robov[160Jand Hua and Wang[145]and in the surveyarticles of Zaremba[365] andNiederreiter[225J;seeNiederreiter[251Jforan updateof the latter paper. Theorem5.5isfromNiederreiter[264].ByaveragingthequantityR1(g, N) in(5.9)oversuitablesetsofpointsgmod Nandusingthefirstinequalityin 146CHAPTER 5 Theorem5.6,improvedexistencetheoremsforthediscrepancyDN(P)of the point set Pin (5.1)are obtained; see Niederreiter [239Jforprimes Nand Nieder-reiter [237]forarbitrary N2.Algorithms for the determination of good lattice pointsmodulopowersof 2weredescribedbyKorobov[161].Computer imple-mentations of the method of good lattice points are discussedinGenz[113]and Kahaner[149].Awidelyavailable softwareimplementation isroutineDOlGCF in the NAGlibrary.Severalopen problems on goodlattice points are stated in Niederreiter[233].Further existence theorems for good lattice points of the form (1,g,g2, ... ,g8-1) can be foundin Larcher[171]and Temlyakov[343J. A report on calculations in connection with Zaremba's conjecture is given in Borosh and Niederreiter [30].For theoretical results related to Zaremba's conjec-ture, see Cusick[46],[47],[48J,Hensley [133],and Sander [300].Cusick[49]con-sidered a higher-dimensional analogue of Zaremba's conjecture.Temlyakov[344] proved the optimality of the two-diniensionallattice point g= (1, F m-l)mod F m fornumerical integrationaclass of functionswith bounded mixedderiv0tive. Thefirststepsinthedirectionof latticerulesweretakenbyFrolov[111]. Sloan[314]and Sloan and Kachoyan[315]againapproachedthe subject,and a systematictheoryof lattice ruleswasdevelopedby SloanandKachoyan[316]. For abackgroundon lattices,wereferto the book of Cassels[38J.Lyness[198] and Sloan and Walsh[320]gave brief surveys of lattice rules fromthe viewpoints of generatormatricesandclassificationtheory,respectively.Lyness[197]dis-cussedvariouswaysof assessingthe quality of lattice rules.Further resultson ranksandinvariantscanbefoundinSloanandLyness[317],andadetailed study of lattice rUlesof r8.nk2 iscarned out inLyneSsandSloan[199].Struc-turalresultsonintegrationlatticeSaridtheirduallatticesm:eestablishedin Lyness,Srevik,and Keast[201 J.For lattice niles of'r3.nk 1, .Theorem 5.34was shownbyZaremba[365],and ina'somewhatweakerform,by Niederieiter .[224](seealsoNiederreiter[234]).Themetli6dof obtainingthe lowerboundfor'thediscrepancy' incanalsobetracedto[234]. Lemmas5.39and5.40arederivedfrom[262J.Fhrtherinforma-tion on copy ruleS,in'oil their be f9undin Sloan and Lyness[317].Joe[148] and Patterson [45]from lattice rulesi to rules. ;Analogues of lattice rilles for integration over JiBSloan and Os-born [319J,and Sugihara (336).A generalization oflattice rules for Haar integrals overcompact groups wasintroduced by Niooerreiter[262]. --"'.,., CHAPTER6 Quasi-MonteCarlo. Methodsfor Opti mization Another basicproblem of numericalanalysis to which quasi-MonteCarlo meth-odscan beappliedisglQbaloptimization.The standard MonteCarlomethod forfindingglobal6ptnha; israndomsearch,anditisemployedinsituations wherethe objectivefunctionhasalowdegreeof regularity,e.g.,in the caseof a nondifferentiableobjectivefunctionin whichthe usual gradientmethods fail. The deterministicanalogue of random search isthe quasi-MonteCarlomethod of quasirandom search.The analysisof quasi-Monte Carlo optimization follows the sameapproachas forquasi-MonteCarlointegration:Wefirstestablishan effectiveerrorboundinte:n:nsof asuitablequantitydependingonthedeter-ministically selectedpoints(inthis case,the relevantquantity isthe dispersion rather than the discrepancy),and then westrive to finddeterministic point sets or sequences that make this quantity assmall as possible. Bothrandomsearchandquasirandomsearchcanbedescribedinaquite general setting, and this is done in 6.LIn the standard case where the objective functionisdefinedonaboundedsubsetofaEuclideanspace,moreconcrete informationcanbe given.Sinceastraightforwardquasirandomsearchmethod isusually inefficient,we also discuss more refined techniques, such as localization of search.In allthesevariantsof quasirandomsearch,abasicroleisplayedby low-dispersionpointsets and sequences,whichare studied in 6.2. 6.1.General theory of quasirandom search methods. LetXbeaseparabletOlpologicalspaceandlettheobjectivefunctionfbea real-valuedfunctionOlnXforwhichwewanttocalculate aglobaloptimum.It suffices,of course,torestrictthe attention to the casewhereweareinterested in the global supremum(ormaximum)of f.Thus weassume that fisbounded fromabove,and weput m(f) =sup f(x). xEX The Monte Carlo method of random search proceeds as follows.Put a probability measure A on X, take a sequence S of independent A-distributed random samples Xl, X2,.EX, andusethe estimate (6.1) 148CHAPTER 6 ThesequencemN(f; S),N=1,2, ...isnondecreasing,andwewouldexpect that it converges to m(f).In fact,this can be shown to happen with probability 1 if Iiscontinuous and the measure >.issuitable. THEOREM6.1.If fiscontinuousonXandif themeasure,\issuchthat '\(A) > 0foreverynon emptyopensubsetAof X,then limmN(JjS) = mU),\oo-a.e. N-+oo Proof.For given c > 0,the nonempty set Ae={xEX: I(x) > mU)