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Numerical Recipes ThFeoHAaDrrevtrpaWrtarSadrBioWt-SimSnnalrfmleeuViiiniPNlaatStchFlloooismnluocfAaaoolPrnoiummnPi.heaiTdrynm.dTsn.CetiCFHceoVrtsereErlu,in.piaaeCftocdekP1iorntnracaritofntrontieooelerilCnle7rsolAfRsUlsrkoin7tnseyrnioyvmpceghrsiypipstyicuestsing visit website http://www.nr.com or call 1-800-872-7423 (North America only),(cid:10)or send email to [email protected] (outside Noreadable files (including this one) to any server(cid:10)computer, is strictly prohibited. To order Numerical Recipes books,(cid:10)diskettes, or CPermission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyinCopyright (C) 1986-1992 by Cambridge University Press.(cid:10)Programs Copyright (C) 1986-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN FORTRAN 77: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43064-X) EXXONResearchandEngineeringCompany rth AmericaDROMsg of machin ). e- PublishedbythePressSyndicateoftheUniversityofCambridge ThePitt Building,TrumpingtonStreet, CambridgeCB21RP 40West 20thStreet, NewYork, NY10011-4211,USA 10StamfordRoad,Oakleigh,Melbourne3166,Australia Copyright(cid:13)c CambridgeUniversityPress 1986,1992 exceptforx13.10,whichisplacedintothepublicdomain, andexceptforallothercomputerprogramsandprocedures,whichare NASiCFNRRSTPTnUecS7lfo“Maasr“coiryhoueeoulooorlxotLir8oi:lmPpppintssrnpmwmeudrifcrse1:faitoedrrlthtynrRcteteexeiwde:iiII1esiuewlEyrrccternnpnnSra8tdieperi.ceisssaoengdittfiitagenc[Bt6@MTWoeeatpcieefcrnNssehiircslchido3ncesdadrtohaNtueoeinsttsiiaict.suontr-idsllCene,oAdnntbinmwIi1cuwnmah(cid:13)omdtbrsRnneteyortvch07PhpRleTio,nioselsif.tneem.atei-3Eeeac.t]ooutroOsafehnct5rchggh.naaa,ib9XeNhprfttrisapei2oUn.aenscuitrmic.epafusam-giwcuebei1fzdatsreosrtirBennin–thosinaomlfni-lvne,ita.rnsinigee4nahrtoccmrgiaoaePtiradriaresre2leenc3igLeieoexonddlgdeeionlsr,dcebon.qv0lrinaa.cirdurbtFy(bstCracfd2buimah6tlbF”tsciiVkepodoySricaotNp4ocee4ooClpoc”Oi(aeoauorskleetnrsho3o-ppmuttanucyhirkeladtNXerRsRmin,ks(.yptbsyismecale,o,.rdpe–fiaT,r.uel1Cenansoaefwf1souiCilfecRcxotmslslgtd1,woa9fetralso7eotihoxrtoeAhi9irimprro9Chlearnac7fceem–r,lfrt9eoNesr6reyeeafodebxNciiiFeo3(cid:13)s:fbep,Ansnrl:coacnorxf,o1rbYe1uRaisfdlStrogntgawmieTdir9oheiowl91tiicfoverntrtAgeihgn8rde1ooar9e9ae)eRtcfaaacrmieaego6o0imrrktnsn7n9fiiarenpwne,liplsco;w5e,taa,s4dAcrspuaiecreoetreM7StlnaCaa,)oNbunsrrideilv(spd,rp3isogeioyetglrpla1mneceuAyp,eoucepfnrNo.ot(t9tafrpmrsoaaahnaohUos1osrfn9iouufmdni0liedcssesr9ferSl5oSSgombeehmdi2oIwer8sstcoe.gcrnAnlsdiosrht2meo.ia6onEcraifoists,)uoriet3)nifmt2r,2atnttshedtw,itiinlini8gel.wcw.tCoc1rd“fiteisf0abtalumatatooRdin9yicSniaita8(oilfirtntltoobnr.ryU,9omercsedehoocdcRneeei2fiPcaASkndoneeobudWieCC,piniiuoneAmrnpnfrrawmcordys1fsobasitrdceedoirb)dPh@optiic9hmpmlmebeose,hgfdhitgehrfu9of)e–xceatiiritpibcreeeeyhtsn7sbwSbaieM.iupurrdmlniesnaesiitibiineUapetititiicrdnsladgicenaaeoonfrrl.enefgtnooyesdtsitnogFoho/dnelge,o(i,rr.kfrenpvhoWsgmmD1nto“m,methueprtU”i9paroawaotirnibuamtflarTl8.ssnttr(lfoltlriiaaptiexy8NNei,iiot,ocibtrasvhn:ccxyo–h@gnitohuomelshny/ir1ai7ie.oemr.–rnmnPpnt/9stn7sCdkoChHcSeireeri9we:ic.teloe1istroacr2y.wuasesiAomTaen9mo.coldPwsdlfihy9maPm.ptnenr.plhqeu2leriudectanu”eceeuessotAaR,rltorSesseeneiesmilsrs.nrcrecsideoerot,csetsecamcpcerpifeeon)iootdu1rnpsioftftsonmot1hahsaeoerea)gsnx0eeeesssrr.t,rams. visit website http://www.nr.com or call 1-800-872-7423 (North America only),(cid:10)or send email to [email protected] (outside Noreadable files (including this one) to any server(cid:10)computer, is strictly prohibited. To order Numerical Recipes books,(cid:10)diskettes, or CPermission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyinCopyright (C) 1986-1992 by Cambridge University Press.(cid:10)Programs Copyright (C) 1986-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN FORTRAN 77: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43064-X) Q531A.92.F49O070R.2NT88R560A563N–1d(C9c29o02mputerprogramlanguage) I.Press, WilliamH. 92-8876 rth AmericaDROMsg of machin AcatalogrecordforthisbookisavailablefromtheBritishLibrary. ). e- ISBN 052143064X Volume1(thisbook) ISBN 0521574390 Volume2 ISBN 0521437210 ExamplebookinFORTRAN ISBN 0521574404 FORTRANdiskette(IBM3.500) ISBN 0521576083 CDROM(IBMPC/Macintosh) ISBN 0521576075 CDROM(UNIX) Contents 12 PCLPSPP111222222222222...............ilrror171004589123612ocaeee10lPSIITIVCELSGGmenunntflfIQrpirarUehiaaaanttsnoirtnmraprruuodRoogagiccooMrdsDssliorutsureddafsse,iieeDeaeealvsg-inuurntAamacJtktmeeLoeccrrttonaohIyccittniooInoorVrmxiiOconremEaoodDnuamiffelrIPpnnalttderlpoeagniaTnuohhpaLmecrrravsnrcesoowonoeEeiSeMdiiymsvnmtnDirl,yigizoesBiaFtSaoeasmepaaiimttrntoa-onctioireaeatiiVonneroindsonamiccrnndsmmiononaaeSttsoitwtdnsDAptnolaiosaouonaniEInNifbltantnsdghmsdaibgdid3ltCAioeBSiyitTPeoEnytoobaponrianclCpodoeErluktlcpartSiisnhedoictluyoiisailactibssnotzSt?tispietonttMEoorimtntuequnasLscttrtiuriounoifacareneeaEntssriqdoEuqanStuisaoetnicostnison xxvxxxx2iii9284182935364v1xi2vi511297872514332ii visit website http://www.nr.com or call 1-800-872-7423 (North America only),(cid:10)or send email to [email protected] (outside Noreadable files (including this one) to any server(cid:10)computer, is strictly prohibited. To order Numerical Recipes books,(cid:10)diskettes, or CPermission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyinCopyright (C) 1986-1992 by Cambridge University Press.(cid:10)Programs Copyright (C) 1986-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN FORTRAN 77: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43064-X) 3 I3n.0teInrtproodulacttiioonnandExtrapolation 9999 rth AmericaDROMsg of machin ). e- 3.1PolynomialInterpolationandExtrapolation 102 3.2RationalFunctionInterpolationandExtrapolation 104 3.3CubicSplineInterpolation 107 3.4HowtoSearchanOrderedTable 110 3.5CoefficientsoftheInterpolatingPolynomial 113 3.6InterpolationinTwoorMoreDimensions 116 v vi Contents 4 IntegrationofFunctions 123 4.0Introduction 123 4.1ClassicalFormulasforEquallySpacedAbscissas 124 4.2ElementaryAlgorithms 130 4.3RombergIntegration 134 4.4ImproperIntegrals 135 56 ES4455555555555555566666666.........................vp557101246891111023456763120134aeGGRNISECQCDIIEIBMBMPnnnnPCBPPEREeolvxceoheeaauuettccouurraolcacvpssrrmureuamemaooiiyodloodissulolestaboaasdtmvmmuyneeibimpiddese´ssrlnynrofidalltlaunoerariuluuapposeanAataeieeiitlbaoFFmtlFccihmcillmnonnndxaiatvoeeFiFmpttuuaceitldlictuiitteneinoiupliBAuQannviooeeezCaiatsnvnlTnanrnccDynnaloensruAGBseohRoolttcchtisoIfdiciiaFeaixtetrAtnBsepoooaeeohffdriCinubiettCmtioplIonnninmmpiraFderailnyvnoFnnranassoupgtmucsaotFoCnaeFtloi,sRuooebrrhuttnontxnftuIBmuoaiiaogiffiiaerncnovnticncPnnlxnersestmvIFtiFstsenaiccusioaintecvEorsuoma,ltaAtalegwteaasiqintnnCentdoDoFrrcnapiiuaeogcsgadontunotudpFliratifleiebnssm,ionsoCtrFrnrSnoOatyiSoanocnrouulcOnescixttCreafPennrblifu,ttnliraoirIhihsacumEddeonotsOnttoeitesehrhimtnavi,bgronreroatsFedoyoIgntiwnCn’oaersnePsscrth’nrhFao,esDteleOuogiARvsbPinrrrss-iiayeocdrtaoatcrsytlpeliniiyuhsorbopF,nreFnunrrBuvoouet,nmiinxnCConccicinohtomaeti,eiimolo-aFFsfnSfitino-easqDcrd,luimSCieFasnpurotutrehlesniaebfcfirutiicctoiiaoenlnnt,s 211121211222111121111221112056805077130594981965228790590319581544950749297399216 visit website http://www.nr.com or call 1-800-872-7423 (North America only),(cid:10)or send email to [email protected] (outside Noreadable files (including this one) to any server(cid:10)computer, is strictly prohibited. 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Sample page from NUMERICAL RECIPES IN FORTRAN 77: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43064-X) 666...1890SFDprehasewnreiscloaInln’HtseIagnrrmtaelgos,rnaCilcossineandSineIntegrals 222544268 rth AmericaDROMsg of machin 6.11EllipticIntegralsandJacobianEllipticFunctions 254 ). e- 6.12HypergeometricFunctions 263 7 RandomNumbers 266 7.0Introduction 266 7.1UniformDeviates 267 Contents vii 7.2TransformationMethod: ExponentialandNormalDeviates 277 7.3RejectionMethod: Gamma, Poisson,BinomialDeviates 281 7.4GenerationofRandomBits 287 7.5RandomSequencesBasedonDataEncryption 290 7.6SimpleMonteCarloIntegration 295 7.7Quasi-(thatis,Sub-)RandomSequences 299 8910 SRM99999788888889991111111111......0.......0000...00000o3342526780140561oi..........3018924567nrHVNQSSNGAISIIDBRonnntiOIGLSPDDCVeetaoreeudeletirmtdtnlcoaniaiarrawwtnaooinaooiemetnaecoorcbFrrpirptncelwxWaenttrikigegkddosadsutoo-atjmzibicatehisilnouuduDennoblnnitnvlrotoaatMnhjrccuigiygfli--igennlodtrtPenItticatRmRietPiiSCnlgaanecgiartttooildMaaietnotoeoshaennonnoihSSIppaecandlgeoAdGnnnenoyeinrghhtrsrnmtvittMdRtdnnrdRiasseorioeaeoooo,onimo(rpaanecBrdnPnpntnFfnemnlgcrhmina–eoSoaMiaEkueeaMMsilxlDlwelLMnidnairneqiasSnenanslttMceeaedteeustriggeigtattrMkhMlNvcioihhaPgSlevkaoohe’MxnrooeothesaeonendchsdM)dtiisltaerdhesnhtehonmi–nlMttUfnloodiothwBdiOol’nhcoddsniiesoririeeBnznnsitstnMdteMnNheheaSnCrfsieMMgoneouCoetiDFtladnmrltDnaeMiauMithsitosrrNtilrm’lpeohdisslisteunnrltooniedoSieteidlMmeshnDvdntxMMiaeioolam,sedeeriMdtitanneurtifioSeshmintvselvnnehytoiFdaeiteoasosorddhinutRntdoiSioefissmvsninnmdyideoEsescdOssntnqeetsonsrmiifuesoo’sEDannMoqstisfumieoaEtethqninooussdniaostinons 333333333334333433443343334332845224823020462145970133729907053007736972122068366372395 visit website http://www.nr.com or call 1-800-872-7423 (North America only),(cid:10)or send email to [email protected] (outside Noreadable files (including this one) to any server(cid:10)computer, is strictly prohibited. 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Sample page from NUMERICAL RECIPES IN FORTRAN 77: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43064-X) 11 E11i.g0eInntrsoydusctteiomn s 444499 rth AmericaDROMsg of machin ). e- 11.1JacobiTransformationsofaSymmetricMatrix 456 11.2ReductionofaSymmetricMatrixtoTridiagonalForm: GivensandHouseholderReductions 462 11.3EigenvaluesandEigenvectorsofaTridiagonalMatrix 469 11.4HermitianMatrices 475 11.5ReductionofaGeneralMatrixtoHessenbergForm 476 viii Contents 11.6TheQRAlgorithmforRealHessenbergMatrices 480 11.7ImprovingEigenvaluesand/orFindingEigenvectorsby InverseIteration 487 12 FastFourierTransform 490 12.0Introduction 490 1143 SF1111111111111111111111111122233342223333344443344433ot..........................a25681121340147903672614535u01t(aFFESDFFFICPPCIANDCLMCLODriAnnnWNaopoFFoosixiioooooroopittnnoedsgueuwweltTTrrnmrnnuttlateeemTooTcriirrriSvtpmTeevmtraaPeiiFtpddoiicwwaneeeorrroannrrwleaououufelaaanrraarloolgSSPClrtuucclTFtolaeFRnltTeT-ieiilppsttrmtrtDA(nwoocDoiSsiineeDrirdiWeelDooTa)ogrorneaatatdceinoccirtleoinnsfrnnnrmMlythiettFaisearStarUrcorressTaFtlrlianioaneuuTtrfafynsenricnprusigdioonubDsmmtatseogabeMcidnhfoerrsrubieronnAiuommsiicior)ostrlEEoDensoontcriettaudoifRfFirosimrnrrsspooneootttfeiAtnttiIaaMoohcbhnfrdiirslnntUslnmmtDcfomenauessDieteLkHDonaeno,RtrDTiaalSmiirlmgii(veSttiaCoseyirnnnFaiAiisrvomvoceinafooesemgtaonFeflrlerrlin:nnalpueaoreeeynisTwspDrrsbMtrttt-lUbpDeoif)lhaieieUyiuPLoioaoylntlfnenesolathDrtynnsonSdiitiaaTgettmSion?sccihnaUntSdUwaoCianahgTmea,ganiamnlsssetcooihVmtiMipntatFtThDnsneeFhiaDilpiwFveogogearFeinMliefiTrdxFoeeTsatfntFeshihedtCFnemDmTFarsreeacTionD?TrabnueFFaddtu,samnaFiFttSnTosaiTTgfokEhronernVrewstmaernrosiDepasniysmc,eesn?sions 656566565655555655556655456403130060336138734501235490293040349910554873209580179424 visit website http://www.nr.com or call 1-800-872-7423 (North America only),(cid:10)or send email to [email protected] (outside Noreadable files (including this one) to any server(cid:10)computer, is strictly prohibited. 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Sample page from NUMERICAL RECIPES IN FORTRAN 77: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43064-X) 15 M14o.8dSeavliintzgky-oGfoDlayaStamoothingFilters 665404 rth AmericaDROMsg of machin ). e- 15.0Introduction 650 15.1LeastSquaresasaMaximumLikelihoodEstimator 651 15.2FittingDatatoaStraightLine 655 15.3Straight-LineDatawithErrorsinBothCoordinates 660 15.4GeneralLinearLeastSquares 665 15.5NonlinearModels 675 Contents ix 15.6ConfidenceLimitsonEstimatedModelParameters 684 15.7RobustEstimation 694 16 IntegrationofOrdinaryDifferentialEquations 701 16.0Introduction 701 16.1Runge-KuttaMethod 704 111789 TIP1111111111111111111111111n6788866667777788888996799wat.........................5434623460135601257017223reotSAIIBAMRSITRAHIFVLMIFMSDIginnnnnnetrlhhoiaPaieduairiutvtttinoauecWcfocellefrrrtnatalfxdexhetdalioofooookgoupdetarSoaixh-ainDlidddSmurtrmslistlfiCsrrraohiadriiuuuneeisEtVnkdvnetaailvloRoue--tcccimfgdesPttegasOGoepEnomqtttEefeodliiirnt,stoSMedBIuqgoooIriqnouEoielnMndEEfntueunnnuMbbrArqetiEoieaagxalEnvaeeedltPpuuareitxleattrtrauFqpMaisrrllnairmtCntmtitootiooiuoloihtrionizMzattincovebanvsaiVppotaoelinatpsnidnetlsayltdaaaehnnBietteCiolswoinlItssaoosmgulMI:eouholnnodmndaeirorneoPuEiSstsnvta,etyfhtdMniaoiptoPhiaatoqangrtnhdihStnferhonlneVieoauoeddiMvMelVttUrnadrbhaReayfoPaSnglsuoeoIleleEidtrueeCsnldurudsemitchlqsteioaotoRdavoeodhuslfrnPiPnunreieacHdKmoAadnPrttrsBitiooiagetioensKrPbrioruernn-otmonlernl-siCsseniinKeiboromosodlsTunrsrlconsiitrehhrsctIea-nmsecSSftioootnsorrgremMyuralMeatitrohePntohodoisndts 787787777777877778787887777771482297747912850154304061849858557455998666163998844820 visit website http://www.nr.com or call 1-800-872-7423 (North America only),(cid:10)or send email to [email protected] (outside Noreadable files (including this one) to any server(cid:10)computer, is strictly prohibited. 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Sample page from NUMERICAL RECIPES IN FORTRAN 77: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43064-X) 1199..54VRFoaeluluarexieaPrtiraoonnbdlMeCmeytschloicdsRfeodruBctoiounndMareythVoadlsuefoPrrBoboluenmdsary 885448 rth AmericaDROMsg of machin 19.6MultigridMethodsforBoundaryValueProblems 862 ). e- 20 Less-NumericalAlgorithms 881 20.0Introduction 881 20.1DiagnosingMachineParameters 881 20.2GrayCodes 886 x Contents 20.3CyclicRedundancyandOtherChecksums 888 20.4HuffmanCodingandCompressionofData 896 20.5ArithmeticCoding 902 20.6ArithmeticatArbitraryPrecision 906 ReferencesforVolume1 916 22BB2BBBB321654213CoIGnneNLIFFIPSEISPIdnnnntniooeepvroructttteeerrxnarreeelmeuetfoolrlgrtciranaouswtpmddeairicslfaaoarouunooieIieltPtnnlccncrifai9FotdraottIadVotnooii0unrilnooeoboffignVRCexnnoolynoLrcuosreoafftitttmMmlaoncoomdiFuFonieaeiuempVuFPsncdatnenoChosia2eraocEsclarr:hAunntta2xteUairNmidllotaloglprtuennMeniDaetllmsesispbee9tPriorept01seraclreosaiLiaanccgtnlafiadfrooEndalerngqmRn2Fuucemoaaicregtitiinsperogaen(FnVsseo9ianl0.tuF1r)oerstran11111119x000000099990vv48750118263ixi3i0390427125ii visit website http://www.nr.com or call 1-800-872-7423 (North America only),(cid:10)or send email to [email protected] (outside Noreadable files (including this one) to any server(cid:10)computer, is strictly prohibited. To order Numerical Recipes books,(cid:10)diskettes, or CPermission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyinCopyright (C) 1986-1992 by Cambridge University Press.(cid:10)Programs Copyright (C) 1986-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN FORTRAN 77: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43064-X) BB78 RSoarntdinogmNumbers 11114617 rth AmericaDROMsg of machin B9 RootFindingandNonlinearSetsofEquations 1182 ). e- B10 MinimizationorMaximizationofFunctions 1201 B11 Eigensystems 1225 B12 FastFourierTransform 1235 Contents xi B13 FourierandSpectralApplications 1253 B14 StatisticalDescriptionofData 1269 B15 ModelingofData 1285 B16 IntegrationofOrdinaryDifferentialEquations 1297 BBBBCCC11213127098 RATGLPLIILnnweiiapeestdssrefponttetesiigeiexnnrPa-rnreNggalooadnDlulfiooicnEmcIiPffnefteqfrUEesdeBsuorxetrfoagiieopxclturinrlaitationytcloVmindAaiMotVsaslllIogrouEnaaydolmqtnuneurVudmdeirltaaefhIeD2altsnumisceove(epsnen1sPersrastrnynoedpdbeTeln2ehacmenioesdrsyn(rVuotli.l)2) 111111111333334343168534342414294735 visit website http://www.nr.com or call 1-800-872-7423 (North America only),(cid:10)or send email to [email protected] (outside Noreadable files (including this one) to any server(cid:10)computer, is strictly prohibited. To order Numerical Recipes books,(cid:10)diskettes, or CPermission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyinCopyright (C) 1986-1992 by Cambridge University Press.(cid:10)Programs Copyright (C) 1986-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN FORTRAN 77: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43064-X) rth AmericaDROMsg of machin ). e- visit website http://www.nr.com or call 1-800-872-7423 (North America only),(cid:10)or send email to [email protected] (outside Noreadable files (including this one) to any server(cid:10)computer, is strictly prohibited. To order Numerical Recipes books,(cid:10)diskettes, or CPermission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyinCopyright (C) 1986-1992 by Cambridge University Press.(cid:10)Programs Copyright (C) 1986-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN FORTRAN 77: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43064-X) rth AmericaDROMsg of machin ). e- xii