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Code Samples - Core Java/Math: functions library
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| public class constants { | |
| public static final double PI_INV = 1.0/StrictMath.PI; | |
| public static final double PI_SL8 = StrictMath.PI * 0.125; | |
| public static final double TWO_PI = 2*StrictMath.PI; | |
| public static final double TWO_PI_INV = 0.5*PI_INV; | |
| public static final double HALF_PI = 0.5*StrictMath.PI; | |
| // Euler-Mascheroni constant (gamma) | |
| public static final double EM = | |
| +0.5772156649015328606065120900824024310421593359; | |
| public static final double oneonlog10 = | |
| 0.4342944819032518276511289189166050822944; | |
| // The arrays DCI(30) can easily be extended to DCI (>30). | |
| // See: High precision coefficients related to the zeta function | |
| // by F.D. Crary and J.Barkley Rosser (Univ.Wisconsin, Report#1344). | |
| // Also see: W.Gabcke, Dissertation, Gottingen, 1979. | |
| // Maple: evalf(series(cos(Pi/8*(4*z^2+3))/cos(Pi*z), z=0,78)); | |
| // Mathematica: N[Series[Cos[Pi/8*(4*z^2+3)]/Cos[Pi*z], {z,0,78}], 100] | |
| public static final double[] DC0 = { | |
| +0.382683432365089771728459984030398866761344562485627041433800635627546033960089692237013785342283547148424288661, | |
| +0.437240468077520449360296467371331987073041501042363031866378610690404508264026362304924073435669639094324286079, | |
| +0.1323765754803435233240352673915105554743229955586736726649426880559566435420738501669778293204183675017943241749, | |
| -0.01360502604767418865498318870909990766070687027421894648326187950903716176818633068962485278356136895679047020906, | |
| -0.01356762197010358088791567058349920618602959696188054686917829168674934087357724499903115819721107719785647775895, | |
| -0.001623725323144465282854625294133649725659201718172548305398980308243971457730734393999432889383665122685176399873, | |
| +0.0002970535373337969078312728339951586690679333334505619667872265698183319369203982805177634566104207720143400458262, | |
| +0.0000794330087952146958801639026487950144873099152560321495071752950226903173395878801705671344477374930716659999123, | |
| +4.65561246145045050370634021603476231240414569015306013316932838935348879193481820928464518626366832313236741193e-7, | |
| -1.432725163095510575408246312062615888246258029228068660731581897805436577213074435556392472287142822769748329508e-6, | |
| -1.035484711231294607500741567738403498882724615888283179447948653075217984108710847647177577896120460086478764151e-7, | |
| +1.235792708386173805612576262312530316510117762098112867236451610214808402930282161805187583832632897001667458326e-8, | |
| +1.788108385795490498566678140706904566454558839254644214800304281524529021049731788926650722192519086137173526827e-9, | |
| -3.391414389927035906940621897884455615248397316288972845006785754803502806891904570703993463969380667640255995188e-11, | |
| -1.632663390256590510137405297104810281346405431822126827413147731265198687738077289595025493824604361493389459115e-11, | |
| -3.78510931854122038285464720018504502639038535531200438309272315224985279176248298240356571256856962726814475742e-13, | |
| +9.32742325920172484566232063986986360002139698116915631937705187400020105037249065099538843442588188664675147467e-14, | |
| +5.22184301597813685531389314785302371037675394827206477433177228406371981681645773099043136444005380798369532226e-15, | |
| -3.350673072744263789515090357947326053042838022398065784111555892808740716677888930078701970602503964936006395104e-16, | |
| -3.412426522811726494080987104562058778608283456254096935869001796289917525710367617748381397622542683846888672015e-17, | |
| +5.75120334143239916033950179516459231161537829470064807231431453813486388545094592961884811901724585064764962844e-19, | |
| +1.489530136321150545475627775734689089370735009741635415160709048603295381414367364949760330404014020213930338814e-19, | |
| +1.256537271702141685330428176609282176536606717440674080535520482356810764118743819183518519587390237199100704507e-21, | |
| -4.72129525014342566895398813667305340706330304767749675702994903089879756535834541307555978776385327369051553303e-22, | |
| -1.32690693630396199927354130926183589457507684264465236751272493873018443686610022790536188919297368265742721815e-23, | |
| +1.105343999512141834453782254227205003182486780211827086005486828970265089129731688133254857575539823725195683922e-24, | |
| +5.49964637752746551114010449998398178325210538268839922240622794248949462106984664855058759404246021418007498035e-26, | |
| -1.823137650231802628064108980945407064129881555145279779343173540693674436633740912616549635547508188700365680468e-27, | |
| -1.568940373772088014686829823192433140971355533415866343941007167915490728982359363699569882728024654791588147859e-28, | |
| +1.583963508823801161065976053779295278390321164235705734225919985692892191820446152269080950811170989257202082911e-30, | |
| +3.4346207254372040220415367076516947462098483922500503564813783179382274162068573263112671201649692317453615905e-31, | |
| +1.702103350031701775318130755271536920853455319760507766090796235001978260644018065519766552941782676624037679422e-33, | |
| -5.99511930495781673363972633565278514210564502273708131802609647284822583102487894658236367081899707846320368051e-34, | |
| -1.04876827540944523668427321724480041261517736197636621170912258890766184956298037903925915158527893326057340676e-35, | |
| +8.42213517834932107854815160735950929394240415164747688538014234387628955118839912704841909631604183775608950552e-37, | |
| +2.584703859771955713160011615401679072208538467158450776918252923628843991474698009272117251533209327471142444888e-38, | |
| -9.34763937488998521367903976236981418745369692048179313134504611480555205177899495504433350362542373205050278136e-40, | |
| -4.56941922524370129765254861055742309974472681838538268606496265105749673629290338481038140628402537476198769926e-4, | |
| +7.5455973947653905212085611627422817804352042098369215797696318278534625566582905601529992674332954823406110951e-43}; | |
| // Maple: evalf(series(diff(cos(Pi/8*(4*z^2+3))/cos(Pi*z), z$3)/(12*Pi^2), z=0,78)); | |
| // Mathematica: N[Series[D[Cos[Pi/8*(4*z^2+3)]/Cos[Pi*z], {z,3}]/(12*Pi^2), {z,0,78}], 80] | |
| public static final double[] DC1 = { | |
| +0.026825102628375347029991403955666749659270472430643220749776752805867359134188269214280683052, | |
| -0.013784773426351853049870452589896162365948225597532513294465798867691807541688152085576031383, | |
| -0.03849125048223508222873641536318936689609880749450906478327860153836847426887901787229257659, | |
| -0.00987106629906207647201214704618854069280421459666950839994786868965757411195190486323582121, | |
| +0.0033107597608584043329090769513006978028020918561175309707694861178448645426345718015500843326, | |
| +0.0014647808577954150824977965619831119780775457722862078933453303143287255048972099659207209621, | |
| +0.000013207940624876963675161447494430967824291835406154372613530235724535011475016069716813294177, | |
| -0.00005922748701847141323223499528189568406802912492160850522823303196158210932726481656944581815, | |
| -5.980242585373448587710835074515858419335890174202825600668342888976410900581355969992952868e-6, | |
| +9.641322456169826352672985329851666875707836639273182783168607374528344107916458627247029283e-7, | |
| +1.8334733722714411760016793657832219080753603339709992766273715660151990962666208683124437739e-7, | |
| -4.467087562717833599560794227150551934657469384377660055067166840151970252514746820463948005e-9, | |
| -2.7096350821772743216926283987091937259316030722995844412753447458314425593181090759150020757e-9, | |
| -7.785288654315851046294823085209610006727820577278800382831566727931867764632910377867374876e-11, | |
| +2.3437626010893688532484550487104512273133964049734205404375275618829488523490151174368404654e-11, | |
| +1.5830172789987521642162226426287421196746979819448013053513508455771578055396438744193146485e-12, | |
| -1.2119941573723791246646344738017572576448530651660888461727279790245362198776996808868093029e-13, | |
| -1.4583781161108307017582854816989993171964777297933249751938379402944425481261805530630616476e-14, | |
| +2.8786305258131917504558212800208760753536483952839345119896953142223473009317304761040685377e-16, | |
| +8.662862902123724122528252887933104042807961436294852178958442068342689422294037865607742406e-17, | |
| +8.43072272713704127156002253146274997727634288682530999309124260276609727264253556324577217e-19, | |
| -3.630807223097346200173246181103281136955871882367415423137780078494511015098068311048451748e-19, | |
| -1.1626698212838296719413888629248324378808802522154009317733711360471269908689980684964899997e-20, | |
| +1.0975486711527531815901832833980075157643671464693844798841337840675643501161428099083592646e-21, | |
| +6.157399020468427103881470790974945857407651738028111860714322669192813619603136094113741154e-23, | |
| -2.2909280067678471513963826309991269307343977096746438696025932017457239554824102031775874946e-24, | |
| -2.2032811748848795343795982704373537544376537775704078003688163099187931890241159108319322546e-25, | |
| +2.476025180040278508285274215182918632587580796467112641727687468810950715262244429469791761e-27, | |
| +5.954277215583657802272682863953454743730467881559988360025759772968743994237177956667902438e-28, | |
| +3.2612020746795952615337563190661874830891135980274339139845795592240636373843954612255006828e-30, | |
| -1.2654035591041162243650179791261536112558798012347744231540654284163522268689446590266925924e-30, | |
| -2.4312846965496981901634636363338047426353348019050697893343812128678861564371131634064246015e-32, | |
| +2.1383011387546953739564195753197140950296216408034685099183785890974590986138498294625891461e-33, | |
| +7.167799413941061690328338683692708749333075099712339753173535550674454000240550835751326567e-35, | |
| -2.8242936072336665615525326594221813904617133033516236406598931104936386534532483771217816036e-36, | |
| -1.5006074196069282189178370454993947599747135926005292679869580497329724359381439103658938812e-37, | |
| +2.687318940531486108260118827557915092685193539193918356025160117503267595083118443609927278e-39, | |
| +2.4904195007933094154169676810430522646696390103991181635576185062669795340798460145976606456e-40, | |
| -1.1605389825678419639763678032899844219760188058814234948874818004278848389077335785169161417e-42}; | |
| // Maple: evalf(series(diff(cos(Pi/8*(4*z^2+3))/cos(Pi*z), z$2)/(16*Pi^2)+diff(cos(Pi/8*(4*z^2+3))/cos(Pi*z), z$6)/(288*Pi^4), z=0, 38)); | |
| // Mathematica: N[Series[D[Cos[Pi/8*(4*z^2+3)]/Cos[Pi*z], {z,2}]/(16*Pi^2)+D[Cos[Pi/8*(4*z^2+3)]/Cos[Pi*z], {z,6}]/(288*Pi^4), {z,0,78}], 80] | |
| public static final double[] DC2 = { | |
| 0.005188542830293168493784581519230959565968684337910516563725522452072126841897897, | |
| 0.00030946583880634746033456743609587882366950030794878439289358718588031842799211307, | |
| -0.011335941078229373382182435255883513410249474890261540938858667942986863079981311, | |
| 0.0022330457419581447720571255275803681570983979981642796951293105725719455469601119, | |
| 0.005196637408862330205116926953068191888515832107618595474609879576976564736555503, | |
| 0.0003439914407620833669465591357991809598418589002147317484417304617440685730817159, | |
| -0.0005910648427470582821732252303077395276588375610173232690309594523280267574834747, | |
| -0.00010229972547935857454427867522727787133943747273471360465084634256262134151394488, | |
| 0.000020888392216992755408073296174175415931186305360443691531185576321472947573577868, | |
| 5.927665493096535957891996484982863335742249862441772318721993759532135400499668e-6, | |
| -1.6423838362436275977690302847783780496161212669336910235253514740438053597308203e-7, | |
| -1.5161199700940682861734605397187381660081084155970349774087695236188185367222967e-7, | |
| -5.907803698206667962922790253978962060716281592010072012867608882567102850848194e-9, | |
| 2.0911514859478188977745555189722580395885704419004240100593948634461745082130407e-9, | |
| 1.7815649583292351053799701878847486656009684349096795930345702159393431817224899e-10, | |
| -1.6164072455353830752855769444473857776802820362311211738301273256347933803317603e-11, | |
| -2.3806962496667615707210740380135849781560242513320681741897741346182257287784557e-12, | |
| 5.398265295542594918182004148336822987325682997025679714114209585960532400298625e-14, | |
| 1.9750142196969515273308733588451725185221802033544774174325263225894059186101604e-14, | |
| 2.3332868732882634831048153005923547597095168404970213704335126869670513579626541e-16, | |
| -1.1187517610048080208200483808971615892736704619952328379264891944667482662087555e-16, | |
| -4.164009488883767188501122836433308161241527720525451717634563860759792248049601e-18, | |
| 4.446081109291883028903043500928743324161025016190532684886900866475208961810096e-19, | |
| 2.8546114783637144545733874269779565433096490605491768968430556415535467088959418e-20, | |
| -1.19132314300378943049718475052661810410453703351011923973272939551201153470687e-21, | |
| -1.2981634360736498946709902313291029349864968257004043970653583551747007944948404e-22, | |
| 1.6123763178033262338779658663222193262651519787452057819826151396265479872321981e-24, | |
| 4.382497519887344059655258424644950704152853920980276818591850390318129841119647e-25, | |
| 2.7186389576555759138820356271448872774947633995904411012405376808492919628590506e-27, | |
| -1.1458896506774580369743945579295750204092666668699824045661401768984212546138787e-27, | |
| -2.4415318181927522978909188671073094149415439741835244151526434958860319358446682e-29, | |
| 2.3505675086790434606664221960472243599181441248526662245549521169413326741733033e-30, | |
| 8.669258995621298717800714563004282859479548850038258850838724790295883213896554e-32, | |
| -3.723977985489462680382745552671764711614778698594250205029269305778538368069886e-33, | |
| -2.1646033266321799468220479128490325581663125661515692147696595451120812363930903e-34, | |
| 4.203457751935555749203270075437513679883585299088583490129455804307114122286297e-36, | |
| 4.244052494804297215797687543359423065727104705085375433170467492601781347640104e-37, | |
| -2.1231392753906157383843052537658427398583379292318669518502402669687755858823324e-39, | |
| -6.813496373118564864349052387612632119903467123774393160507783275039315832205749e-40}; | |
| // Maple: evalf(series(diff(cos(Pi/8*(4*z^2+3))/cos(Pi*z), z)/(32*Pi^2)+diff(cos(Pi/8*(4*z^2+3))/cos(Pi*z), z$5)/(120*Pi^4)+diff(cos(Pi/8*(4*z^2+3))/cos(Pi*z), z$9)/(10368*Pi^6), z=0, 38)); | |
| // Mathematica: N[Series[D[Cos[Pi/8*(4*z^2+3)]/Cos[Pi*z], z]/(32*Pi^2)+D[Cos[Pi/8*(4*z^2+3)]/Cos[Pi*z], {z,5}]/(120*Pi^4)+D[Cos[Pi/8*(4*z^2+3)]/Cos[Pi*z], {z,9}]/(10368*Pi^6), {z,0,78}], 80] | |
| public static final double[] DC3 = { | |
| 0.0013397160907194569042698357299452281238563539531678386569289925594833800871811798, | |
| -0.003744215136379393704664161864462396581284315042446001020488345395501319713045551, | |
| 0.001330317891932146812031854722402410509897088246099457996340123183515196888263071, | |
| 0.0022654660765471787114760319905210068874119513448871865405689358390986959711214106, | |
| -0.0009548499998506730415112255157650113355104637663298519791633424726946992806658956, | |
| -0.0006010038458963603912075805875795611286932555907537579329769784072390871266471527, | |
| 0.00010128858286776621953344349418087858288813181266544865585870025117392863374779861, | |
| 0.00006865733449299825642457428364865218534328592530073865273869547284481941105060958, | |
| -5.985366791538598159305933853289474476033254319520777140355079450311296076398332e-7, | |
| -3.33165985123994712904355366983830793171285955443637487051019057060568872956295e-6, | |
| -2.1919289102435081057184842192253694457056301094787176328502199928079472310677209e-7, | |
| 7.890884245681494410555248261568885233534195350876097985016870653592807297722941e-8, | |
| 9.41468508129526215165246515670888721434440703062285787060390697195816617233733e-9, | |
| -9.570116210883480301880722847736899414920424990844021810115468872811044469914093e-10, | |
| -1.8763137453470662796812970577763318771497261610950370939934010326350481628028716e-10, | |
| 4.437837679323399327464708984967982039427175136450134427877701103829299216728465e-12, | |
| 2.2426738505617353248411068573063743908847573555967383640946296048845725322165884e-12, | |
| 3.627686865735243689408255637923200993091627857438909649329814867831900563004191e-14, | |
| -1.7639809550821581607831121498067405612829056369595996874090447022130186603011647e-14, | |
| -7.960765246786777757290345179277877672969068290389354910445419703993693619905826e-16, | |
| 9.419651490589690763914895025694423958555439910507587271684011367592679056089096e-17, | |
| 7.133103854569657824556667924637208733076137253777613086648835921659278195880674e-18, | |
| -3.289910584554624321179665258492719604389867767706730329578387560456883113849552e-19, | |
| -4.180730374898459291362924870562363545456098499893508469921618986694281682179048e-20, | |
| 5.550542071646333789782116402662976359558940175392938033703401483591298082182587e-22, | |
| 1.7870441906260123858717636353127488582980187518683339734105072221029271545889173e-22, | |
| 1.331280396465609428629734301456596911495766449193494655875984603950395330646233e-24, | |
| -5.818610611090987516179216596088520206267870242969322822752311337488525847711228e-25, | |
| -1.4019036088526555374364967097960412447326079695863802828805235344426680330368205e-26, | |
| 1.4641320211626254148997752501860798089334733220831445783786891177657679796555688e-27, | |
| 6.02332655108914231894545302168540534775476763292822981011798811478407558676743e-29, | |
| -2.8064472319113607480413277200041961058924502875599284388627977593567952984627978e-30, | |
| -1.8065060055924548468166679975772190204281290392132976714058746943222867824547636e-31, | |
| 3.779508331934081109538275143960178072816211713033459437067101283961906998805713e-33, | |
| 4.214558052947562754928267311962627941300036513813646793136217341959757887604597e-34, | |
| -2.2110619283398807703089110108495251373369608027039372389420927090703989012463229e-36, | |
| -7.977857191491540240197869206644086240230293731794895306614670662176495711819498e-37, | |
| -5.134879815416697465299965219246691353718718876508043721203193132000758105946362e-39, | |
| 1.2486406302153718700908292104194463596234890838129830954576262777222039185771739e-39}; | |
| /* | |
| * Stieltjes Constants from 0 to 78, to 256 digits each. | |
| * ref: http://pi.lacim.uqam.ca/piDATA/stieltjesgamma.txt | |
| * also called the gamma constants defined by | |
| * | |
| * | |
| * / m \ | |
| * |----- n| (n + 1) | |
| * | \ ln(k) | ln(m) | |
| * lim | ) ------| - ------------ = gamma(n) | |
| * m -> infinity | / k | n + 1 | |
| * |----- | | |
| * \k = 1 / | |
| * | |
| */ | |
| public static final double[] Y = { | |
| .5772156649015328606065120900824024310421593359399235988057672348848677267776646709369470632917467495146314472498070824809605040144865428362241739976449235362535003337429373377376739427925952582470949160087352039481656708532331517766115286211995015079847937, | |
| -.7281584548367672486058637587490131913773633833433795259900655974140143357151148487808692824484401460407720727888674475946468021218957925934357889512942805222349515279549606111502095344684131064742443502359366559538469614380259381045738025493664557651677028e-1, | |
| -.9690363192872318484530386035212529359065806101340749880701365451850755382280414171978197381374537319286223858587946823125346497950962571972526493648255902282233425225376030952454650839481623874159222094228123258327703499005518661324301612485127438367384026e-2, | |
| .2053834420303345866160046542753384285715804445410618245481483336913834492112970053570557166228566702972851647965455643681856998286631518491198828868652927507879634329919223749694521937588516086128932900100887582794153232910617743621951263465535646918550222e-2, | |
| .2325370065467300057468170177526068000904469413784850990758040907124841005315521900301678059039306360827843551103225522196456664007682437918861215754578332362815081585973614343456254349981107648311254210559424139592687296840743156065574648306992970376230434e-2, | |
| .7933238173010627017533348774444448307315394045848870757342562698231482118017152023797200635876308162719168578623728950693993277932862610560981556563334528255145467565669705027727800328021356905705583506896973325036121197655863337023278457363824591927681467e-3, | |
| -.2387693454301996098724218419080042777837151563580786314764253073910675599929638714368611141285111024780673075909533038020656912269595124699717764644539646672994359920016312185594700697656216552224200467213493615215357142501593344546382399161688323344633459e-3, | |
| -.5272895670577510460740975054788582819962534729698953310134042268856827324651411821440413807979996094255347137014747313651473070509002813384872173815912447148812297961891402918386808326741994669807197014678473723857395756624902062083941316454527809794259124e-3, | |
| -.3521233538030395096020521650012087417291805337923503566573315073642817765060653010801409187200115151480243957720152737286032860943501971136758373996071542419096556138574571993689232705602040409831370056942984921502852878319887965827056475310881183605291025e-3, | |
| -.3439477441808804817791462379822739062078953859444162975929190484315010334446152837095754389345718990812939708751106559999723650791217087757476384733602840643363955402889741500455608351851986205711990705391789569312430556033918346536209762876458392962080569e-4, | |
| .2053328149090647946837222892370653029598537741667643038402087143530090240710691751984960510609028168654630716692437652938609455701636670314763888351107401248788037757774478758676467051586842465830033648294891261556376188509817851906503013130840649822003101e-3, | |
| .2701844395439035266729020820679556738278420586884025039737358031367999909642929802378444607202380543341892065433898331362430539345163022251660977905645827208257807545662673885178807828093358372485533868496985893326451504654464211037386959473080948618383043e-3, | |
| .1672729121051401933535015433411834466078066328055658280477909376512195970327407625539042834685454860827918008130473096673223305416815315435148622550674827969874566633280326548214334209784161729704490822545521988490380808808815410064684131411221199776802669e-3, | |
| -.2746380660376015886000760369335518152678533767039553609283308916757051860700887286766227018014141226063021263616344526017942940219226384963138784380546201092996086073554616887396220116191432159038003288891331098518783382994530877461169787303572031210085377e-4, | |
| -.2092092620592999458371396973445849578315442115060695624342083257187577618413479238849643630363170068978952103732270941025164202162406217483524889032192155723532675386434812392932022134837802510909806951284079809204130775034571304740902331856694608730746506e-3, | |
| -.2834686553202414466429344749971269770687029807176752539699432929676256905331671583502753305306099359983296641743658578448670204816395094978958650260714071371637925247680241849293162055953372734373236183678636434144439111527617109957199974180603721462395347e-3, | |
| -.1996968583089697747077845632032403919157649740340612798596671625543805947413858376185052901045229228152050749963179893268016715236270037554630588742666500589733359678401153567212058999564688677646468080389369421514423787946034894284978610156860905576783237e-3, | |
| .2627703710991833669946659763051012281607869292911406079711751835228318283659619892378138323561685615911623560201540792781969494575153922147852554290415989679216348097499078673743176921693516030034263511744512517839540600579140351515313919384582936117575383e-4, | |
| .3073684081492528265927547519486256455238112907314616910811036523148083902097289937656175556967012404059095414514102941951119674092289417995555858007852262090239169157743457758869727907667465462418873038971056792798849296000885600799035155113134156928909482e-3, | |
| .5036054530473556290555964377171600353212698076494978373237909270104380947646223189527024912899340799999802379685872863039429327259976022887863525558688062335605764854361061913564269294911630304700024202875309737842304685438543203315544401816345818487684388e-3, | |
| .4663435615115594494005948244335505251131434739256889976707266280985445821300329007010634781743483037676276776893978664788490376098061008113591441797256336823512267157959337408681128870085117167176652714745827612534802328769644422488731525593967185587090795e-3, | |
| .1044377697560001158107956743677204910444282507055467478343714867390804411994132201971857629970557612477080276816174297098442819046793799873530097156786216211170154344305780995660062733094223612700160596820798224374455841978660825820242682142793915007842212e-3, | |
| -.5415995822039977016551961731741055845438609287007488018391913163842120274727846608325326997754074546503753106668640607229437483993682169764132200837068069923525151709033207834551632712814364366899723899724930759778803264577711209436089086578419457274846364e-3, | |
| -.1243962090408245779299741599537165809147028113964637716532971108378003227300447476688043928770660618906589245188584132576689392153326370316493290511316121734714766939840656684143340621090548028988852205331608344386554756230080950750377804966175863406283901e-2, | |
| -.1588511278903561561906196611521115857318722822144129067478194125480950383379748723977182688294225715704915149922400750354649555457261240500959167389177194357862621241155775647357272837529119488611489255817396821745210322189035753496755771351316733341251102e-2, | |
| -.1074591952738488824724291987353173089273979331453170361409902581781368852879725083510487626853780543878731137936481194598448964551855691857004868428077492623871354826190616571737826265422531415866113138027144599171360216617932498664012392456911554072571146e-2, | |
| .6568035186371544315047730033562152488860650604775373760992825250891170965034776663910579342625757859715232062111062915856525206876176042963292134187616474914067354036499371011291177322406733034202470236023762935497158074335393463790043477652282574744043603e-3, | |
| .3477836913618538209007359574258811547662915663885919292269369089163601152516386090184507808325259934643247527620919916504590034421507603998744879095106816233978110202439753349023737964681719231706674211339160697022354031902450736661875203742977403043595923e-2, | |
| .6400068531700629458107228221945863666637198144588475220590330625837093213897134487336264083586702624780911771466881249158837079615155947036988364224391544873825936603455801551951739436482654980238076705978473654568449071615955207841466403251011797640636588e-2, | |
| .7371151770472239134412402423559402157841327488512840153180311593260094964080124902874271317941480450169928391633082168004655647838467136598965148773299440893911326321952210876313787678510143647647900141023350228313526634026921869524825211534867202695510451e-2, | |
| .3557728855573160947913537748908402610809650649522125076131381742604895375347768268411760486644490119698820111281432172758850410765212613111042978612558180006655129918249475202003030282427138985184474816070832944619701532581247303970119806166784524080913113e-2, | |
| -.7513325997815228933135160081576145616636587418058437829820653897726656851364608736093406575498735199953375557645227668649877454170501745904952874352569833828340361752923325058378630036421872568935701095205574194985993332448759010555982110176282553449891202e-2, | |
| -.2570372910842040179348788378034991655408420213709272996040553488073373925092915194869402974674313960730940827885805986079503641578184306369509419528819255356778241434806326040117004842490643574687237090396478809065180424485158191278852413572924082385053692e-1, | |
| -.4510673410808021990498284969956376278181476720118751914813317935089473799365486780079426205395715048231336142622172311472080346051129198009798869840229135326813527696281135231523051833221278728337033589841096529250915588639261001568781069882956884511058368e-1, | |
| -.5112692802150846442507582003801215757213099693370947031283705080222204757165161391299384814318941989350085938186440763932854104808109240365643509465403766309112448633408990761899675968647705676410731088908908770878621830867075270142641783507042859178782092e-1, | |
| -.2037304360386131270575189730254735763855514540127568404226904414244063724030988503350463213893919466606979495327877395077626203087646069193281918128083913334868246722812020110757918056153158263567517772557098298030827761720020741589113319214710546324868855e-1, | |
| .7248215881681133373380044422036406713494156746516600301109791139709584630115609076184768490496514066910360733891536130570598692290165869195273978736246742908618723458316488977615459103966319968155863913816498268488587418654995797804155697655649958847357895e-1, | |
| .2360263822743015027209817621993795633447977318546636606795696524336922591383506319852601197035035699203623244114544400619644905611166062223529956334728610781278059048805132264984375254618970256448247285280843274150107341682480156517967257924219186788145330, | |
| .4289634463848091527368615465389604104951855560836298508881511467174782165406052598052578837703479513149605271888650666596964536012997065102084281647972845368239473903765489789230269010448415666019284057957391635619051037937314897060412487116366199455816836, | |
| .5179218426929237189788930575162080963032869116382440388334761465713068965780043079965859932817502694727319695029057167172255798781972489650526343539907081526696209591694159811807914277744490491038137744209287798148227736083094114236529502428204747874931539, | |
| .2487215593946154650844919104403834212137661679529456385350509654042314398657765133858509357834519851660462686780038493302932372007287500470374521580805939391035285372957104115822097373824399880947615656580184641803366839892091737835298438895206210595752375, | |
| -.7195748469013003506888739112196854239261748831904649437785449263509786356293101897746613034720604271142999671606071191146849003123224065620009891082439621461705375083652807042878644516387056361921714149226264455652409110895061126053434853106969343380563178, | |
| -2.638794927335734535788281675648811310351782987035211499393243485907022239259505477040826570395791839498801399715723709634295474856557592397489594583251684015695249594247301673922474234026168332480339091413459364363081675429428395332081292613817734336565494, | |
| -5.264930312355023828811032859580336606977042993956806905911825107752096822582823647571129789899774007449077183706697896011698161033442136588696911829176438099711663253191742572376731229116446238288895074259071273499782678262582441265329247305947655986144330, | |
| -7.188745889503527282342094824577664215932574859281338805833810538667337501527011967915372049590948156198532169549971970898863800579381929252662783608393442431305718388587696952733141648850458812574971021050868587199490079795392192132425737225940758618726619, | |
| -5.072344589916372492298940404798877527548898547920341433576114007970208835005861247222957450067358896625318542197749232212662512002598509927706415764052961176009738698115190727795940434449848894227264093298941761645790234028006626255339596450726680973432486, | |
| 6.609915609096965813839975106591453976121624142676258962466161263230138247955950680501240606225730339947058738765837375315012758818853754069379249379465003451583201983190721503025985727108804064481134263041435677655221268593105524179459213774579064542886341, | |
| 34.03977498215874824766115211222887543481585062040728845880766195415091850111097098102234038334063582255146136304735662018166909676792633649172593400903096852662977956340472884566589366552943189154254333330735508711818454888919225114589304435439875918866231, | |
| 78.68247976324258495603848420938847587592859021606368068347311248131245045816477978354222279127347778484079585199133838207604437335391341578617869858249504960215493811609322620525071575891947562789820590429774578807866291149245067581168982559771813175765951, | |
| 125.8443876319784690933640869642634548210830426469704980639576282947523289773055415013393636381818820365174280709651417996658374631556841875987387400798841785568945454815405004576022607269753629090377417076709892483708691350430892879564872582173161084734278, | |
| 126.8236026513227165967252536486575555384835759448901701980956993469732544041271419075472839202481342000655527466907773263202970132939623809589591513912865283594558393685656496366403863160270130724104654611609335672863609966882450434670623623037390894982865, | |
| -19.19691187302785580049992278889461000488617138521125094955804692768632497077431894010336425512641087536957362345904530571260115521167316919248856770615789826449707900700147413646023179865419434659545981369663884566897807702752807188599160262261877584723048, | |
| -463.1889230267168108085343582392362931366116957180006901624675536855778259610180865548529674806333795445784720929015557429014573440317655361459554522755159806774751854061019646746026001962717761365205206395869989959129560170027580385380148222072396574437614, | |
| -1340.659144376892189727101963668908904813680425823094277360983924970123545905951320804164099765412339187206633285729948584751525184352621143592938185171889188832432976967119173814211291335944639480016298975541814145863037680692144453474530167024849384966632, | |
| -2572.454740404435516763574381743194537144349074129635190025656675261689968560135282533680795608528937664815193978544465137589608940353479663547865399362584761794796720486829475401161510062152263672635951091948051219547961625385885682988539181752335571182500, | |
| -3457.141208645389953809462307608548197267392571262945382531901198419835265449139114139752724233485244785535339119102051777751947298714580782741473753259866854277965873748702018770556207132668621431246868940695237459978570056725932042526772704517088887009050, | |
| -2055.275816231974304572825156345659341714378586681402878963865132625780739633561673788853189681295671720531311167392431020178456949604170241477352885604106843954614982380859102212646287075146135017827460000558232950054762819030973497931729190368748362000642, | |
| 5372.282213203191284181833545959014329555896642905224954409370153194551639289251004905214862802153702614891093139187266989686294985772286939348850809250994703278188368030520223602114426496922514344077289177925193404535132017379204347643038237561684778349535, | |
| 24019.38937760698818204861035985736409058929002953145915799769330693343585546963061640387520046045702663578653894052569085070940959579119305407994509817271201272546721845612222693895881739318046735749910581720490135833336177178720592740120790968916641082098, | |
| 57424.31929696407550657104394311255386400824370256752620991647966953367902572223109625676082012930154382556432735615395354528456397119780281123421908197993868853546446676051766559695367292479004717927284373185020644824412329349572910288157946256744189407140, | |
| 98543.25459014604211093211428941711531650104882341554197930622783608489036287602715386941804646663591106764763550763071863725791968111497783011529453607923260637137817777048517149527447460321197286999081445397775020004411932959031469803884925185616975390011, | |
| 111670.9578149410793387893403211563974853622621861860070492558245106321152197978896356784986709659246398447028682342515620904866717282773920131267534144689106562607331771415956711044043588809723539995371146389025477672830672938535201055031258424853698622521, | |
| 5333.665210500764343403713072261136718914593293565738489694853347359352589030809494678259291885891979613064318888829142263676324267800058016249399128627054877573090524750544029388489126257672223385886733854495868967399358045126339726416940001003776331780669, | |
| -390972.6873133963957775428164347301012865910198508746441638769290837847013047456217619412022961983123163124272712579711729696456704271694618158088390339217519866909714445427931569604142174394070839878737071843151424174567821468441597847504269831119565319909, | |
| -1303180.712532519808573578260908777678044948332570067129963421421848029593041641458697133568977980576700883270277981455463933985088650175081474094621803307108859599115906956541023262677524307005797980356671824629484862625887654008181910920206214383524051245, | |
| -2845076.552608612116523243100237370870167655675924216498192869458404187329953640177299560183025897558495297082006826021694564155486292072145777913453704712722953138646764731475253810351084080951541575328521843225182807547286142617638437012547115115046403931, | |
| -4540526.609737724100509220867003971251048957338640918478864644401431771260187110948708426225730696673484910608169992239039359208562183905400100070421973273676747789972100524122177059432652884762115926751661713705281886452057166399772740063853326419021621033, | |
| -4341905.139001516198415670264383693842260958150127609160422048895643211404947195718257471803919996372167852148973302363686896828779612377060187735478109745765995707365214973502468316183453023901759425272748017178186339655775494652737519189536753913543805680, | |
| 2871566.945972460509905357306147525690106604257063853632662792335691123095953849148759187546180165016153744782976866579448776215707008569628936807475513691359949909887658409137518529362425919225421409999326669652492263291715447395177074780931961343784615131, | |
| 26604908.54668677092425521912451936480795299150646426379923783074812038437400291966231913301672803147593025933983520666997287407005167498500146308540816725205740171309465570488989279619206587140609236965698637546585522770792996633444924580867051639968204913, | |
| 79321663.11929906059211001875506986109078678605929299821733167638956228811094702496866006501239037925742409911928107256247873459490010385482553313165643858925233075421940693131005923399599328970140465301608533561129008200715549168186165980180247842021792257, | |
| 166215134.0468254355839825083764775026922493156873793437567024004040720676074041053803255270535873330605351293137839271608605683647864515142266769019278900135634372572577287785416673156462232216791854730153546523662690892362166893419994060430067089918268485, | |
| 255153258.3082389934909848078277768987902985514378076503823427021825601645614601862599207385980737560906754309716204100869275599148675590710569782697323107792348115172035957067240704301036234325777779946135648388560892047021038514968620844931091344338610286, | |
| 212655631.6918540369512019248627158396680614022195978324871483336108798420688927107279671233050594954058347952103381180030465863386349837153315936212092445328817755413706057708197992558132570343461837062889786366607647313628680221334335862993688051677647064, | |
| -298767089.4311661838592139845456232588222876086992170490259044012663702238324435706623905293153630798969724334390118682919837048033136392454203199753628085167331437818516158612262299723479084451715092046370319754099231070567632457508936168937828037690257857, | |
| -1919487427.732802396229627757729327898023244830602085729304679210428118470905660169503320644717346154413202421403219588184732013410426970230132448744061230223078726672201628641965688174011380164991661246163863144544756484425183731247748949286364250669027383, | |
| -5515574258.129220269427426037306452066936170169171072411797916588895750116790116120042606241013411261739570410392314324043817608753827749236198030823299522969460857196981095390178299797003363459101202590130435740820533907989476579686558014505421730147475846, | |
| -11483450987.92625603025962308750354819358989636858236343418922095146003529652099454932564500092640974706164032396286399805090630030297913074096080965875780082646299473016444439508070872009844923178137154154440180991389231196962826618318021120909673249402227, | |
| -17570152277.77726386163608615618525785039204899986664599956286229502140896936467083284521308786084527317826035812378168370902350943468812664885992884478165346500022035578688139196146186122588937724007564250471442030547825810788316951289967818338033762342923 | |
| }; | |
| private constants () { | |
| // private constructor to prevent instantiation | |
| } | |
| } |
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| import java.io.BufferedReader; | |
| import java.io.FileReader; | |
| import java.io.FileNotFoundException; | |
| import java.io.IOException; | |
| public class functions { | |
| private static double rootpi = StrictMath.sqrt(StrictMath.PI); | |
| //private static double root2pi = StrictMath.sqrt(2*StrictMath.PI); | |
| private static double oneonrootpi = 1/rootpi; | |
| private static double twoonrootpi = 2/rootpi; | |
| private static double logtwo = StrictMath.log(2); | |
| private static double pi2on12 = StrictMath.pow(StrictMath.PI, 2)/12.; | |
| private static int INFINITY = 100000; | |
| private static double PRECISION = 1E-12; | |
| private static final int[] PRIMES = | |
| {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59, | |
| 61,67,71,73,79,83,89,97,101,103,107,109,113,127, | |
| 131,137,139,149,151,157,163,167,173,179,181,191, | |
| 193,197,199,211,223,227,229,233,239,241,251,257, | |
| 263,269,271,277,281,283,293,307,311,313,317,331, | |
| 337,347,349,353,359,367,373,379,383,389,397,401, | |
| 409,419,421,431,433,439,443,449,457,461,463,467, | |
| 479,487,491,499,503,509,521,523,541,547,557,563, | |
| 569,571,577,587,593,599,601,607,613,617,619,631, | |
| 641,643,647,653,659,661,673,677,683,691,701,709, | |
| 719,727,733,739,743,751,757,761,769,773,787,797, | |
| 809,811,821,823,827,829,839,853,857,859,863,877, | |
| 881,883,887,907,911,919,929,937,941,947,953,967, | |
| 971,977,983,991,997}; | |
| private static int[] primes = PRIMES; // default values in case of file error | |
| static { | |
| // load primes array with PARI/GP generated file (5000 primes) | |
| try { | |
| BufferedReader pf = | |
| new BufferedReader(new FileReader("primes.txt")); | |
| String ps = pf.readLine(); | |
| String[] pe = ps.split(", "); | |
| int pel = pe.length; | |
| primes = new int[pel]; | |
| for (int i=0; i<pel; i++) { | |
| primes[i] = Integer.parseInt(pe[i]); | |
| } | |
| } catch (FileNotFoundException e) { | |
| // TODO Auto-generated catch block | |
| e.printStackTrace(); | |
| } catch (IOException e) { | |
| // TODO Auto-generated catch block | |
| e.printStackTrace(); | |
| } | |
| } | |
| // Taylor coefficients for 1/gamma(1+x)-x... | |
| private static final double c[]= { | |
| +0.5772156649015328606065120900824024310421593359, | |
| -0.6558780715202538810770195151453904812797663805, | |
| -0.0420026350340952355290039348754298187113945004, | |
| +0.1665386113822914895017007951021052357177815022, | |
| -0.0421977345555443367482083012891873913016526841, | |
| -0.0096219715278769735621149216723481989753629422, | |
| +0.0072189432466630995423950103404465727099048009, | |
| -0.0011651675918590651121139710840183886668093337, | |
| -0.0002152416741149509728157299630536478064782419, | |
| +0.0001280502823881161861531986263281643233948920, | |
| -0.0000201348547807882386556893914210218183822948, | |
| -0.0000012504934821426706573453594738330922423226, | |
| +0.0000011330272319816958823741296203307449433240, | |
| -0.0000002056338416977607103450154130020572836512, | |
| +0.0000000061160951044814158178624986828553428672, | |
| +0.0000000050020076444692229300556650480599913030, | |
| -0.0000000011812745704870201445881265654365055777, | |
| +1.0434267116911005104915403323122501914007098231E-10, | |
| +7.7822634399050712540499373113607772260680861813E-12, | |
| -3.6968056186422057081878158780857662365709634513E-12, | |
| +5.1003702874544759790154813228632318027268860697E-13, | |
| -2.0583260535665067832224295448552374197460910808E-14, | |
| -5.3481225394230179823700173187279399489897154781E-15, | |
| +1.2267786282382607901588938466224224281654557504E-15, | |
| -1.1812593016974587695137645868422978312115572918E-16, | |
| +1.1866922547516003325797772429286740710884940796E-18, | |
| +1.4123806553180317815558039475667090370863507503E-18, | |
| -2.2987456844353702065924785806336992602845059314E-19, | |
| +1.7144063219273374333839633702672570668126560625E-20, | |
| +1.3373517304936931148647813951222680228750594717E-22, | |
| -2.0542335517666727893250253513557337966820379352E-22, | |
| +2.7360300486079998448315099043309820148653116958E-23, | |
| -1.7323564459105166390574284515647797990697491087E-24, | |
| -2.3606190244992872873434507354275310079264135521E-26, | |
| +1.8649829417172944307184131618786668989458684290E-26, | |
| +2.2180956242071972043997169136268603797317795006E-27, | |
| +1.2977819749479936688244144863305941656194998646E-28, | |
| +1.1806974749665284062227454155099715185596846378E-30, | |
| -1.1245843492770880902936546742614395121194117955E-30, | |
| +1.2770851751408662039902066777511246477487720656E-31, | |
| -7.3914511696151408234612893301085528237105689924E-33, | |
| +1.1347502575542157609541652594693063930086121959E-35, | |
| +4.6391346410587220299448049079522284630579686797E-35 | |
| }; | |
| private functions () { | |
| // prevent instantiation | |
| } | |
| /** | |
| * error function | |
| * ref: Java implementation of C function from zetagrid code (math.cpp) | |
| * @param x | |
| * @return | |
| */ | |
| public static double erf (double x) { | |
| double cut = 1.5; | |
| double y, r = 0.0; | |
| y = functions.fabs(x); | |
| if (y < cut) { | |
| // power series expansion | |
| double ap = 0.5; | |
| double s, t, x2; | |
| s = t = 2.0; | |
| x2 = StrictMath.sqrt(x); | |
| for (int i = 0; i<INFINITY; i++) { | |
| ap += 1; | |
| t *= x2/ap; | |
| s += t; | |
| if (functions.fabs(t) < 1e-35 * functions.fabs(s)) { | |
| r = x * functions.oneonrootpi * s/StrictMath.exp(x2); | |
| break; | |
| } | |
| } | |
| } else { | |
| // continued fractions | |
| double an, small = 1e-300; | |
| double b,c,d,h,del,x2; | |
| x2 = StrictMath.sqrt(x); | |
| b = x2+0.5; | |
| c = 1.0e300; | |
| d = functions.recip(b); | |
| h = d; | |
| for (int i=0; i<INFINITY; i++) { | |
| an = i*(0.5 - i); | |
| b += 2.0; | |
| d = an*d+b; | |
| if (functions.fabs(d) < small) | |
| d = small; | |
| c = b+an/c; | |
| if (functions.fabs(c) < small) | |
| c = small; | |
| d = functions.recip(d); | |
| del = c*d; | |
| h *= del; | |
| if ((del - frac(del)) == 1.0 && del < 1.0e-300) | |
| break; | |
| } | |
| r = 1.0 - functions.oneonrootpi * StrictMath.sqrt(x2) / | |
| StrictMath.exp(x2) * h; | |
| } | |
| if (x > 0.0) | |
| return r; | |
| else | |
| return -r; | |
| } | |
| /** | |
| * complementary error function | |
| * ref: Java implementation of C function from zetagrid code (math.cpp) | |
| * @param x | |
| * @return | |
| */ | |
| public static double erfc (double x) { | |
| double cut = 1.5; | |
| double y, r = 0.0; | |
| y = functions.fabs(x); | |
| if (y < cut) { | |
| // power series expansion | |
| double ap = 0.5; | |
| double s, t, x2; | |
| s = t = 2.0; | |
| x2 = StrictMath.sqrt(x); | |
| for (int i = 0; i<INFINITY; i++) { | |
| ap += 1; | |
| t *= x2/ap; | |
| s += t; | |
| if (functions.fabs(t) < 1e-35 * functions.fabs(s)) { | |
| r = 1.0 - x * functions.oneonrootpi * s/StrictMath.exp(x2); | |
| break; | |
| } | |
| } | |
| } else { | |
| // continued fractions | |
| double an, small = 1e-300; | |
| double b,c,d,h,del,x2; | |
| x2 = StrictMath.sqrt(x); | |
| b = x2+0.5; | |
| c = 1.0e300; | |
| d = functions.recip(b); | |
| h = d; | |
| for (int i=0; i<INFINITY; i++) { | |
| an = i*(0.5 - i); | |
| b += 2.0; | |
| d = an*d+b; | |
| if (functions.fabs(d) < small) | |
| d = small; | |
| c = b+an/c; | |
| if (functions.fabs(c) < small) | |
| c = small; | |
| d = functions.recip(d); | |
| del = c*d; | |
| h *= del; | |
| if ((del - frac(del)) == 1.0 && del < 1.0e-300) | |
| break; | |
| } | |
| r = functions.oneonrootpi * StrictMath.sqrt(x2) / | |
| StrictMath.exp(x2) * h; | |
| } | |
| if (x > 0.0) | |
| return r; | |
| else | |
| return -r; | |
| } | |
| // test erf,c for x > 0 | |
| public static double erfg(double x) | |
| { | |
| return oneonrootpi * | |
| incGamma(.5, StrictMath.pow(x,2),0); | |
| } | |
| public static double erfgc(double x) | |
| { | |
| return incGamma(.5, StrictMath.pow(x,2),0)/ | |
| rootpi; | |
| } | |
| public static double erf2(double x) | |
| { | |
| double sum = 0.0; | |
| for (int k=0; k<INFINITY; k++) | |
| { | |
| sum += StrictMath.pow(-1, k)*StrictMath.pow(x, 2*k+1)/ | |
| factorial(k)*(2*k+1); | |
| } | |
| return twoonrootpi * sum; | |
| } | |
| public static double erf2c(double x) | |
| { | |
| return q_gamma(.5, StrictMath.pow(x,2),0)/ | |
| rootpi; | |
| } | |
| /** | |
| * reciprocal function | |
| * @param x | |
| * @return | |
| */ | |
| public static double recip(double x) { | |
| if (x == 0.0) | |
| return 0.0; | |
| return 1.0/x; | |
| } | |
| /** | |
| * combine StrictMath.floor and StrictMath.abs to implement c++ fabs | |
| * @param x | |
| * @return | |
| */ | |
| public static double fabs(double x) { | |
| return StrictMath.floor(StrictMath.abs(x)); | |
| } | |
| /** | |
| * fractional function {x} = x - [x] | |
| * @param x | |
| * @return fractional part of x | |
| * | |
| * caveat: this function does not seem to play well with negative input values | |
| * | |
| */ | |
| public static double frac(double x) | |
| { | |
| return x - StrictMath.floor(x); | |
| } | |
| /** | |
| * check if argument is an even integer | |
| * @param n | |
| * @return 1 if n is even, else -1 | |
| */ | |
| public static int even (int n) { | |
| if (n%2 == 0) | |
| return 1; | |
| else | |
| return -1; | |
| } | |
| /** | |
| * Riemann-Siegel theta function using Stirling series | |
| * @param t | |
| * @return | |
| */ | |
| public static double theta (double t) { | |
| return (t/2.0 * StrictMath.log(t/2.0/StrictMath.PI) - t/2.0 | |
| - StrictMath.PI/8.0 + 1.0/48.0/t + 7.0/5760.0/t/t/t); | |
| } | |
| /** | |
| * moebius function | |
| * implemented as series representation | |
| * ref: http://functions.wolfram.com/13.07.06.0002.01 | |
| * @param n | |
| * @return | |
| * 0, if n has >= repeated prime factors | |
| * 1, if n = 1 | |
| * (-1)^k, if n is a product of k distinct primes | |
| */ | |
| public static long moebiusmu (int n) { | |
| // test for zero value | |
| if (n == 0) | |
| return 0; | |
| double m = 0.0; | |
| for (int k=1; k <= n; k++) { | |
| m += (functions.kdelta(functions.gcd(n, k), 1) | |
| * (StrictMath.cos((2*StrictMath.PI*k)/n))); | |
| } | |
| return (long) StrictMath.round(m); | |
| } | |
| /** | |
| * Mangoldt function | |
| * @param n | |
| * @return 0 if n is not a power of a prime p, | |
| * ln(p) otherwise | |
| */ | |
| public static double mangoldtLambda (int n) { | |
| int plen = primes.length; | |
| int klim = 200; | |
| double powerof = 0.0; | |
| for (int p = 0; p < plen; p++) { | |
| for (int k = 1; k < klim; k++) { | |
| if (n == StrictMath.pow(primes[p], k)) { | |
| powerof = primes[p]; | |
| break; | |
| } | |
| } | |
| } | |
| if (powerof > 0) { | |
| return StrictMath.log(powerof); | |
| } else { | |
| return 0.0; | |
| } | |
| } | |
| /** | |
| * Summatory mangoldt function | |
| * @param x | |
| * @return double s | |
| */ | |
| public static double psi(int x) { | |
| double s = 0.0; | |
| for (int n = 1; n < x; n++) { | |
| s += functions.mangoldtLambda(n); | |
| } | |
| return s; | |
| } | |
| /** | |
| * repetitive prime factor counting function | |
| * @param x | |
| * @return double s | |
| */ | |
| public static int bigOmega(long n) | |
| { | |
| int factors = 0; | |
| // for each potential factor i | |
| for (long i = 2; i <= n / i; i++) { | |
| // if i is a factor of N, repeatedly divide it out | |
| while (n % i == 0) { | |
| factors++; | |
| n = n / i; | |
| } | |
| } | |
| // if biggest factor occurs only once, n > 1 | |
| if (n > 1) factors++; | |
| return factors; | |
| } | |
| /** | |
| * Liouville Lambda | |
| * @param long n | |
| * @return int | |
| */ | |
| public static int LiouvilleLambda(long n) | |
| { | |
| return (int)StrictMath.pow(-1, bigOmega(n)); | |
| } | |
| /** | |
| * Summatory Liouville function | |
| * @param long n | |
| * @return int | |
| */ | |
| public static int L(int n) | |
| { | |
| int sum = 0; | |
| for (int k=1; k<=n; k++) | |
| { | |
| sum += LiouvilleLambda(k); | |
| } | |
| return sum; | |
| } | |
| /** | |
| * another Summatory Liouville function? | |
| * @param long n | |
| * @return int | |
| */ | |
| public static int T(int n) | |
| { | |
| int sum = 0; | |
| for (int k=1; k<=n; k++) | |
| { | |
| sum += (LiouvilleLambda(k)/k); | |
| } | |
| return sum; | |
| } | |
| /** | |
| * Kronecker delta | |
| * @param n1 | |
| * @param n2 | |
| * @return 1 if n1 == n2, 0 otherwise | |
| */ | |
| public static int kdelta (int n1, int n2) { | |
| if (n1 == n2) | |
| return 1; | |
| else | |
| return 0; | |
| } | |
| /** | |
| * gcd | |
| * @param int m | |
| * @param int n | |
| * @return gcd of m,n | |
| */ | |
| public static int gcd(int m, int n) { | |
| if (m < n) { | |
| int t = m; | |
| m = n; | |
| n = t; | |
| } | |
| int r = m % n; | |
| if (r == 0) { | |
| return n; | |
| } else { | |
| return gcd(n, r); | |
| } | |
| } | |
| /** | |
| * Mertens summary function M(n) | |
| * @param n | |
| * @return | |
| */ | |
| public static int M (int n) { | |
| int m = 0; | |
| for (int k = 1; k <= n; k++) { | |
| m += functions.moebiusmu(k); | |
| } | |
| return m; | |
| } | |
| /** | |
| * Binomial function | |
| * ref: http://functions.wolfram.com/06.03.02.0001.01 | |
| * @param n | |
| * @param k | |
| * @return | |
| */ | |
| public static double Binomial(int n, int k) | |
| { | |
| return factorial(n)/(factorial(k)*factorial(n-k)); | |
| } | |
| /** | |
| * factorial function | |
| * @param n | |
| * @return | |
| */ | |
| public static double factorial(int N) | |
| { | |
| int n = 1; | |
| double f = 1.0; | |
| do | |
| { | |
| f *= n; | |
| n++; | |
| } while (n <= N); | |
| return f; | |
| } | |
| /** | |
| * return a Stirling number of the first kind | |
| * @param n | |
| * @param k - partitions | |
| * @return | |
| */ | |
| public static double Stirling1K(int n, int k) | |
| { | |
| // special value 1 if n, k = 0 | |
| if (n == 0.0 && k == 0.0) | |
| return 1.0; | |
| return factorial(n) * Binomial(k, n); | |
| } | |
| /** | |
| * logarithmic integral function | |
| * @param z | |
| * @return | |
| * | |
| * ref: http://functions.wolfram.com/06.36.06.0015.01 | |
| */ | |
| public static double Li(double z) | |
| { | |
| double z1 = z - 1; | |
| double l2 = (StrictMath.log(z - 1) - StrictMath.log(1/z1))/2; | |
| double s1 = 0.0; | |
| double s2 = 0.0; | |
| for (int k=0; k < INFINITY; k++) | |
| { | |
| s1 += (StrictMath.pow(-1, k))/factorial(k+1); | |
| s2 = 0.0; | |
| for (int j=1; j <= k+1; j++) | |
| { | |
| s2 += (BernoulliB(j)*Stirling1K(k, j-1))/j; | |
| } | |
| s1 *= s2; | |
| s1 *= StrictMath.pow(1-z, k+1); | |
| } | |
| return l2 + constants.EM + s1; | |
| } | |
| /** | |
| * extended Gamma function | |
| * @param s | |
| * @param N | |
| * @return | |
| */ | |
| public static double PI(int s, int N) | |
| { | |
| double fn = functions.factorial(N); | |
| double fs = functions.factorial(s+N); | |
| return (fn/fs)*(StrictMath.pow(N+1, s)); | |
| } | |
| /** | |
| * extended Gamma function for double arguments | |
| * @param s | |
| * @return | |
| */ | |
| public static double PI(double s) | |
| { | |
| int n=1; | |
| double result = | |
| (StrictMath.pow(n, 1-s)*StrictMath.pow(n+1, s))/(s+n); | |
| for (n=2;n<INFINITY;n++) | |
| { | |
| result *= (StrictMath.pow(n, 1-s)*StrictMath.pow(n+1, s))/(s+n); | |
| } | |
| return result; | |
| } | |
| /** | |
| * Gamma function | |
| * ref: java implementation of zetagrid c++ code (math.cpp) | |
| * @param z | |
| * @return | |
| */ | |
| public static double Gamma(double z) { | |
| double ss = z; | |
| double f = 1.0; | |
| double sum = 0.0; | |
| double one = 1.0; | |
| int n = 43; | |
| while (ss > one) { | |
| ss -= 1; | |
| f *= ss; | |
| } | |
| while (ss < one) { | |
| f/= ss; | |
| ss += 1; | |
| } | |
| if (ss == one) { | |
| return f; | |
| } | |
| ss -= 1.0; | |
| for (int i = n-1; i >= 0; i--) { | |
| sum = c[i] + ss * sum; | |
| } | |
| return f/ (ss * sum+1); | |
| } | |
| /** | |
| * Beta function | |
| * ref: Weisstein, Eric W. "Beta Function." | |
| * From MathWorld--A Wolfram Web Resource. | |
| * http://mathworld.wolfram.com/BetaFunction.html | |
| * @param p | |
| * @param q | |
| * @return | |
| */ | |
| public static double beta(double p, double q) | |
| { | |
| return (Gamma(p)*Gamma(q))/Gamma(p+q); | |
| } | |
| /** | |
| * Generalized Incomplete Gamma (3 arg) | |
| * ref: http://functions.wolfram.com/06.07.06.0002.01 | |
| * @param a | |
| * @param z1 | |
| * @param z2 | |
| * @return | |
| */ | |
| public static double incGamma(double a, double z1, double z2) | |
| { | |
| double z1a = StrictMath.pow(z1,a); | |
| double z2a = StrictMath.pow(z2,a); | |
| double sum2 = 0.0; | |
| // check zero values for z2,z1 to avoid NaN, +/- Infinity | |
| if (z2 != 0.0) | |
| { | |
| for (int k=0;k<INFINITY;k++) | |
| { | |
| sum2 += (StrictMath.pow(-z2, k))/ | |
| ((a+k)*factorial(k)); | |
| } | |
| } | |
| double sum1 = 0.0; | |
| if (z1 != 0.0) | |
| { | |
| for (int k=0;k<INFINITY;k++) | |
| { | |
| sum1 += (StrictMath.pow(-z1, k))/ | |
| ((a+k)*factorial(k)); | |
| } | |
| } | |
| double result = (z2a*sum2); | |
| if (result != 0.0) | |
| result -= (z1a*sum1); | |
| else | |
| result = (z1a*sum1); | |
| return result; | |
| //return (z2a*sum2) - (z1a*sum1); | |
| } | |
| /** | |
| * Generalized Incomplete Gamma (2 args) | |
| * | |
| * @param a | |
| * @param z1 | |
| * @return | |
| */ | |
| public static double incGamma(double a, double z1) | |
| { | |
| if (z1 == 0.0) | |
| return Gamma(a); | |
| double result = StrictMath.pow(z1,a); | |
| double sum = 0.0; | |
| for (int k=0; k<100; k++) | |
| { | |
| sum += | |
| ((StrictMath.pow(-z1,k)/(a+k)*factorial(k))); | |
| } | |
| return Gamma(a) - (result*sum); | |
| } | |
| /** | |
| * Generalized Incomplete Gamma (2 args) | |
| * alternate experimental version | |
| * ref: Spanier & Oldham | |
| * @param v | |
| * @param x | |
| * @return | |
| */ | |
| public static double incGamma2(double v, double x) | |
| { | |
| double g,p,j,f = 0.0; | |
| g = p = 1.0; | |
| double vl = v+1; | |
| while(vl <= 2.) | |
| { | |
| g *= x; | |
| p *= vl; | |
| p += g; | |
| vl++; | |
| } | |
| double j2 = (5*(3+StrictMath.abs(x))/2); | |
| j = j2 - frac(j2); | |
| f = 1/(j+vl-x); | |
| do { | |
| j -= 1; | |
| f = (f*x+1)/(j+vl); | |
| } while (j > 0.0); | |
| p += (f*g*x); | |
| double vl2 = StrictMath.pow(vl, 2); | |
| g = ((1-2/7*vl2)*(1-2/3*vl2))/(30*vl2); | |
| g = ((g-1)/12*vl) - vl*(StrictMath.log(vl)-1); | |
| f = p*StrictMath.exp(g-x)*StrictMath.sqrt(vl/(2*StrictMath.PI)); | |
| return f; | |
| } | |
| /** | |
| * Regularized Incomplete Gamma | |
| * ref: http://functions.wolfram.com/06.09.02.0001.01 | |
| * @param a | |
| * @param z1 | |
| * @param z2 | |
| * @return | |
| */ | |
| public static double regIncGamma(double a, double z1, double z2) | |
| { | |
| return incGamma(a, z1, z2)/Gamma(a); | |
| } | |
| /** | |
| * Incomplete gamma function | |
| * 1 / Gamma(a) * Int_0^x exp(-t) t^(a-1) dt | |
| * ref: | |
| * erf.c - public domain implementation of error function erf(3m) | |
| * reference - Haruhiko Okumura: C-gengo niyoru saishin algorithm jiten | |
| * New Algorithm handbook in C language) (Gijyutsu hyouronsha, Tokyo, 1991) p.227 [in Japanese] | |
| * | |
| * @param a | |
| * @param x | |
| * @param loggamma_a | |
| * @return | |
| */ | |
| public static double p_gamma(double a, double x, double loggamma_a) | |
| { | |
| int k; | |
| double result, term, previous; | |
| //if (x >= 1 + a) | |
| // return 1 - q_gamma(a, x, loggamma_a); | |
| if (x == 0) | |
| return 0; | |
| result = term = StrictMath.exp(a * StrictMath.log(x) - x - loggamma_a) / a; | |
| for (k = 1; k < 1000; k++) { | |
| term *= x / (a + k); | |
| previous = result; | |
| result += term; | |
| if (result == previous) | |
| return result; | |
| } | |
| System.out.println("functions.java:%d:p_gamma() could not converge."); | |
| return result; | |
| } | |
| /** | |
| * Incomplete gamma function | |
| * 1 / Gamma(a) * Int_x^inf exp(-t) t^(a-1) dt | |
| * ref: | |
| * erf.c - public domain implementation of error function erf(3m) | |
| * reference - Haruhiko Okumura: C-gengo niyoru saishin algorithm jiten | |
| * New Algorithm handbook in C language) (Gijyutsu hyouronsha, Tokyo, 1991) p.227 [in Japanese] | |
| * | |
| * @param a | |
| * @param x | |
| * @param loggamma_a | |
| * @return | |
| */ | |
| public static double q_gamma(double a, double x, double loggamma_a) | |
| { | |
| int k; | |
| double result, w, temp, previous; | |
| double la = 1, lb = 1 + x - a; /* Laguerre polynomial */ | |
| //if (x < 1 + a) | |
| // return 1 - p_gamma(a, x, loggamma_a); | |
| w = StrictMath.exp(a * StrictMath.log(x) - x - loggamma_a); | |
| result = w / lb; | |
| for (k = 2; k < 1000; k++) { | |
| temp = ((k - 1 - a) * (lb - la) + (k + x) * lb) / k; | |
| la = lb; lb = temp; | |
| w *= (k - 1 - a) / k; | |
| temp = w / (la * lb); | |
| previous = result; | |
| result += temp; | |
| if (result == previous) | |
| return result; | |
| } | |
| System.out.println("functions.java:%d:q_gamma() could not converge."); | |
| return result; | |
| } | |
| /** | |
| * return nth Bernoulli number in rational form | |
| * @param n | |
| * @return | |
| */ | |
| public static double BernoulliB(int n) | |
| { | |
| double Bn = 0.0; // return 0 if n is not even, >= 2 | |
| if ((n % 2) != 0) | |
| return Bn; | |
| if (n == 0) | |
| return 1; | |
| if (n == 1) | |
| return -0.5; | |
| double K = (2*factorial(n))/StrictMath.pow((2*StrictMath.PI), n); | |
| double d = 0.0; | |
| int pl = primes.length; | |
| int prod = 1; | |
| for (int i = 0; i < pl; i++) | |
| { | |
| if ((n % (primes[i]-1)) == 0) | |
| prod *= primes[i]; | |
| } | |
| d = prod; | |
| // intermediate work field for debugging | |
| //double nwork = StrictMath.pow((K*d), 1.0/(n-1)); | |
| double N = StrictMath.ceil(StrictMath.pow((K*d), 1.0/(n-1))); | |
| double z = 1.0; | |
| for (int i = 0; i < pl; i++) | |
| { | |
| if (primes[i] > N) | |
| break; | |
| if (i == 0) | |
| { | |
| z = StrictMath.pow(1-(StrictMath.pow(primes[i], -n)), -1); | |
| }else { | |
| z *= StrictMath.pow(1-(StrictMath.pow(primes[i], -n)), -1); | |
| } | |
| } | |
| double a = StrictMath.pow((-1), (n/2)+1)*StrictMath.ceil(d*K*z); | |
| Bn = a/d; | |
| return Bn; | |
| } | |
| /** | |
| * return nth Bernoulli polynomial of x using Fourier expansion | |
| * ref: Abramowitz & Stegun 23.1.16 pp 805 | |
| * @param n | |
| * @param x | |
| * @return | |
| */ | |
| public static double BernoulliP (int n, double x) | |
| { | |
| // for n>1, 1>=x>=0 | |
| // for n=1, 1>x>0 | |
| // for x=0 return the nth Bernoulli number | |
| if (x==0.0) { | |
| return BernoulliB(n); | |
| } | |
| double K = -2*(factorial(n))/StrictMath.pow((2*StrictMath.PI), n); | |
| double S = 0.0; | |
| for (int k=1; k<100001; k++) | |
| { | |
| S += | |
| StrictMath.cos((2*StrictMath.PI*k*x)-(.5*StrictMath.PI*n))/ | |
| StrictMath.pow(k, n); | |
| } | |
| return K*S; | |
| } | |
| /** | |
| * Periodic Bernoulli function - Bn({x}) | |
| * @param n | |
| * @param x | |
| * @return | |
| */ | |
| public static double BBn(int n, double x) { | |
| return BernoulliP(n, frac(x)); | |
| } | |
| /** | |
| * return nth Euler number in rational form | |
| * ref: http://functions.wolfram.com/04.12.06.0001.01 | |
| * @param n | |
| * @return | |
| */ | |
| public static double EulerE(int n) | |
| { | |
| // return fixed point values | |
| if (n == 0) | |
| return 1; | |
| else if (even(n) != 1) | |
| return 0; | |
| double K = (StrictMath.pow(2,n+2)*factorial(n)) / | |
| StrictMath.pow(StrictMath.PI, n+1); | |
| double S = 0.0; | |
| for (int k=0; k<100001; k++) | |
| { | |
| S += (1./StrictMath.pow(2*k+1,n+1)) * | |
| StrictMath.cos((k*StrictMath.PI)-((n*StrictMath.PI)/2)); | |
| } | |
| return K*S; | |
| } | |
| /** | |
| * return nth Euler polynomial of x using Fourier expansion | |
| * ref: | |
| * - Abramowitz & Stegun 23.1.16 pp 805, table 23.1 | |
| * - http://functions.wolfram.com/05.13.06.0001.01 | |
| * @param n | |
| * @param x | |
| * @return | |
| */ | |
| public static double EulerP(int n, double x) | |
| { | |
| // for n>0, 1>=x>=0 | |
| // for n=0, 1>x>0 | |
| // for small n (<=15) calculate using coefficients - A&S table 23.1 pp 809 | |
| if (n == 0) | |
| return 1.0; | |
| else if (n == 1) | |
| return x - .5; | |
| else if (n == 2) | |
| return x*x - x; | |
| else if (n == 3) | |
| return StrictMath.pow(x,3) - | |
| (StrictMath.pow(x,2)*1.5) + .25; | |
| else if (n == 4) | |
| return StrictMath.pow(x,4) - | |
| (2*StrictMath.pow(x,3)) + x; | |
| else if (n == 5) | |
| return StrictMath.pow(x,5) - | |
| (StrictMath.pow(x,4)*2.5) + (StrictMath.pow(x,2)*2.5) - .5; | |
| else if (n == 6) | |
| return StrictMath.pow(x, 6) - (3*StrictMath.pow(x, 5)) + | |
| (5*StrictMath.pow(x, 3)) - 3*x; | |
| else if (n == 7) | |
| return StrictMath.pow(x, 7) - (StrictMath.pow(x, 6)*3.5) + | |
| (StrictMath.pow(x, 4)*8.75) - | |
| (StrictMath.pow(x, 2)*10.5) + 2.125; | |
| else if (n == 8) | |
| return StrictMath.pow(x, 8) - (4*StrictMath.pow(x, 7)) + | |
| (14*StrictMath.pow(x, 5)) - | |
| (28*StrictMath.pow(x, 3)) + 17*x; | |
| else if (n == 9) | |
| return StrictMath.pow(x, 9) - (StrictMath.pow(x, 8)*4.5) + | |
| (21*StrictMath.pow(x, 6)) - | |
| (63*StrictMath.pow(x, 4)) + | |
| (StrictMath.pow(x, 2)*76.5) - 15.5; | |
| else if (n == 10) | |
| return StrictMath.pow(x, 10) - (5*StrictMath.pow(x, 9)) + | |
| (30*StrictMath.pow(x, 7)) - | |
| (126*StrictMath.pow(x, 5)) + | |
| (255*StrictMath.pow(x, 3)) - | |
| (155*x); | |
| else if (n == 11) | |
| return StrictMath.pow(x, 11) - (StrictMath.pow(x, 10)*5.5) + | |
| (StrictMath.pow(x, 8)*41.25) - | |
| (231*StrictMath.pow(x, 6)) + | |
| (StrictMath.pow(x, 4)*701.25) - | |
| (StrictMath.pow(x, 2)*852.5) + 172.75; | |
| else if (n == 12) | |
| return StrictMath.pow(x, 12) - (6*StrictMath.pow(x, 11)) + | |
| (55*StrictMath.pow(x, 9)) - | |
| (396*StrictMath.pow(x, 7)) + | |
| (1683*StrictMath.pow(x, 5)) - | |
| (3410*StrictMath.pow(x, 3)) + 2073*x; | |
| else if (n == 13) | |
| return StrictMath.pow(x, 13) - (StrictMath.pow(x, 12)*6.5) + | |
| (StrictMath.pow(x, 10)*71.5) - | |
| (StrictMath.pow(x, 8)*643.5) + | |
| (StrictMath.pow(x, 6)*3646.5) - | |
| (StrictMath.pow(x, 4)*11082.5) + | |
| (StrictMath.pow(x, 2)*13474.5) - 2730.5; | |
| else if (n == 14) | |
| return StrictMath.pow(x, 14) - (7*StrictMath.pow(x, 13)) + | |
| (91*StrictMath.pow(x, 11)) - | |
| (1001*StrictMath.pow(x, 9)) + | |
| (7293*StrictMath.pow(x, 7)) - | |
| (31031*StrictMath.pow(x, 5)) + | |
| (62881*StrictMath.pow(x, 3)) - 38227*x; | |
| else if (n == 15) | |
| return StrictMath.pow(x, 15) - (StrictMath.pow(x, 14)*7.5) + | |
| (StrictMath.pow(x, 12)*113.75) - | |
| (StrictMath.pow(x, 10)*1501.5) + | |
| (StrictMath.pow(x, 8)*13674.375) - | |
| (StrictMath.pow(x, 6)*77577.5) + | |
| (StrictMath.pow(x, 4)*235803.75) - | |
| (StrictMath.pow(x, 2)*286702.5) + 58098.0625; | |
| double K = 4*(factorial(n))/StrictMath.pow(StrictMath.PI, n+1); | |
| double S = 0.0; | |
| for (int k=0; k<100001; k++) | |
| { | |
| S += | |
| StrictMath.sin((2*k+1)*((StrictMath.PI*x)-.5*(StrictMath.PI*n)))/ | |
| StrictMath.pow(2*k+1, n+1); | |
| } | |
| return K*S; | |
| } | |
| /** | |
| * ref: Spanier & Oldham (?) | |
| * @param n | |
| * @param z | |
| * @return | |
| */ | |
| public static double Ep(int n, double z) | |
| { | |
| double s,f,g,j,x; | |
| x = z; | |
| s = f = 1; | |
| f = g = 0; | |
| while(x>1) { | |
| x--; | |
| g += 2*s*StrictMath.pow(x,n); | |
| s *= -1; | |
| } | |
| do { | |
| g += 2*s*StrictMath.pow(x,n); | |
| s *= -1; | |
| x++; | |
| } while (x<0); | |
| j = 1.5 + StrictMath.floor(30/n); | |
| do { | |
| f += StrictMath.sin((StrictMath.PI/2)*(4*j*x - n))/StrictMath.pow(j,n+1); | |
| j--; | |
| } while (j>0); | |
| f *= (4*s*factorial(n))/StrictMath.pow(2*StrictMath.PI, n+1); | |
| f += g; | |
| return f; | |
| } | |
| /** | |
| * Euler totient function | |
| * ref: | |
| * http://functions.wolfram.com/13.06.02.0001.01 | |
| * @param n | |
| * @return number of integers not exceeding n and (m,n)=1 | |
| */ | |
| public static int EulerPhi(int n) | |
| { | |
| int s=0; | |
| for (int k=1; k<=n; k++) | |
| { | |
| s += kdelta(gcd(n,k), 1); | |
| } | |
| return s; | |
| } | |
| /** | |
| * Lambert W(z) funtion - implemented using Lagrange Inversion Theorem | |
| * for |z| < 1/e | |
| * @param z | |
| * @return | |
| */ | |
| public static double LambertW1e(double z) | |
| { | |
| double S = 0.0; | |
| for (int k=1; k <= INFINITY; k++) | |
| { | |
| S += | |
| (StrictMath.pow(-k, k-1) * StrictMath.pow(z,k)) | |
| / factorial(k); | |
| } | |
| return S; | |
| } | |
| /** | |
| * Lambert W(z) function - Series approximation | |
| * ref: implemented from Python code found somewhere in the web | |
| * @param z | |
| * @return | |
| */ | |
| public static double LambertW(double z) | |
| { | |
| double S = 0.0; | |
| for (int n=1; n <= 100; n++) | |
| { | |
| double Se = S * StrictMath.pow(StrictMath.E, S); | |
| double S1e = (S+1) * | |
| StrictMath.pow(StrictMath.E, S); | |
| if (PRECISION > StrictMath.abs((z-Se)/S1e)) | |
| { | |
| return S; | |
| } | |
| S -= | |
| (Se-z) / (S1e - (S+2) * (Se-z) / (2*S+2)); | |
| } | |
| return S; | |
| } | |
| /** | |
| * return the generalized nth Laguerre polynomial | |
| * ref: Spanier & Oldham | |
| * @param n | |
| * @param m // zero will return regular Laguerre polynomial | |
| * @param x | |
| * @return | |
| */ | |
| public static double Lp(int n, double m, double x) | |
| { | |
| double h, f, j; | |
| h = f = j = 1; | |
| if (n == 0) | |
| return f; | |
| f = 1 + m - x; | |
| if (n == 1) | |
| return f; | |
| do { | |
| j++; | |
| double g = f; | |
| f = ((2*j+m - 1 - x)*g - (j+m - 1)*h)/j; | |
| h = g; | |
| } while (j<n); | |
| return f; | |
| } | |
| /** | |
| * return the nth Laguerre polynomial | |
| * | |
| * @param n | |
| * @param x | |
| * @return | |
| */ | |
| public static double Lp(int n, double x) | |
| { | |
| return Lp(n, 0, x); | |
| } | |
| /** | |
| * return the nth Pochhammer symbol (rising factorial) | |
| * @param n | |
| * @param x | |
| * @return | |
| */ | |
| public static double P(int n, double x) | |
| { | |
| return Gamma(n+x)/Gamma(x); | |
| } | |
| /** | |
| * falling factorial | |
| * @param n | |
| * @param x | |
| * @return | |
| */ | |
| public static double Pf(int n, int x) | |
| { | |
| return factorial(x)/factorial(x-n); | |
| } | |
| /** | |
| * Dirichlet eta fuction (aka Zeta alternating series) | |
| * ref: Weisstein, Eric W. "Dirichlet Eta Function." | |
| * From MathWorld--A Wolfram Web Resource. | |
| * http://mathworld.wolfram.com/DirichletEtaFunction.html | |
| * @param s | |
| * @return | |
| */ | |
| public static double eta(double s) | |
| { | |
| // return precomputed values | |
| if (s == 0) | |
| return .5; | |
| else if (s == 1) | |
| return logtwo; | |
| else if (s == 2) | |
| return pi2on12; | |
| else if (s > 125) | |
| // any value > 125 converges to 1 | |
| return 1.0; | |
| double result = 0.0; | |
| for (int k=1; k<INFINITY; k++) | |
| { | |
| result += StrictMath.pow(-1, k-1)/StrictMath.pow(k, s); | |
| } | |
| return result; | |
| } | |
| /** | |
| * Chebyshev polynomial of the 1st kind | |
| * ref: http://functions.wolfram.com/05.04.02.0001.01 | |
| * @param n | |
| * @param z | |
| * @return | |
| */ | |
| public static double Tn(int n, double z) | |
| { | |
| if (n == 0) | |
| return 1; | |
| else if (n == 1) | |
| return z; | |
| if (z == 0) | |
| if (even(n) == 1) | |
| return StrictMath.pow(-1, n); | |
| else | |
| return 0.0; | |
| double K = kdelta(n, 0)/2. + n/2.; | |
| double S = 0.0; | |
| double n2 = StrictMath.floor(n/2.); | |
| for (int k=1; k<=n2; k++) | |
| { | |
| S += ((StrictMath.pow(-1, k)*factorial(n-k-1)*StrictMath.pow(2*z, n-2*k))/ | |
| factorial(k)*factorial(n-2*k)) + | |
| StrictMath.pow(2, n-1)*StrictMath.pow(z, n); | |
| } | |
| return K*S; | |
| } | |
| /** | |
| * Chebyshev polynomial of the 2nd kind | |
| * ref: http://functions.wolfram.com/05.05.02.0001.01 | |
| * @param n | |
| * @param z | |
| * @return | |
| */ | |
| public static double Un(int n, double z) | |
| { | |
| if (n == 0) | |
| return 1; | |
| else if (n == 1) | |
| return z*2; | |
| if (z == 0) | |
| if (even(n) == 1) | |
| return StrictMath.pow(-1, n); | |
| else | |
| return 0.0; | |
| double S = 0.0; | |
| double n2 = StrictMath.floor(n/2.); | |
| for (int k=1; k<=n2; k++) | |
| { | |
| S += ((StrictMath.pow(-1, k)*factorial(n-k-1)*StrictMath.pow(2*z, n-2*k))/ | |
| factorial(k)*factorial(n-2*k)); | |
| } | |
| return S; | |
| } | |
| /** | |
| * Chebyshev polynomial of the 1st kind | |
| * ref: Spanier & Oldham (??) | |
| * @param n | |
| * @param x | |
| * @return | |
| */ | |
| public static double ChebyshevT(int n, double x) | |
| { | |
| int j = 0; | |
| double f,g,h; | |
| f = g = x; /* 2x fo Un(x) */ | |
| if (n == 1) | |
| return f; | |
| f = h = j = 1; | |
| if (n == 0) | |
| return f; | |
| do { | |
| j++; | |
| f = 2*x*g - h; | |
| if (j == n) | |
| break; | |
| h = g; | |
| g = f; | |
| } while (1==1); | |
| return f; | |
| } | |
| /** | |
| * Chebyshev polynomial of the 2nd kind | |
| * ref: Spanier & Oldham (??) | |
| * @param n | |
| * @param x | |
| * @return | |
| */ | |
| public static double ChebyshevU(int n, double x) | |
| { | |
| int j = 0; | |
| double f,g,h; | |
| f = g = 2*x; /* 2x fo Un(x) */ | |
| if (n == 1) | |
| return f; | |
| f = h = j = 1; | |
| if (n == 0) | |
| return f; | |
| do { | |
| j++; | |
| f = 2*x*g - h; | |
| if (j == n) | |
| break; | |
| h = g; | |
| g = f; | |
| } while (1==1); | |
| return f; | |
| } | |
| /** | |
| * @param n | |
| * @return | |
| */ | |
| public static double StieltjesGp(int n) | |
| { | |
| // lim m->infinity | |
| int m = 101; | |
| double sum = 0.0; | |
| for (int k=1; k<=m; k++) | |
| { | |
| sum += (StrictMath.pow(-1, k)/k) * | |
| functions.BernoulliP( | |
| n+1,(StrictMath.log(k)/StrictMath.log(2))); | |
| } | |
| return (StrictMath.pow(StrictMath.log(2),n)/n+1) * sum; | |
| } | |
| /** | |
| * @param n | |
| * @return | |
| */ | |
| public static double StieltjesG(int n) | |
| { | |
| // lim m->infinity | |
| int m = 101; | |
| double sum = 0.0; | |
| double t2 = (StrictMath.pow(StrictMath.log(m),n+1)/n+1); | |
| for (int k=1; k<=m; k++) | |
| { | |
| sum += (StrictMath.pow(StrictMath.log(k), n)/k); | |
| } | |
| return (StrictMath.pow(-1,n)/factorial(n)) * (sum - t2); | |
| } | |
| /** | |
| * diGamma Psi asymptotic approximation using Bernoulli numbers | |
| * @param x | |
| * @return | |
| */ | |
| public static double PolyGamma(double x) | |
| { | |
| double sum = 0.0; | |
| // special values - for x=1, return -EM constant | |
| if (x == 1.0) | |
| return -1*constants.EM; | |
| for (int n=1; n<10; n++) | |
| { | |
| sum += functions.BernoulliB(2*n)/(2*n*(StrictMath.pow(x, 2*n))); | |
| } | |
| return (StrictMath.log(x)) - (1.0/(2.0*x)) - sum; | |
| } | |
| /** | |
| * diGamma Psi | |
| * ref: Spanier & Oldham | |
| * @param x | |
| * @return psi(x) | |
| */ | |
| public static double PolyGammaP(double x) | |
| { | |
| double f = StrictMath.pow(10, 99); | |
| double g = 0.0; | |
| if (x == 0.0) | |
| return f; | |
| do | |
| { | |
| g += 1/x; | |
| x++; | |
| } while(x<5); | |
| f = 1.0 + ((((0.46/StrictMath.pow(x, 2))-1)))/(10*StrictMath.pow(x, 2)); | |
| f = -((f/(6*x)+1)/(2*x)) + StrictMath.log(x) - g; | |
| return f; | |
| } | |
| /** | |
| * PolyGamma Psi | |
| * ref: Spanier & Oldham | |
| * @param n | |
| * @param x | |
| * @return | |
| */ | |
| public static double PolyGamma(int n, double x) | |
| { | |
| double f = StrictMath.pow(10, 99); | |
| double g = 0.0; | |
| double h = 0.0; | |
| int j = 0; | |
| if (x == 0.0) | |
| return f; | |
| do { | |
| h = -1/x; | |
| j = n; | |
| do { | |
| h = -(h*j)/x; | |
| j--; | |
| } while (j != 0); | |
| g += h; | |
| x += 1; | |
| } while (x<5); | |
| f = (1 - ((n+4)*(n+5))/(45.0*StrictMath.pow(x, 2)))/(60.0*StrictMath.pow(x, 2)); | |
| f = ((n+1)*(1-(f*(n+2)*(n+3))))/(6*x); | |
| f = -((1/n)+((f+1)/(2*x))); | |
| j = n; | |
| do { | |
| f = (-f*j)/x; | |
| j--; | |
| } while(j != 0); | |
| f = f+g; | |
| return f; | |
| } | |
| /** | |
| * PolyGamma Psi | |
| * ref: http://functions.wolfram.com/06.15.02.0002.02 | |
| * @param n (N+) | |
| * @param z | |
| * @return | |
| */ | |
| public static double PolyGamma2(int n, double z) | |
| { | |
| double t1 = StrictMath.pow(-1, n+1); | |
| double t2 = factorial(n); | |
| double sum = 0.0; | |
| //TODO | |
| for (int k=0; k<INFINITY; k++) | |
| { | |
| sum += 1/StrictMath.pow(k+z,n+1); | |
| } | |
| return t1*t2*sum; | |
| } | |
| /** | |
| * simple algorithm for generating an arbitrary polynomial of degree n | |
| * that does not vanish at -1 | |
| * ref: "An Efficient Algorithm for the Riemann Zeta Function", P. Borwein | |
| * @param n | |
| * @param x | |
| * @return | |
| */ | |
| public static double Pn(int n, double x) { | |
| return StrictMath.pow(x, n) * StrictMath.pow(1-x, n); | |
| } | |
| /** | |
| * @param args | |
| */ | |
| public static void main(String[] args) { | |
| // TODO Auto-generated method stub | |
| //double x = Double.parseDouble(args[0]); | |
| int n = 1; | |
| try { | |
| n = Integer.parseInt(args[0]); | |
| } catch (NumberFormatException e) { | |
| // TODO Auto-generated catch block | |
| } | |
| for (int i=1; i<=n; i++) { | |
| System.out.println( | |
| "moebiusmu for " + i + "=" + functions.moebiusmu(i)); | |
| } | |
| // for (int i=0; i<100; i++) { | |
| // int n; | |
| // try { | |
| // n = i; | |
| // } catch (NumberFormatException e) { | |
| // break; | |
| // } | |
| // System.out.println( | |
| // "exp(lambda(n)) for " + i + "=" | |
| // + StrictMath.round(StrictMath.exp(functions.mangoldtLambda(n)))); | |
| // } | |
| // for (int i=0; i<100; i++) { | |
| // int n = i; | |
| // | |
| // System.out.println( | |
| // "psi for " + i + "=" + functions.psi(n)); | |
| // } | |
| //System.out.println("e^pi: " + StrictMath.pow(StrictMath.E, StrictMath.PI)); | |
| // test lambda for the first 95 values | |
| // for (int i=0; i<96; i++) { | |
| // System.out.println("lambda(" + i + "): " + mangoldtLambda(i)); | |
| // } | |
| // test psi for first 10 values | |
| //for (int i=0; i<10; i++) { | |
| // System.out.println("psi(" + i + "): " + psi(i)); | |
| //} | |
| // test mobius mu for the first 20 values | |
| // for (int i=0; i<40; i++) { | |
| // System.out.println("mobiusmu(" + i + "): " + moebiusmu(i)); | |
| // } | |
| // test mertens M for some large n | |
| //int n = Integer.parseInt(args[0]); | |
| // for (int n=0; n<82; n++) { | |
| // System.out.println("M(" + n + "):" + M(n)); | |
| // } | |
| // int n = c | |
| // System.out.println("factorial(" + n + "): " + factorial(n)); | |
| // | |
| // int N = Integer.parseInt(args[0]); | |
| // double s = Double.parseDouble(args[1]); | |
| // System.out.println("PI(" + s + "," + N + "): " + PI(s,N)); | |
| /* | |
| int n = 50; | |
| for (int i=0; i<=n; i++) | |
| { | |
| if (even(i) == 1) | |
| { | |
| System.out.print("BernoulliB(" + i + "):" + BernoulliB(i)); | |
| System.out.print("\n"); | |
| } | |
| } | |
| */ | |
| //double x = Double.parseDouble(args[1]); | |
| //System.out.print("BernoulliP(" + n + ", " + x + "):" + BernoulliP(n, x)); | |
| // test Lambert W function | |
| // double z = Double.parseDouble(args[0]); | |
| // System.out.println("LambertW(" + z + "): " + LambertW(z)); | |
| // try a gram point approximation using the Lambert W function | |
| // double gn = 2*StrictMath.PI*StrictMath.exp(1+LambertW((8*z+1)/8*StrictMath.E)); | |
| // System.out.println("gram point ~~ " + gn); | |
| // test beta function | |
| //System.out.println("beta(2,4): " + beta(2.,4.)); | |
| // test erf,erfc | |
| //System.out.println("erf(.15): " + erf(.15)); | |
| //System.out.println("erfc(.15): " + erfc(.15)); | |
| //System.out.println("erf+erfc(.15): " + (erf(.15)+erfc(.15))); | |
| // test gamma | |
| /* | |
| System.out.println("Gamma(.5): " + Gamma(.5)); | |
| System.out.println("Gamma(0.6): " + Gamma(0.6)); | |
| System.out.println("Gamma(7.4): " + Gamma(7.4)); | |
| System.out.println("Gamma(-4.2): " + Gamma(-4.2)); | |
| */ | |
| // test inc gamma | |
| /* | |
| System.out.println("incGamma(.5,4.84): " + incGamma(.5,4.84)); | |
| System.out.println("incGamma(.5,4.84,0): " + incGamma(.5,4.84,0)); | |
| System.out.println("p_gamma(.5,4.84): " + p_gamma(.5,4.84,0)); | |
| System.out.println("q_gamma(.5,4.84): " + q_gamma(.5,4.84,0)); | |
| System.out.println("incGamma2(.5,-0.49): " + incGamma2(.5,-0.49)); | |
| System.out.println("incGamma2(-3.9,0): " + incGamma2(-3.9,0)); | |
| System.out.println("incGamma2(1,Pi): " + incGamma2(1,StrictMath.PI)); | |
| */ | |
| // test inc gamma | |
| //System.out.println("incGamma(.5,4.84,Gamma(.5)): " + incGamma(.5,4.84,Gamma(.5))); | |
| // test reg inc gamma | |
| //System.out.println("regIncGamma(.5,0,1): " + regIncGamma(.5,0,1)); | |
| // test first n laguerre polynomials | |
| //double x = Double.parseDouble(args[0]); | |
| //for (int i=0; i<11; i++) { | |
| // int n = i; | |
| /* | |
| System.out.println( | |
| "L" + 0 + "(" + 2 + ")=" + functions.Lp(0,2)); | |
| System.out.println( | |
| "L" + 1 + "(" + 9 + ")=" + functions.Lp(1,9)); | |
| System.out.println( | |
| "L" + 6 + "(" + 8.5 + ")=" + functions.Lp(6,8.5)); | |
| System.out.println( | |
| "L" + 3 + "(2)(" + 1 + ")=" + functions.Lp(3,2,1)); | |
| */ | |
| //} | |
| // test pochhammer | |
| //System.out.println("P" + 1 + "(" + .5 + ")=" + functions.P(1,.5)); | |
| // test eta function 0-42 | |
| /* | |
| for (double s = 0; s<43; s++) | |
| { | |
| System.out.println("eta("+ s + "): " + eta(s)); | |
| } | |
| */ | |
| // test Euler number & polynomials | |
| //double x = 0.25; | |
| //for (n = 1; n<16; n++) | |
| //{ | |
| //System.out.print("\nEulerE(" + n + "):" + EulerE(n)); | |
| // System.out.print("\nEulerP(" + n + ", " + x + "):" + EulerP(n, x)); | |
| // System.out.print("\n(-1)^" + n + " * EulerP(" + n + "," + (1-x) + "):" + | |
| // StrictMath.pow(-1,n)*EulerP(n, 1-x)); | |
| //} | |
| // more Euler polynomial tests - Spanier & Oldham | |
| /* | |
| System.out.print("\nEulerP(" + 6 + ", " + 0.4 + "):" + EulerP(6, 0.4)); | |
| System.out.print("\nEulerP(" + 13 + ", " + 1 + "):" + EulerP(13, 1.)); | |
| System.out.print("\nEulerP(" + 8 + ", " + -1.5 + "):" + EulerP(8, -1.5)); | |
| System.out.print("\nEulerP(" + 9 + ", " + 3.14159 + "...):" + EulerP(9, StrictMath.PI)); | |
| */ | |
| // test Chebyshev polynomials | |
| /* | |
| double z = .2; | |
| for (int n = 0; n<13; n++) | |
| { | |
| //System.out.print("\nTn(" + n + ", " + z + "):" + Tn(n, z)); | |
| System.out.print("\nChebyshevT(" + n + ", " + z + "):" + ChebyshevT(n, z)); | |
| //System.out.print("\nUn(" + n + ", " + z + "):" + Un(n, z)); | |
| System.out.print("\nChebyshevU(" + n + ", " + z + "):" + ChebyshevU(n, z)); | |
| } | |
| System.out.print("\nChebyshevU(" + 5 + ", " + 0.5 + "):" + ChebyshevU(5, 0.5)); | |
| System.out.print("\nChebyshevU(" + 11 + ", " + -0.6 + "):" + ChebyshevU(11, -0.6)); | |
| */ | |
| // test Stieltjes constants 0-43 | |
| /* | |
| for (int n = 0; n<10; n++) | |
| { | |
| System.out.print("\nStieltjesG("+ n + "): " + StieltjesG(n)); | |
| } | |
| */ | |
| // test binomial | |
| //System.out.println("\n(n k): 5,3: " + Binomial(5,3)); | |
| // test PI | |
| //System.out.println("5!: " + factorial(5)); | |
| //System.out.println("PI(5):" + PI(5,100)); | |
| //System.out.println("PI(5.):" + PI(5.)); | |
| //System.out.println("0!=" + factorial(0)); | |
| // test bigOmega | |
| //System.out.println("factors of 2008:" + bigOmega(2008)); | |
| // test Liouville function 1-40 | |
| /* | |
| for (long l = 1; l<41; l++) | |
| { | |
| System.out.println("LLambda(" + l + "): " + LiouvilleLambda(l)); | |
| } | |
| // test L(n) function 1-80 | |
| for (int l = 1; l<81; l++) | |
| { | |
| System.out.println("L(" + l + "): " + L(l)); | |
| } | |
| */ | |
| // test polygamma | |
| /* | |
| for (double x = 1; x<10; x++) | |
| { | |
| System.out.println("PolyGamma(" + x + "): " + PolyGamma(x)); | |
| } | |
| System.out.println("PolyGamma(" + .5 + "): " + PolyGamma(.5)); | |
| for (int x = 1; x<10; x++) | |
| { | |
| System.out.println("PolyGammaP(" + x + "): " + PolyGammaP(x)); | |
| } | |
| System.out.println("PolyGammaP(" + .5 + "): " + PolyGammaP(.5)); | |
| System.out.println("PolyGammaP(" + -1.05 + "): " + PolyGammaP(-1.05)); | |
| System.out.println("PolyGammaP(" + 1.461632145 + "): " + PolyGammaP(1.461632145)); | |
| // test PolyGamma with derivatives | |
| System.out.println("PolyGamma(1,1): " + PolyGamma(1,1)); | |
| System.out.println("PolyGamma(1,5.5): " + PolyGamma(1,5.5)); | |
| System.out.println("PolyGamma(2,-0.5): " + PolyGamma(2,-0.5)); | |
| System.out.println("PolyGamma(2,5): " + PolyGamma(2,5)); | |
| System.out.println("PolyGamma(3,9): " + PolyGamma(3,9)); | |
| System.out.println("PolyGamma2(1,1): " + PolyGamma2(1,1)); | |
| System.out.println("PolyGamma2(1,5.5): " + PolyGamma2(1,5.5)); | |
| System.out.println("PolyGamma2(2,-0.5): " + PolyGamma2(2,-0.5)); | |
| System.out.println("PolyGamma2(2,5): " + PolyGamma2(2,5)); | |
| System.out.println("PolyGamma2(3,9): " + PolyGamma2(3,9)); | |
| */ | |
| // test stirling | |
| // for (int n=0; n <= 9; n++) | |
| // { | |
| // for (int k=0; k <= 9; k++) | |
| // { | |
| // System.out.println("Stirling(" + n + "," + k +") : " + Stirling1K(n,k)); | |
| // } | |
| // } | |
| } | |
| } |
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