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mXparser is a super easy, rich, fast and highly flexible math expression parser library (parser and evaluator of mathematical expressions / formulas provided as plain text / string). Software delivers easy to use API for JAVA, Android and C# .NET/MONO (Common Language Specification compliant: F#, Visual Basic, C++/CLI). *** If you find the software useful donation is something you might consider: https://mathparser.org/donate/ *** Scalar Scientific Calculator, Charts and Scripts, Scalar Lite: https://play.google.com/store/apps/details?id=org.mathparser.scalar.lite *** Scalar Pro: https://play.google.com/store/apps/details?id=org.mathparser.scalar.pro *** ScalarMath.org: https://scalarmath.org/ *** MathSpace.pl: https://mathspace.pl/ ***
/*
* @(#)Coefficients.java 4.4.3 2022-05-28
*
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package org.mariuszgromada.math.mxparser.mathcollection;
/**
* Coefficients - various coefficients supporting numerical computation.
*
* @author Mariusz Gromada
* [email protected]
* MathSpace.pl
* MathParser.org - mXparser project page
* mXparser on GitHub
* mXparser on SourceForge
* mXparser on Bitbucket
* mXparser on CodePlex
* Janet Sudoku - project web page
* Janet Sudoku on GitHub
* Janet Sudoku on CodePlex
* Janet Sudoku on SourceForge
* Janet Sudoku on BitBucket
* Scalar Free
* Scalar Pro
* ScalarMath.org
*
* @version 4.2.0
*/
final class Coefficients {
/*
* --------------------------------------
* COEFFICIENTS FOR METHOD erfImp
* --------------------------------------
*/
/**
* Polynomial coefficients for a numerator of erfImp
* calculation for erf(x) in the interval [1e-10, 0.5].
*/
static final double[] erfImpAn = {
0.00337916709551257388990745
,-0.00073695653048167948530905
,-0.374732337392919607868241
,0.0817442448733587196071743
,-0.0421089319936548595203468
,0.0070165709512095756344528
,-0.00495091255982435110337458
,0.000871646599037922480317225
};
/**
* Polynomial coefficients for adenominator of erfImp
* calculation for erf(x) in the interval [1e-10, 0.5].
*/
static final double[] erfImpAd = {
1
,-0.218088218087924645390535
,0.412542972725442099083918
,-0.0841891147873106755410271
,0.0655338856400241519690695
,-0.0120019604454941768171266
,0.00408165558926174048329689
,-0.000615900721557769691924509
};
/**
* Polynomial coefficients for a numerator in erfImp
* calculation for erfc(x) in the interval [0.5, 0.75].
*/
static final double[] erfImpBn = {
-0.0361790390718262471360258
,0.292251883444882683221149
,0.281447041797604512774415
,0.125610208862766947294894
,0.0274135028268930549240776
,0.00250839672168065762786937
};
/**
* Polynomial coefficients for a denominator in erfImp
* calculation for erfc(x) in the interval [0.5, 0.75].
*/
static final double[] erfImpBd = {
1
,1.8545005897903486499845
,1.43575803037831418074962
,0.582827658753036572454135
,0.124810476932949746447682
,0.0113724176546353285778481
};
/**
* Polynomial coefficients for a numerator in erfImp
* calculation for erfc(x) in the interval [0.75, 1.25].
*/
static final double[] erfImpCn = {
-0.0397876892611136856954425
,0.153165212467878293257683
,0.191260295600936245503129
,0.10276327061989304213645
,0.029637090615738836726027
,0.0046093486780275489468812
,0.000307607820348680180548455
};
/**
* Polynomial coefficients for a denominator in erfImp
* calculation for erfc(x) in the interval [0.75, 1.25].
*/
static final double[] erfImpCd = {
1
,1.95520072987627704987886
,1.64762317199384860109595
,0.768238607022126250082483
,0.209793185936509782784315
,0.0319569316899913392596356
,0.00213363160895785378615014
};
/**
* Polynomial coefficients for a numerator in erfImp
* calculation for erfc(x) in the interval [1.25, 2.25].
*/
static final double[] erfImpDn = {
-0.0300838560557949717328341
,0.0538578829844454508530552
,0.0726211541651914182692959
,0.0367628469888049348429018
,0.00964629015572527529605267
,0.00133453480075291076745275
,0.778087599782504251917881e-4
};
/**
* Polynomial coefficients for a denominator in erfImp
* calculation for erfc(x) in the interval [1.25, 2.25].
*/
static final double[] erfImpDd = {
1
,1.75967098147167528287343
,1.32883571437961120556307
,0.552528596508757581287907
,0.133793056941332861912279
,0.0179509645176280768640766
,0.00104712440019937356634038
,-0.106640381820357337177643e-7
};
/**
* Polynomial coefficients for a numerator in erfImp
* calculation for erfc(x) in the interval [2.25, 3.5].
*/
static final double[] erfImpEn = {
-0.0117907570137227847827732
,0.014262132090538809896674
,0.0202234435902960820020765
,0.00930668299990432009042239
,0.00213357802422065994322516
,0.00025022987386460102395382
,0.120534912219588189822126e-4
};
/**
* Polynomial coefficients for a denominator in erfImp
* calculation for erfc(x) in the interval [2.25, 3.5].
*/
static final double[] erfImpEd = {
1
,1.50376225203620482047419
,0.965397786204462896346934
,0.339265230476796681555511
,0.0689740649541569716897427
,0.00771060262491768307365526
,0.000371421101531069302990367
};
/**
* Polynomial coefficients for a numerator in erfImp
* calculation for erfc(x) in the interval [3.5, 5.25].
*/
static final double[] erfImpFn = {
-0.00546954795538729307482955
,0.00404190278731707110245394
,0.0054963369553161170521356
,0.00212616472603945399437862
,0.000394984014495083900689956
,0.365565477064442377259271e-4
,0.135485897109932323253786e-5
};
/**
* Polynomial coefficients for a denominator in erfImp
* calculation for erfc(x) in the interval [3.5, 5.25].
*/
static final double[] erfImpFd = {
1
,1.21019697773630784832251
,0.620914668221143886601045
,0.173038430661142762569515
,0.0276550813773432047594539
,0.00240625974424309709745382
,0.891811817251336577241006e-4
,-0.465528836283382684461025e-11
};
/**
* Polynomial coefficients for a numerator in erfImp
* calculation for erfc(x) in the interval [5.25, 8].
*/
static final double[] erfImpGn = {
-0.00270722535905778347999196
,0.0013187563425029400461378
,0.00119925933261002333923989
,0.00027849619811344664248235
,0.267822988218331849989363e-4
,0.923043672315028197865066e-6
};
/**
* Polynomial coefficients for a denominator in erfImp
* calculation for erfc(x) in the interval [5.25, 8].
*/
static final double[] erfImpGd = {
1
,0.814632808543141591118279
,0.268901665856299542168425
,0.0449877216103041118694989
,0.00381759663320248459168994
,0.000131571897888596914350697
,0.404815359675764138445257e-11
};
/**
* Polynomial coefficients for a numerator in erfImp
* calculation for erfc(x) in the interval [8, 11.5].
*/
static final double[] erfImpHn = {
-0.00109946720691742196814323
,0.000406425442750422675169153
,0.000274499489416900707787024
,0.465293770646659383436343e-4
,0.320955425395767463401993e-5
,0.778286018145020892261936e-7
};
/**
* Polynomial coefficients for a denominator in erfImp
* calculation for erfc(x) in the interval [8, 11.5].
*/
static final double[] erfImpHd = {
1, 0.588173710611846046373373
,0.139363331289409746077541
,0.0166329340417083678763028
,0.00100023921310234908642639
,0.24254837521587225125068e-4
};
/**
* Polynomial coefficients for a numerator in erfImp
* calculation for erfc(x) in the interval [11.5, 17].
*/
static final double[] erfImpIn = {
-0.00056907993601094962855594
,0.000169498540373762264416984
,0.518472354581100890120501e-4
,0.382819312231928859704678e-5
,0.824989931281894431781794e-7
};
/**
* Polynomial coefficients for a denominator in erfImp
* calculation for erfc(x) in the interval [11.5, 17].
*/
static final double[] erfImpId = {
1
,0.339637250051139347430323
,0.043472647870310663055044
,0.00248549335224637114641629
,0.535633305337152900549536e-4
,-0.117490944405459578783846e-12
};
/**
* Polynomial coefficients for a numerator in erfImp
* calculation for erfc(x) in the interval [17, 24].
*/
static final double[] erfImpJn = {
-0.000241313599483991337479091
,0.574224975202501512365975e-4
,0.115998962927383778460557e-4
,0.581762134402593739370875e-6
,0.853971555085673614607418e-8
};
/**
* Polynomial coefficients for a denominator in erfImp
* calculation for erfc(x) in the interval [17, 24].
*/
static final double[] erfImpJd = {
1
,0.233044138299687841018015
,0.0204186940546440312625597
,0.000797185647564398289151125
,0.117019281670172327758019e-4
};
/**
* Polynomial coefficients for a numerator in erfImp
* calculation for erfc(x) in the interval [24, 38].
*/
static final double[] erfImpKn = {
-0.000146674699277760365803642
,0.162666552112280519955647e-4
,0.269116248509165239294897e-5
,0.979584479468091935086972e-7
,0.101994647625723465722285e-8
};
/**
* Polynomial coefficients for a denominator in erfImp
* calculation for erfc(x) in the interval [24, 38].
*/
static final double[] erfImpKd = {
1
,0.165907812944847226546036
,0.0103361716191505884359634
,0.000286593026373868366935721
,0.298401570840900340874568e-5
};
/**
* Polynomial coefficients for a numerator in erfImp
* calculation for erfc(x) in the interval [38, 60].
*/
static final double[] erfImpLn = {
-0.583905797629771786720406e-4
,0.412510325105496173512992e-5
,0.431790922420250949096906e-6
,0.993365155590013193345569e-8
,0.653480510020104699270084e-10
};
/**
* Polynomial coefficients for a denominator in erfImp
* calculation for erfc(x) in the interval [38, 60].
*/
static final double[] erfImpLd = {
1
,0.105077086072039915406159
,0.00414278428675475620830226
,0.726338754644523769144108e-4
,0.477818471047398785369849e-6
};
/**
* Polynomial coefficients for a numerator in erfImp
* calculation for erfc(x) in the interval [60, 85].
*/
static final double[] erfImpMn = {
-0.196457797609229579459841e-4
,0.157243887666800692441195e-5
,0.543902511192700878690335e-7
,0.317472492369117710852685e-9
};
/**
* Polynomial coefficients for a denominator in erfImp
* calculation for erfc(x) in the interval [60, 85].
*/
static final double[] erfImpMd = {
1
,0.052803989240957632204885
,0.000926876069151753290378112
,0.541011723226630257077328e-5
,0.535093845803642394908747e-15
};
/**
* Polynomial coefficients for a numerator in erfImp
* calculation for erfc(x) in the interval [85, 110].
*/
static final double[] erfImpNn = {
-0.789224703978722689089794e-5
,0.622088451660986955124162e-6
,0.145728445676882396797184e-7
,0.603715505542715364529243e-10
};
/**
* Polynomial coefficients for a denominator in erfImp
* calculation for erfc(x) in the interval [85, 110].
*/
static final double[] erfImpNd = {
1
,0.0375328846356293715248719
,0.000467919535974625308126054
,0.193847039275845656900547e-5
};
/*
*
* --------------------------------------
* COEFFICIENTS FOR METHOD erfInvImp
* --------------------------------------
*/
/**
* Polynomial coefficients for a numerator of erfInvImp
* calculation for erf^-1(z) in the interval [0, 0.5].
*/
static final double[] ervInvImpAn = {
-0.000508781949658280665617
,-0.00836874819741736770379
,0.0334806625409744615033
,-0.0126926147662974029034
,-0.0365637971411762664006
,0.0219878681111168899165
,0.00822687874676915743155
,-0.00538772965071242932965
};
/**
* Polynomial coefficients for a denominator of erfInvImp
* calculation for erf^-1(z) in the interval [0, 0.5].
*/
static final double[] ervInvImpAd = {
1
,-0.970005043303290640362
,-1.56574558234175846809
,1.56221558398423026363
,0.662328840472002992063
,-0.71228902341542847553
,-0.0527396382340099713954
,0.0795283687341571680018
,-0.00233393759374190016776
,0.000886216390456424707504
};
/**
* Polynomial coefficients for a numerator of erfInvImp
* calculation for erf^-1(z) in the interval [0.5, 0.75].
*/
static final double[] ervInvImpBn = {
-0.202433508355938759655
,0.105264680699391713268
,8.37050328343119927838
,17.6447298408374015486
,-18.8510648058714251895
,-44.6382324441786960818
,17.445385985570866523
,21.1294655448340526258
,-3.67192254707729348546
};
/**
* Polynomial coefficients for a denominator of erfInvImp
* calculation for erf^-1(z) in the interval [0.5, 0.75].
*/
static final double[] ervInvImpBd = {
1
,6.24264124854247537712
,3.9713437953343869095
,-28.6608180499800029974
,-20.1432634680485188801
,48.5609213108739935468
,10.8268667355460159008
,-22.6436933413139721736
,1.72114765761200282724
};
/**
* Polynomial coefficients for a numerator of erfInvImp
* calculation for erf^-1(z) in the interval [0.75, 1] with x less than 3.
*/
static final double[] ervInvImpCn = {
-0.131102781679951906451
,-0.163794047193317060787
,0.117030156341995252019
,0.387079738972604337464
,0.337785538912035898924
,0.142869534408157156766
,0.0290157910005329060432
,0.00214558995388805277169
,-0.679465575181126350155e-6
,0.285225331782217055858e-7
,-0.681149956853776992068e-9
};
/**
* Polynomial coefficients for a denominator of erfInvImp
* calculation for erf^-1(z) in the interval [0.75, 1] with x less than 3.
*/
static final double[] ervInvImpCd = {
1
,3.46625407242567245975
,5.38168345707006855425
,4.77846592945843778382
,2.59301921623620271374
,0.848854343457902036425
,0.152264338295331783612
,0.01105924229346489121
};
/**
* Polynomial coefficients for a numerator of erfInvImp
* calculation for erf^-1(z) in the interval [0.75, 1] with x between 3 and 6.
*/
static final double[] ervInvImpDn = {
-0.0350353787183177984712
,-0.00222426529213447927281
,0.0185573306514231072324
,0.00950804701325919603619
,0.00187123492819559223345
,0.000157544617424960554631
,0.460469890584317994083e-5
,-0.230404776911882601748e-9
,0.266339227425782031962e-11
};
/**
* Polynomial coefficients for a denominator of erfInvImp
* calculation for erf^-1(z) in the interval [0.75, 1] with x between 3 and 6.
*/
static final double[] ervInvImpDd = {
1
,1.3653349817554063097
,0.762059164553623404043
,0.220091105764131249824
,0.0341589143670947727934
,0.00263861676657015992959
,0.764675292302794483503e-4
};
/**
* Polynomial coefficients for a numerator of erfInvImp
* calculation for erf^-1(z) in the interval [0.75, 1] with x between 6 and 18.
*/
static final double[] ervInvImpEn = {
-0.0167431005076633737133
,-0.00112951438745580278863
,0.00105628862152492910091
,0.000209386317487588078668
,0.149624783758342370182e-4
,0.449696789927706453732e-6
,0.462596163522878599135e-8
,-0.281128735628831791805e-13
,0.99055709973310326855e-16
};
/**
* Polynomial coefficients for a denominator of erfInvImp
* calculation for erf^-1(z) in the interval [0.75, 1] with x between 6 and 18.
*/
static final double[] ervInvImpEd = {
1, 0.591429344886417493481
,0.138151865749083321638
,0.0160746087093676504695
,0.000964011807005165528527
,0.275335474764726041141e-4
,0.282243172016108031869e-6
};
/**
* Polynomial coefficients for a numerator of erfInvImp
* calculation for erf^-1(z) in the interval [0.75, 1] with x between 18 and 44.
*/
static final double[] ervInvImpFn = {
-0.0024978212791898131227
,-0.779190719229053954292e-5
,0.254723037413027451751e-4
,0.162397777342510920873e-5
,0.396341011304801168516e-7
,0.411632831190944208473e-9
,0.145596286718675035587e-11
,-0.116765012397184275695e-17
};
/**
* Polynomial coefficients for a denominator of erfInvImp
* calculation for erf^-1(z) in the interval [0.75, 1] with x between 18 and 44.
*/
static final double[] ervInvImpFd = {
1
,0.207123112214422517181
,0.0169410838120975906478
,0.000690538265622684595676
,0.145007359818232637924e-4
,0.144437756628144157666e-6
,0.509761276599778486139e-9
};
/**
* Polynomial coefficients for a numerator of erfInvImp
* calculation for erf^-1(z) in the interval [0.75, 1] with x greater than 44.
*/
static final double[] ervInvImpGn = {
-0.000539042911019078575891
,-0.28398759004727721098e-6
,0.899465114892291446442e-6
,0.229345859265920864296e-7
,0.225561444863500149219e-9
,0.947846627503022684216e-12
,0.135880130108924861008e-14
,-0.348890393399948882918e-21
};
/**
* Polynomial coefficients for a denominator of erfInvImp
* calculation for erf^-1(z) in the interval [0.75, 1] with x greater than 44.
*/
static final double[] ervInvImpGd = {
1
,0.0845746234001899436914
,0.00282092984726264681981
,0.468292921940894236786e-4
,0.399968812193862100054e-6
,0.161809290887904476097e-8
,0.231558608310259605225e-11
};
/**
* Supporting function
* while Exponential integral function Ei(x) calculation
*/
static final double[] EI = {
1.915047433355013959531e2
,4.403798995348382689974e2
,1.037878290717089587658e3
,2.492228976241877759138e3
,6.071406374098611507965e3
,1.495953266639752885229e4
,3.719768849068903560439e4
,9.319251363396537129882e4
,2.349558524907683035782e5
,5.955609986708370018502e5
,1.516637894042516884433e6
,3.877904330597443502996e6
,9.950907251046844760026e6
,2.561565266405658882048e7
,6.612718635548492136250e7
,1.711446713003636684975e8
,4.439663698302712208698e8
,1.154115391849182948287e9
,3.005950906525548689841e9
,7.842940991898186370453e9
,2.049649711988081236484e10
,5.364511859231469415605e10
,1.405991957584069047340e11
,3.689732094072741970640e11
,9.694555759683939661662e11
,2.550043566357786926147e12
,6.714640184076497558707e12
,1.769803724411626854310e13
,4.669055014466159544500e13
,1.232852079912097685431e14
,3.257988998672263996790e14
,8.616388199965786544948e14
,2.280446200301902595341e15
,6.039718263611241578359e15
,1.600664914324504111070e16
,4.244796092136850759368e16
,1.126348290166966760275e17
,2.990444718632336675058e17
,7.943916035704453771510e17
,2.111342388647824195000e18
,5.614329680810343111535e18
,1.493630213112993142255e19
,3.975442747903744836007e19
,1.058563689713169096306e20
};
/**
* Coefficients for Lanchos Gamma function approximation
*/
static final double[] lanchosGamma = {
0.99999999999980993
,676.5203681218851
,-1259.1392167224028
,771.32342877765313
,-176.61502916214059
,12.507343278686905
,-0.13857109526572012
,9.9843695780195716e-6
,1.5056327351493116e-7
};
/**
* Coefficients for Log Gamma function approximation - A
*/
static final double[] logGammaA = {
8.11614167470508450300E-4
,-5.95061904284301438324E-4
,7.93650340457716943945E-4
,-2.77777777730099687205E-3
,8.33333333333331927722E-2
};
/**
* Coefficients for Log Gamma function approximation - B
*/
static final double[] logGammaB = {
-1.37825152569120859100E3
,-3.88016315134637840924E4
,-3.31612992738871184744E5
,-1.16237097492762307383E6
,-1.72173700820839662146E6
,-8.53555664245765465627E5
};
/**
* Coefficients for Log Gamma function approximation - C
*/
static final double[] logGammaC = {
-3.51815701436523470549E2
,-1.70642106651881159223E4
,-2.20528590553854454839E5
,-1.13933444367982507207E6
,-2.53252307177582951285E6
,-2.01889141433532773231E6
};
/**
* Coefficients for Lambert W function, series for q near zero
*/
static final double[] lambertWqNearZero = {
-1.0
,2.331643981597124203363536062168
,-1.812187885639363490240191647568
,1.936631114492359755363277457668
,-2.353551201881614516821543561516
,3.066858901050631912893148922704
,-4.175335600258177138854984177460
,5.858023729874774148815053846119
,-8.401032217523977370984161688514
,12.250753501314460424
,-18.100697012472442755
,27.029044799010561650
};
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