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GaussKronrodNodesWeights.h
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GaussKronrodNodesWeights.h
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/**
* \file GaussKronrodNodesWeights.h
* \sa R Piessens, E de Doncker-Kapenger, C Ueberhuber, D Kahaner, QUADPACK, A Subroutine Package
* for Automatic Integration, Springer Verlag, 1983.
*/
#ifndef EIGEN_QUADRATURE_KRONROD_H
#define EIGEN_QUADRATURE_KRONROD_H
namespace Eigen
{
/**
* \brief The abscissae and weights are given for the interval (-1,1).
* Because of symmetry, only the positive abscissae and their
* corresponding weights are given.
*
* \param abscissaeGaussKronrod15 The abscissae of the 15 point kronrod rule.
* \param abscissaeGaussKronrod21 The abscissae of the 21 point kronrod rule.
* \param abscissaeGaussKronrod31 The abscissae of the 31 point kronrod rule.
* \param abscissaeGaussKronrod41 The abscissae of the 41 point kronrod rule.
* \param abscissaeGaussKronrod51 The abscissae of the 51 point kronrod rule.
* \param abscissaeGaussKronrod61 The abscissae of the 61 point kronrod rule.
* \param abscissaeGaussKronrod71 The abscissae of the 71 point kronrod rule.
* \param abscissaeGaussKronrod81 The abscissae of the 81 point kronrod rule.
* \param abscissaeGaussKronrod91 The abscissae of the 91 point kronrod rule.
* \param abscissaeGaussKronrod101 The abscissae of the 101 point kronrod rule.
* \param abscissaeGaussKronrod121 The abscissae of the 121 point kronrod rule.
* \param abscissaeGaussKronrod201 The abscissae of the 201 point kronrod rule.
*
* \param weightsGaussKronrod15 The weights of the 15 point kronrod rule.
* \param weightsGaussKronrod21 The weights of the 21 point kronrod rule.
* \param weightsGaussKronrod31 The weights of the 31 point kronrod rule.
* \param weightsGaussKronrod41 The weights of the 41 point kronrod rule.
* \param weightsGaussKronrod51 The weights of the 51 point kronrod rule.
* \param weightsGaussKronrod61 The weights of the 61 point kronrod rule.
* \param weightsGaussKronrod71 The weights of the 71 point kronrod rule.
* \param weightsGaussKronrod81 The weights of the 81 point kronrod rule.
* \param weightsGaussKronrod91 The weights of the 91 point kronrod rule.
* \param weightsGaussKronrod101 The weights of the 101 point kronrod rule.
* \param weightsGaussKronrod121 The weights of the 121 point kronrod rule.
* \param weightsGaussKronrod201 The weights of the 201 point kronrod rule.
*
* \param abscissaeGauss15 The abscissae of the 7 point gauss rule.
* \param abscissaeGauss21 The abscissae of the 10 point gauss rule.
* \param abscissaeGauss31 The abscissae of the 15 point gauss rule.
* \param abscissaeGauss41 The abscissae of the 20 point gauss rule.
* \param abscissaeGauss51 The abscissae of the 25 point gauss rule.
* \param abscissaeGauss61 The abscissae of the 30 point gauss rule.
* \param abscissaeGauss71 The abscissae of the 35 point gauss rule.
* \param abscissaeGauss81 The abscissae of the 40 point gauss rule.
* \param abscissaeGauss91 The abscissae of the 45 point gauss rule.
* \param abscissaeGauss101 The abscissae of the 50 point gauss rule.
* \param abscissaeGauss121 The abscissae of the 60 point gauss rule.
* \param abscissaeGauss201 The abscissae of the 100 point gauss rule.
*
* \param weightsGauss15 The weights of the 7 point gauss rule.
* \param weightsGauss21 The weights of the 10 point gauss rule.
* \param weightsGauss31 The weights of the 15 point gauss rule.
* \param weightsGauss41 The weights of the 20 point gauss rule.
* \param weightsGauss51 The weights of the 25 point gauss rule.
* \param weightsGauss61 The weights of the 30 point gauss rule.
* \param weightsGauss71 The weights of the 35 point gauss rule.
* \param weightsGauss81 The weights of the 40 point gauss rule.
* \param weightsGauss91 The weights of the 45 point gauss rule.
* \param weightsGauss101 The weights of the 50 point gauss rule.
* \param weightsGauss121 The weights of the 60 point gauss rule.
* \param weightsGauss201 The weights of the 100 point gauss rule.
*/
template <typename Scalar>
class QuadratureKronrod
{
public:
static Array<Scalar, 8, 1> abscissaeGaussKronrod15;
static Array<Scalar, 8, 1> weightsGaussKronrod15;
static Array<Scalar, 4, 1> abscissaeGauss15;
static Array<Scalar, 4, 1> weightsGauss15;
static Array<Scalar, 11, 1> abscissaeGaussKronrod21;
static Array<Scalar, 11, 1> weightsGaussKronrod21;
static Array<Scalar, 5, 1> abscissaeGauss21;
static Array<Scalar, 5, 1> weightsGauss21;
static Array<Scalar, 16, 1> abscissaeGaussKronrod31;
static Array<Scalar, 16, 1> weightsGaussKronrod31;
static Array<Scalar, 8, 1> abscissaeGauss31;
static Array<Scalar, 8, 1> weightsGauss31;
static Array<Scalar, 21, 1> abscissaeGaussKronrod41;
static Array<Scalar, 21, 1> weightsGaussKronrod41;
static Array<Scalar, 10, 1> abscissaeGauss41;
static Array<Scalar, 10, 1> weightsGauss41;
static Array<Scalar, 26, 1> abscissaeGaussKronrod51;
static Array<Scalar, 26, 1> weightsGaussKronrod51;
static Array<Scalar, 13, 1> abscissaeGauss51;
static Array<Scalar, 13, 1> weightsGauss51;
static Array<Scalar, 31, 1> abscissaeGaussKronrod61;
static Array<Scalar, 31, 1> weightsGaussKronrod61;
static Array<Scalar, 15, 1> abscissaeGauss61;
static Array<Scalar, 15, 1> weightsGauss61;
static Array<Scalar, 36, 1> abscissaeGaussKronrod71;
static Array<Scalar, 36, 1> weightsGaussKronrod71;
static Array<Scalar, 18, 1> abscissaeGauss71;
static Array<Scalar, 18, 1> weightsGauss71;
static Array<Scalar, 41, 1> abscissaeGaussKronrod81;
static Array<Scalar, 41, 1> weightsGaussKronrod81;
static Array<Scalar, 20, 1> abscissaeGauss81;
static Array<Scalar, 20, 1> weightsGauss81;
static Array<Scalar, 46, 1> abscissaeGaussKronrod91;
static Array<Scalar, 46, 1> weightsGaussKronrod91;
static Array<Scalar, 23, 1> abscissaeGauss91;
static Array<Scalar, 23, 1> weightsGauss91;
static Array<Scalar, 51, 1> abscissaeGaussKronrod101;
static Array<Scalar, 51, 1> weightsGaussKronrod101;
static Array<Scalar, 25, 1> abscissaeGauss101;
static Array<Scalar, 25, 1> weightsGauss101;
static Array<Scalar, 61, 1> abscissaeGaussKronrod121;
static Array<Scalar, 61, 1> weightsGaussKronrod121;
static Array<Scalar, 30, 1> abscissaeGauss121;
static Array<Scalar, 30, 1> weightsGauss121;
static Array<Scalar, 101, 1> abscissaeGaussKronrod201;
static Array<Scalar, 101, 1> weightsGaussKronrod201;
static Array<Scalar, 50, 1> abscissaeGauss201;
static Array<Scalar, 50, 1> weightsGauss201;
static void computeNodesAndWeights()
{
QuadratureKronrod::computeForRule<7>(abscissaeGaussKronrod15, weightsGaussKronrod15, abscissaeGauss15, weightsGauss15);
QuadratureKronrod::computeForRule<10>(abscissaeGaussKronrod21, weightsGaussKronrod21, abscissaeGauss21, weightsGauss21);
QuadratureKronrod::computeForRule<15>(abscissaeGaussKronrod31, weightsGaussKronrod31, abscissaeGauss31, weightsGauss31);
QuadratureKronrod::computeForRule<20>(abscissaeGaussKronrod41, weightsGaussKronrod41, abscissaeGauss41, weightsGauss41);
QuadratureKronrod::computeForRule<25>(abscissaeGaussKronrod51, weightsGaussKronrod51, abscissaeGauss51, weightsGauss51);
QuadratureKronrod::computeForRule<30>(abscissaeGaussKronrod61, weightsGaussKronrod61, abscissaeGauss61, weightsGauss61);
QuadratureKronrod::computeForRule<35>(abscissaeGaussKronrod71, weightsGaussKronrod71, abscissaeGauss71, weightsGauss71);
QuadratureKronrod::computeForRule<40>(abscissaeGaussKronrod81, weightsGaussKronrod81, abscissaeGauss81, weightsGauss81);
QuadratureKronrod::computeForRule<45>(abscissaeGaussKronrod91, weightsGaussKronrod91, abscissaeGauss91, weightsGauss91);
QuadratureKronrod::computeForRule<50>(abscissaeGaussKronrod101, weightsGaussKronrod101, abscissaeGauss101, weightsGauss101);
QuadratureKronrod::computeForRule<60>(abscissaeGaussKronrod121, weightsGaussKronrod121, abscissaeGauss121, weightsGauss121);
QuadratureKronrod::computeForRule<100>(abscissaeGaussKronrod201, weightsGaussKronrod201, abscissaeGauss201, weightsGauss201);
}
template <Index N>
static void computeForRule(Array<Scalar, N+1, 1>& kronrodAbscissae, Array<Scalar, N+1, 1>& kronrodWeights,
Array<Scalar, (N+1)/2, 1>& gaussAbscissae, Array<Scalar, (N+1)/2, 1>& gaussWeights)
{
Eigen::Array<Scalar, Eigen::Dynamic, 1> xGK;
Eigen::Array<Scalar, Eigen::Dynamic, 1> wGK;
Eigen::Array<Scalar, Eigen::Dynamic, 1> xG;
Eigen::Array<Scalar, Eigen::Dynamic, 1> wG;
Eigen::LaurieGautschi<Scalar>::computeAbscissaeAndWeights(N,xGK,wGK,xG,wG);
//Eigen::Piessens<Scalar>::computeAbscissaeAndWeights(N,xGK,wGK,xG,wG);
//Eigen::Monegato<Scalar>::computeAbscissaeAndWeights(N,xGK,wGK,xG,wG);
for(size_t i=0; i<N+1; ++i)
{
kronrodAbscissae(i) = xGK(i);
kronrodWeights(i) = wGK(i);
}
for(size_t i=0; i<(N+1)/2; ++i)
{
gaussAbscissae(i) = xG(i);
gaussWeights(i) = wG(i);
}
}
};
// Nodes and Weights - Rule 15
template <typename Scalar>
Array<Scalar, 8, 1> QuadratureKronrod<Scalar>::abscissaeGaussKronrod15 =
(Array<Scalar, 8, 1>() <<
0.99145537112081263920685469752632851664204433837033,
0.94910791234275852452618968404785126240077093767062,
0.86486442335976907278971278864092620121097230707409,
0.74153118559939443986386477328078840707414764714139,
0.58608723546769113029414483825872959843678075060436,
0.40584515137739716690660641207696146334738201409937,
0.20778495500789846760068940377324491347978440714517,
0.00000000000000000000000000000000000000000000000000
).finished();
template <typename Scalar>
Array<Scalar, 8, 1> QuadratureKronrod<Scalar>::weightsGaussKronrod15 =
(Array<Scalar, 8, 1>() <<
0.02293532201052922496373200805896959199356081127575,
0.06309209262997855329070066318920428666507115721155,
0.10479001032225018383987632254151801744375665421383,
0.14065325971552591874518959051023792039988975724800,
0.16900472663926790282658342659855028410624490030294,
0.19035057806478540991325640242101368282607807545536,
0.20443294007529889241416199923464908471651760418072,
0.20948214108472782801299917489171426369776208022370
).finished();
template <typename Scalar>
Array<Scalar, 4, 1> QuadratureKronrod<Scalar>::abscissaeGauss15 =
(Array<Scalar, 4, 1>() <<
0.94910791234275852452618968404785126240077093767062,
0.74153118559939443986386477328078840707414764714139,
0.40584515137739716690660641207696146334738201409937,
0.00000000000000000000000000000000000000000000000000
).finished();
template <typename Scalar>
Array<Scalar, 4, 1> QuadratureKronrod<Scalar>::weightsGauss15 =
(Array<Scalar, 4, 1>() <<
0.12948496616886969327061143267908201832858740225995,
0.27970539148927666790146777142377958248692506522660,
0.38183005050511894495036977548897513387836508353386,
0.41795918367346938775510204081632653061224489795918
).finished();
// Nodes and Weights - Rule 21
template <typename Scalar>
Array<Scalar, 11, 1> QuadratureKronrod<Scalar>::abscissaeGaussKronrod21 =
(Array<Scalar, 11, 1>() <<
0.99565716302580808073552728068900284792126058721948,
0.97390652851717172007796401208445205342826994669238,
0.93015749135570822600120718005950834622516790998194,
0.86506336668898451073209668842349304852754301496533,
0.78081772658641689706371757834504237716340752029816,
0.67940956829902440623432736511487357576929471183481,
0.56275713466860468333900009927269414084301388194197,
0.43339539412924719079926594316578416220007183765625,
0.29439286270146019813112660310386556616268662515696,
0.14887433898163121088482600112971998461756485942069,
0.00000000000000000000000000000000000000000000000000
).finished();
template <typename Scalar>
Array<Scalar, 11, 1> QuadratureKronrod<Scalar>::weightsGaussKronrod21 =
(Array<Scalar, 11, 1>() <<
0.01169463886737187427806439606219204839621733248193,
0.03255816230796472747881897245938976061738893984566,
0.05475589657435199603138130024458017637372111405833,
0.07503967481091995276704314091619000939521938200091,
0.09312545458369760553506546508336634439001882888076,
0.10938715880229764189921059032580496027181329983435,
0.12349197626206585107795810983107415951230034952865,
0.13470921731147332592805400177170683276099191300856,
0.14277593857706008079709427313871706088597905653191,
0.14773910490133849137484151597206804552373162548521,
0.14944555400291690566493646838982120374523631668747
).finished();
template <typename Scalar>
Array<Scalar, 5, 1> QuadratureKronrod<Scalar>::abscissaeGauss21 =
(Array<Scalar, 5, 1>() <<
0.97390652851717172007796401208445205342826994669238,
0.86506336668898451073209668842349304852754301496533,
0.67940956829902440623432736511487357576929471183481,
0.43339539412924719079926594316578416220007183765625,
0.14887433898163121088482600112971998461756485942069
).finished();
template <typename Scalar>
Array<Scalar, 5, 1> QuadratureKronrod<Scalar>::weightsGauss21 =
(Array<Scalar, 5, 1>() <<
0.06667134430868813759356880989333179285786483432016,
0.14945134915058059314577633965769733240255663966943,
0.21908636251598204399553493422816319245877187052268,
0.26926671930999635509122692156946935285975993846088,
0.29552422471475287017389299465133832942104671702685
).finished();
// Nodes and Weights - Rule 31
template <typename Scalar>
Array<Scalar, 16, 1> QuadratureKronrod<Scalar>::abscissaeGaussKronrod31 =
(Array<Scalar, 16, 1>() <<
0.99800229869339706028517284015227120907340644231556,
0.98799251802048542848956571858661258114697281712376,
0.96773907567913913425734797878433722528335733730013,
0.93727339240070590430775894771020947124399627351530,
0.89726453234408190088250965645449588283177871149443,
0.84820658341042721620064832077421685136625617473699,
0.79041850144246593296764929481794734686214051995698,
0.72441773136017004741618605461393800963089929458410,
0.65099674129741697053373589531327469254694822609260,
0.57097217260853884753722673725391064123838639628275,
0.48508186364023968069365574023235061286633893089407,
0.39415134707756336989720737098104546836275277615870,
0.29918000715316881216678002426638896266160338274382,
0.20119409399743452230062830339459620781283645446264,
0.10114206691871749902707423144739233878745105740164,
0.00000000000000000000000000000000000000000000000000
).finished();
template <typename Scalar>
Array<Scalar, 16, 1> QuadratureKronrod<Scalar>::weightsGaussKronrod31 =
(Array<Scalar, 16, 1>() <<
0.00537747987292334898779205143012764981830804024313,
0.01500794732931612253837476307580726809463943643739,
0.02546084732671532018687400101965335939727174504686,
0.03534636079137584622203794847836004812263067899242,
0.04458975132476487660822729937327969022325664966792,
0.05348152469092808726534314723943029677155476094712,
0.06200956780067064028513923096080293219040000421033,
0.06985412131872825870952007709914747578604543514067,
0.07684968075772037889443277748265900672210910116795,
0.08308050282313302103828924728610378960155418825337,
0.08856444305621177064727544369377430321226673269066,
0.09312659817082532122548687274734571856192788132132,
0.09664272698362367850517990762758933513665656863050,
0.09917359872179195933239317348460313105956726081671,
0.10076984552387559504494666261756972191634838013536,
0.10133000701479154901737479276749254677092627259660
).finished();
template <typename Scalar>
Array<Scalar, 8, 1> QuadratureKronrod<Scalar>::abscissaeGauss31 =
(Array<Scalar, 8, 1>() <<
0.98799251802048542848956571858661258114697281712376,
0.93727339240070590430775894771020947124399627351530,
0.84820658341042721620064832077421685136625617473699,
0.72441773136017004741618605461393800963089929458410,
0.57097217260853884753722673725391064123838639628275,
0.39415134707756336989720737098104546836275277615870,
0.20119409399743452230062830339459620781283645446264,
0.00000000000000000000000000000000000000000000000000
).finished();
template <typename Scalar>
Array<Scalar, 8, 1> QuadratureKronrod<Scalar>::weightsGauss31 =
(Array<Scalar, 8, 1>() <<
0.03075324199611726835462839357720441772174814483343,
0.07036604748810812470926741645066733846670803275433,
0.10715922046717193501186954668586930341554371575810,
0.13957067792615431444780479451102832252085027531551,
0.16626920581699393355320086048120881113090018009841,
0.18616100001556221102680056186642282450622601227793,
0.19843148532711157645611832644383932481869255995754,
0.20257824192556127288062019996751931483866215800948
).finished();
// Nodes and Weights - Rule 41
template <typename Scalar>
Array<Scalar, 21, 1> QuadratureKronrod<Scalar>::abscissaeGaussKronrod41 =
(Array<Scalar, 21, 1>() <<
0.99885903158827766383831557654586300999957020432630,
0.99312859918509492478612238847132027822264713090166,
0.98150787745025025919334299472021694456725093981024,
0.96397192727791379126766613119727722191206032780619,
0.94082263383175475351998272221244338027429557377965,
0.91223442825132590586775244120329811304918479742369,
0.87827681125228197607744299511307846671124526828251,
0.83911697182221882339452906170152068532962936506564,
0.79504142883755119835063883327278794295938959911578,
0.74633190646015079261430507035564159031073067956918,
0.69323765633475138480549071184593153338642585141021,
0.63605368072651502545283669622628593674338911679937,
0.57514044681971031534294603658642513281381264014772,
0.51086700195082709800436405095525099842549132920243,
0.44359317523872510319999221349264010784010101082300,
0.37370608871541956067254817702492723739574632170568,
0.30162786811491300432055535685859226061539650501373,
0.22778585114164507808049619536857462474308893768293,
0.15260546524092267550522024102267752791167622481842,
0.07652652113349733375464040939883821100479626681350,
0.00000000000000000000000000000000000000000000000000
).finished();
template <typename Scalar>
Array<Scalar, 21, 1> QuadratureKronrod<Scalar>::weightsGaussKronrod41 =
(Array<Scalar, 21, 1>() <<
0.00307358371852053150121829324603098748803350468825,
0.00860026985564294219866178795010234725212892276671,
0.01462616925697125298378796030886835616388105016225,
0.02038837346126652359801023143275470512283862794019,
0.02588213360495115883450506709615314299947911804867,
0.03128730677703279895854311932380073788776928036281,
0.03660016975820079803055724070721100848745349674750,
0.04166887332797368626378830593689473804396084315301,
0.04643482186749767472023188092610751684212707100708,
0.05094457392372869193270767005034494866483636580926,
0.05519510534828599474483237241977732919475345622815,
0.05911140088063957237496722064859421713641936597704,
0.06265323755478116802587012217425498058581974469890,
0.06583459713361842211156355696939794314722350634338,
0.06864867292852161934562341188536780171548970495824,
0.07105442355344406830579036172321016741291215932221,
0.07303069033278666749518941765891311276062684523455,
0.07458287540049918898658141836248752861611649357209,
0.07570449768455667465954277537661655826336315590041,
0.07637786767208073670550283503806100180080103676495,
0.07660071191799965644504990153010174082793250062867
).finished();
template <typename Scalar>
Array<Scalar, 10, 1> QuadratureKronrod<Scalar>::abscissaeGauss41 =
(Array<Scalar, 10, 1>() <<
0.99312859918509492478612238847132027822264713090166,
0.96397192727791379126766613119727722191206032780619,
0.91223442825132590586775244120329811304918479742369,
0.83911697182221882339452906170152068532962936506564,
0.74633190646015079261430507035564159031073067956918,
0.63605368072651502545283669622628593674338911679937,
0.51086700195082709800436405095525099842549132920243,
0.37370608871541956067254817702492723739574632170568,
0.22778585114164507808049619536857462474308893768293,
0.07652652113349733375464040939883821100479626681350
).finished();
template <typename Scalar>
Array<Scalar, 10, 1> QuadratureKronrod<Scalar>::weightsGauss41 =
(Array<Scalar, 10, 1>() <<
0.01761400713915211831186196235185281636214310554334,
0.04060142980038694133103995227493210987909063998995,
0.06267204833410906356950653518704160635160107657844,
0.08327674157670474872475814322204620610017782858316,
0.10193011981724043503675013548034987616669165602339,
0.11819453196151841731237737771138228700504121954897,
0.13168863844917662689849449974816313491611051114698,
0.14209610931838205132929832506716493303451541339202,
0.14917298647260374678782873700196943669267990408137,
0.15275338713072585069808433195509759349194864511238
).finished();
// Nodes and Weights - Rule 51
template <typename Scalar>
Array<Scalar, 26, 1> QuadratureKronrod<Scalar>::abscissaeGaussKronrod51 =
(Array<Scalar, 26, 1>() <<
0.99926210499260983419345748654034059370452496042280,
0.99555696979049809790878494689390161725756264940481,
0.98803579453407724763733101457740622707248415209161,
0.97666392145951751149831538647959406774537055531441,
0.96161498642584251241813003366016724169212642963710,
0.94297457122897433941401116965847053190520157060899,
0.92074711528170156174634608454633063157457035996277,
0.89499199787827536885104200678280495417455484975358,
0.86584706529327559544899696958834008820284409402824,
0.83344262876083400142102110869356956946096411382352,
0.79787379799850005941041090499430656940863230009338,
0.75925926303735763057728286520436097638752201889833,
0.71776640681308438818665407977329778059771167555516,
0.67356636847346836448512063324762217588341672807275,
0.62681009901031741278812268162451788101954628995069,
0.57766293024122296772368984161265406739573503929152,
0.52632528433471918259962377815801017803683252320191,
0.47300273144571496052218211500919204133181773846163,
0.41788538219303774885181439459457248709336998140070,
0.36117230580938783773582173012764066742207834704338,
0.30308953893110783016747890998033932920041937876655,
0.24386688372098843204519036279745158640563315632598,
0.18371893942104889201596988875952841578528447834991,
0.12286469261071039638735981880803680553220534604978,
0.06154448300568507888654639236679663128172434803982,
0.00000000000000000000000000000000000000000000000000
).finished();
template <typename Scalar>
Array<Scalar, 26, 1> QuadratureKronrod<Scalar>::weightsGaussKronrod51 =
(Array<Scalar, 26, 1>() <<
0.00198738389233031592650785188284340988942998042825,
0.00556193213535671375804023690106552207017692954963,
0.00947397338617415160720771052365532387164532684837,
0.01323622919557167481365640584697623807757808499786,
0.01684781770912829823151666753633631584040265462471,
0.02043537114588283545656829223593897367875800609767,
0.02400994560695321622009248916488108139293152820966,
0.02747531758785173780294845551781107861479601328871,
0.03079230016738748889110902021522858560087716239329,
0.03400213027432933783674879522955120322567052825005,
0.03711627148341554356033062536761987599599780268805,
0.04008382550403238207483928446707564640141054926659,
0.04287284502017004947689579243949516110199950419988,
0.04550291304992178890987058475266039304370776893570,
0.04798253713883671390639225575691475498359220742327,
0.05027767908071567196332525943344008444058763060478,
0.05236288580640747586436671213787271488735155072371,
0.05425112988854549014454337045987560682607683844126,
0.05595081122041231730824068638274734682027103511277,
0.05743711636156783285358269393950647199483285682390,
0.05868968002239420796197417585678776413979564625483,
0.05972034032417405997909929193256185383536304547619,
0.06053945537604586294536026751756542716231236571046,
0.06112850971705304830585903041629271192267855232196,
0.06147118987142531666154413196526417758653796287689,
0.06158081806783293507875982424006455319043693690314
).finished();
template <typename Scalar>
Array<Scalar, 13, 1> QuadratureKronrod<Scalar>::abscissaeGauss51 =
(Array<Scalar, 13, 1>() <<
0.99555696979049809790878494689390161725756264940481,
0.97666392145951751149831538647959406774537055531441,
0.94297457122897433941401116965847053190520157060899,
0.89499199787827536885104200678280495417455484975358,
0.83344262876083400142102110869356956946096411382352,
0.75925926303735763057728286520436097638752201889833,
0.67356636847346836448512063324762217588341672807275,
0.57766293024122296772368984161265406739573503929152,
0.47300273144571496052218211500919204133181773846163,
0.36117230580938783773582173012764066742207834704338,
0.24386688372098843204519036279745158640563315632598,
0.12286469261071039638735981880803680553220534604978,
0.00000000000000000000000000000000000000000000000000
).finished();
template <typename Scalar>
Array<Scalar, 13, 1> QuadratureKronrod<Scalar>::weightsGauss51 =
(Array<Scalar, 13, 1>() <<
0.01139379850102628794790296411323477360332052629291,
0.02635498661503213726190181529529914493596328170332,
0.04093915670130631265562348771164595366084578336410,
0.05490469597583519192593689154047332416010998555311,
0.06803833381235691720718718565670796855470949435464,
0.08014070033500101801323495966911130229022573285368,
0.09102826198296364981149722070289165338099255895933,
0.10053594906705064420220689039268582698846609452814,
0.10851962447426365311609395705011661934007758798672,
0.11485825914571164833932554586955580864093619166818,
0.11945576353578477222817812651290104739017670141373,
0.12224244299031004168895951894585150583505924756306,
0.12317605372671545120390287307905014243823362751815
).finished();
// Nodes and Weights - Rule 61
template <typename Scalar>
Array<Scalar, 31, 1> QuadratureKronrod<Scalar>::abscissaeGaussKronrod61 =
(Array<Scalar, 31, 1>() <<
0.99948441005049063757132589570581081946887394701851,
0.99689348407464954027163005091869528334088203811775,
0.99163099687040459485862836610948572485050033374616,
0.98366812327974720997003258160566280194031785470971,
0.97311632250112626837469386842370688488763796428344,
0.96002186496830751221687102558179766293035921740392,
0.94437444474855997941583132403743912158564371496498,
0.92620004742927432587932427708047400408647453682533,
0.90557330769990779854652255892595831956897536366223,
0.88256053579205268154311646253022559005668914714648,
0.85720523354606109895865851065894385682080017062360,
0.82956576238276839744289811973250191643906869617034,
0.79972783582183908301366894232268324073569842937778,
0.76777743210482619491797734097450313169488361723291,
0.73379006245322680472617113136952764566938172775469,
0.69785049479331579693229238802664006838235380065395,
0.66006106412662696137005366814927075303835037480883,
0.62052618298924286114047755643118929920736469282953,
0.57934523582636169175602493217254049590705158881215,
0.53662414814201989926416979331107279416417800693030,
0.49248046786177857499369306120770879564426564096319,
0.44703376953808917678060990032285400016240759386142,
0.40040125483039439253547621154266063361104593297078,
0.35270472553087811347103720708937386065363100802143,
0.30407320227362507737267710719925655353115778980946,
0.25463692616788984643980512981780510788278930330252,
0.20452511668230989143895767100202470952410426459556,
0.15386991360858354696379467274325592041855197124434,
0.10280693796673703014709675131800059247190133296516,
0.05147184255531769583302521316672257374914145366657,
0.00000000000000000000000000000000000000000000000000
).finished();
template <typename Scalar>
Array<Scalar, 31, 1> QuadratureKronrod<Scalar>::weightsGaussKronrod61 =
(Array<Scalar, 31, 1>() <<
0.00138901369867700762455159122675969968104884129196,
0.00389046112709988405126720184451550327851514298489,
0.00663070391593129217331982636975016813362838821778,
0.00927327965951776342844114689202436042127002493819,
0.01182301525349634174223289885325059289626440625061,
0.01436972950704580481245143244358001019584189989500,
0.01692088918905327262757228942032209236856670378384,
0.01941414119394238117340895105012845585142101419143,
0.02182803582160919229716748573833899340150729605683,
0.02419116207808060136568637072523202676039137782818,
0.02650995488233310161060170933507541436651757952275,
0.02875404876504129284397878535433421114467916054207,
0.03090725756238776247288425294309227263527045852381,
0.03298144705748372603181419101685392751059929121386,
0.03497933802806002413749967073146787509722691279482,
0.03688236465182122922391106561713596773695516478103,
0.03867894562472759295034865153228105025092362982155,
0.04037453895153595911199527975246811421612606212603,
0.04196981021516424614714754128596975779008865671899,
0.04345253970135606931683172811707325807460330863170,
0.04481480013316266319235555161672324375743139279637,
0.04605923827100698811627173555937358059469287557182,
0.04718554656929915394526147818109948648288480730063,
0.04818586175708712914077949229830459260579923610843,
0.04905543455502977888752816536723817360588740529530,
0.04979568342707420635781156937994232853920960281370,
0.05040592140278234684089308565358502890219701825162,
0.05088179589874960649229747304980469185338491426092,
0.05122154784925877217065628260494420825114695242525,
0.05142612853745902593386287921578125982955203486240,
0.05149472942945156755834043364709930753273688039646
).finished();
template <typename Scalar>
Array<Scalar, 15, 1> QuadratureKronrod<Scalar>::abscissaeGauss61 =
(Array<Scalar, 15, 1>() <<
0.99689348407464954027163005091869528334088203811775,
0.98366812327974720997003258160566280194031785470971,
0.96002186496830751221687102558179766293035921740392,
0.92620004742927432587932427708047400408647453682533,
0.88256053579205268154311646253022559005668914714648,
0.82956576238276839744289811973250191643906869617034,
0.76777743210482619491797734097450313169488361723291,
0.69785049479331579693229238802664006838235380065395,
0.62052618298924286114047755643118929920736469282953,
0.53662414814201989926416979331107279416417800693030,
0.44703376953808917678060990032285400016240759386142,
0.35270472553087811347103720708937386065363100802143,
0.25463692616788984643980512981780510788278930330252,
0.15386991360858354696379467274325592041855197124434,
0.05147184255531769583302521316672257374914145366657
).finished();
template <typename Scalar>
Array<Scalar, 15, 1> QuadratureKronrod<Scalar>::weightsGauss61 =
(Array<Scalar, 15, 1>() <<
0.00796819249616660561546588347467362245048069658715,
0.01846646831109095914230213191204726909620653396818,
0.02878470788332336934971917961129204363958889454629,
0.03879919256962704959680193644634769203320097676640,
0.04840267283059405290293814042280751781527180919737,
0.05749315621761906648172168940205612879712067072176,
0.06597422988218049512812851511596236123744295365666,
0.07375597473770520626824385002219073415377052603705,
0.08075589522942021535469493846052973087589280370844,
0.08689978720108297980238753071512570257675332874355,
0.09212252223778612871763270708761876719691323441823,
0.09636873717464425963946862635180986509640646143016,
0.09959342058679526706278028210356947652986926366670,
0.10176238974840550459642895216855404463270628948713,
0.10285265289355884034128563670541504386837555706493
).finished();
// Nodes and Weights - Rule 71
template <typename Scalar>
Array<Scalar, 36, 1> QuadratureKronrod<Scalar>::abscissaeGaussKronrod71 =
(Array<Scalar, 36, 1>() <<
0.99961929856058782041953501908431046312122344098794,
0.99770656909960029726016313931209472519327463717101,
0.99382029303890921225840752201784028500756549314159,
0.98793576444385149803511708918548598531300829571417,
0.98013165785134098801985990365260988816688816643335,
0.97043761603922983321507048258477014252488123931238,
0.95883869699584307892126219427392217538678025386427,
0.94534514820782732953872598552997464486281125124787,
0.93000375305070609922503535864064081717372299765533,
0.91285426135931761446493706355576423048806101362556,
0.89391630583904940482411214013268317995163371208619,
0.87321912502522233152328234914138526664433890227937,
0.85081354468109158704203280179571293221562523873360,
0.82674989909222540683405061274855757904689061798231,
0.80106721312570571470358184335837821672915468266178,
0.77381025228691255526742300920988748101181007297155,
0.74503897566640677164430817613858088537216200738750,
0.71481450155662878326440863122444674696204396852173,
0.68319041848815657677750427209465627245583162169621,
0.65022436466589038867579280898455926336976232592837,
0.61598571048722183053960576573763958060437543282625,
0.58054534474976450993450200818969007093207020356433,
0.54396835169625813820629328275772195267411067648783,
0.50632277324148861502429755583732737658608793863637,
0.46768618346152964902238333071075817099648473017136,
0.42813754151781425418762061300147985878471844648853,
0.38775069602784231260916836932812549811185119580936,
0.34660155443081394587697983493023823180840876937729,
0.30477400147105037961997631656077667943504232291863,
0.26235294120929605797089520045558051348630439864578,
0.21941825841501800318906027538425963000063532656757,
0.17605106116598956997430365644505992967542971232976,
0.13233927061341662561114208274719221355992727169910,
0.08837134327565926360092943349754880342411743476510,
0.04423040796047631902497910724648175796711437032383,
0.00000000000000000000000000000000000000000000000000
).finished();
template <typename Scalar>
Array<Scalar, 36, 1> QuadratureKronrod<Scalar>::weightsGaussKronrod71 =
(Array<Scalar, 36, 1>() <<
0.00102550911074666801006956437298132913634921030706,
0.00287226001447070188175126346596935559720130883926,
0.00489809089031614709134489995355345719632799389497,
0.00685548721878420013489017671059314269258714775681,
0.00874803476789701226762937017103524588617649633070,
0.01064412676080364548520974626298521833239102927401,
0.01255213863161942856328562549320926369826006957553,
0.01442614862529363429700928603045290106973954647845,
0.01624977199984979251411432365271640468771985708338,
0.01804665112955870366183510500927265777645403257829,
0.01982463073192568430864633416826825982695039151445,
0.02156072900282074084433582290901980023837200422120,
0.02324181089546663502483166236397397407862232471778,
0.02487938986497896142316498112220908754540096768188,
0.02647872983924452154290308599311750111721575798899,
0.02802485927048032502869308406000651692913135080711,
0.02950731294048380503159847508360512949719541602423,
0.03093298569089254124584577413599566869200123205647,
0.03230574967486032561801474409387789273907226946487,
0.03361454962779494244108331458137284946324198895395,
0.03485077628981657862883631795167583760136994493590,
0.03601932106443251160286472895539991131861935891999,
0.03712347803674947651991054179868437562892996972235,
0.03815455393845179802844670519848738586957355006794,
0.03910531516466664309273221987791124541581708553489,
0.03997983486093488779275160637760618088208667334682,
0.04078134475859293797125394533241037525076355046502,
0.04150279114110496636106397836894150439113925391415,
0.04213802289742381614517983543503522074367479896450,
0.04269093484449388936791278778195848585871259590150,
0.04316504612011059575016094844755100917095935153618,
0.04355454541697310298508166546078403157623950944932,
0.04385415492459730857308029613026789252900261518504,
0.04406798346693598707683395305061859923249071421063,
0.04420009752589896947154443219235054970162309288894,
0.04424566572105622843217879601220347725717507881493
).finished();
template <typename Scalar>
Array<Scalar, 18, 1> QuadratureKronrod<Scalar>::abscissaeGauss71 =
(Array<Scalar, 18, 1>() <<
0.99770656909960029726016313931209472519327463717101,
0.98793576444385149803511708918548598531300829571417,
0.97043761603922983321507048258477014252488123931238,
0.94534514820782732953872598552997464486281125124787,
0.91285426135931761446493706355576423048806101362556,
0.87321912502522233152328234914138526664433890227937,
0.82674989909222540683405061274855757904689061798231,
0.77381025228691255526742300920988748101181007297155,
0.71481450155662878326440863122444674696204396852173,
0.65022436466589038867579280898455926336976232592837,
0.58054534474976450993450200818969007093207020356433,
0.50632277324148861502429755583732737658608793863637,
0.42813754151781425418762061300147985878471844648853,
0.34660155443081394587697983493023823180840876937729,
0.26235294120929605797089520045558051348630439864578,
0.17605106116598956997430365644505992967542971232976,
0.08837134327565926360092943349754880342411743476510,
0.00000000000000000000000000000000000000000000000000
).finished();
template <typename Scalar>
Array<Scalar, 18, 1> QuadratureKronrod<Scalar>::weightsGauss71 =
(Array<Scalar, 18, 1>() <<
0.00588343342044308497575389624011262403075431779823,
0.01365082834836149226640400292051638397399403784142,
0.02132297991148358088343798396620507832380964471994,
0.02882926010889425404871603971448491626152772662959,
0.03611011586346338053271696964754993830853577896999,
0.04310842232617021878230645937490820342280658716399,
0.04976937040135352980519967608499495859449495319457,
0.05604081621237012857832774716510096444034853490456,
0.06187367196608018888701413878868865226537122580410,
0.06722228526908690396430550874814855854709311914960,
0.07204479477256006466546190978527775614726739302743,
0.07630345715544205353865853788422622552708008189564,
0.07996494224232426293266208098504575632883549943009,
0.08300059372885658837992652821617700348517450381345,
0.08538665339209912522594398739111756764119098828375,
0.08710444699718353424332203160554093668505671509385,
0.08814053043027546297073880759309664142545431727449,
0.08848679490710429063820738777761566918240915001061
).finished();
// Nodes and Weights - Rule 81
template <typename Scalar>
Array<Scalar, 41, 1> QuadratureKronrod<Scalar>::abscissaeGaussKronrod81 =
(Array<Scalar, 41, 1>() <<
0.99970755925870001652122454213597471358259546611434,
0.99823770971055920034962270242058649233577038159505,
0.99525057344607275036560954016723640396135078520144,
0.99072623869945700645305435222137215496222208135109,
0.98472283986425006102933341488331281956810416420996,
0.97725994998377426266337028371290380697866793203799,
0.96832312685414990090376748802208053929226225295460,
0.95791681921379165580454099945275928509488349060275,
0.94607183716250003820183480835310183364449049027233,
0.93281280827867653336085216684520571643475357528269,
0.91814954307289887682909109264899900900889565336112,
0.90209880696887429672825333086849310358448808105767,
0.88469200870108974596916731568481615247744988567172,
0.86595950321225950382078180835461996357054655301110,
0.84592398558731071742075270176063153327242355905011,
0.82461223083331166319632023066609877390724038424299,
0.80206056614025212716548248052306700513426840614442,
0.77830565142651938769497154550649484802069131612689,
0.75337980343894219817195210861389792731211452767941,
0.72731825518992710328099645175493054855737867353333,
0.70016297748732993103067821243443810794745752806774,
0.67195668461417954837935451496149410997032598138385,
0.64273952430557996253724399164382458078430653187326,
0.61255388966798023795261245023069487738012378168315,
0.58144706582913000652989501202985449166039158983665,
0.54946712509512820207593130552951797023397510159565,
0.51666060738638370597736752352686730669850255178962,
0.48307580168617871290856657424482300459902239553311,
0.44876451363816376391523113354600061846497447744503,
0.41377920437160500152487974580371368297409962405292,
0.37817143547358892456847074204267036554749984414768,
0.34199409082575847300749248117919431006695362002733,
0.30530244173524671953945499726037915149778023393627,
0.26815218500725368114118434480859618342480437323625,
0.23059852188071949700436103627672748185567094334317,
0.19269758070137109971551685206514989481409202110521,
0.15450687937939447709272996582455060888266557312076,
0.11608407067525520848345128440802411376872853085422,
0.07748658833128284169115486612617194705087430189926,
0.03877241750605082193319344402462329467936463438314,
0.00000000000000000000000000000000000000000000000000
).finished();
template <typename Scalar>
Array<Scalar, 41, 1> QuadratureKronrod<Scalar>::weightsGaussKronrod81 =
(Array<Scalar, 41, 1>() <<
0.00078786332389437149872027155489686731646438112543,
0.00220748573572677796216880922531121147539447921678,
0.00376522867934192207419437277688852248174719104712,
0.00527194271488547391100911400749610401795296325231,
0.00673181348520739996342079325197287731781780936164,
0.00819757638675148244956105325366150439252190701325,
0.00967540148401718791503549171298574769106769313863,
0.01113132166402750374938616623733449344824035103306,
0.01255438476851726603177494941601931549647320859678,
0.01396255986698061404257329276730788872218488725171,
0.01536132635910245297306719365346221049653776351247,
0.01673453247500258319616665389692844862618082103791,
0.01807386840881819058019116416727673799123477315931,
0.01938764589431774100483071215654747790839959650200,
0.02067904327352817531538698511576519576908963214656,
0.02193818733583309346140081267327422472167322967377,
0.02315893101337702414441542631657291426916138363685,
0.02434569018227335927008045325065084644250141492898,
0.02550021760313012760411536558132073210208609486171,
0.02661573749902468675858406158693142228991791577345,
0.02768762611106109151543416219022982991880587861249,
0.02871838684109212328774430242495038364855427947406,
0.02970892727777659464157767791486382641598503632608,
0.03065436089141152537823603492997426208913617114202,
0.03155122361911536248171493595674374505113742718348,
0.03240098250760594428516927456513374062015435029682,
0.03320404434125756040053598377665629314697494328677,
0.03395686283420980625135217300229063782965844230293,
0.03465693584349753394613505048683965340397811449698,
0.03530514470862184103888891975079794872037448391518,
0.03590160278362810442749426164234802328728310552020,
0.03644382653034092475806451684526800902468673825359,
0.03693016953404855460457701491128601575795938601292,
0.03736118002546921808817250104671655660126038378354,
0.03773680126309354415257237969171128676159927920230,
0.03805546377885242099072980354540709765269970008461,
0.03831632400517485967847680297981699802520907959292,
0.03851974174995072693620904485019618182205811434106,
0.03866555543914104039741924425967757530593426930461,
0.03875302937875238614021147436365714157701740560300,
0.03878210476428280538640259652566010072577182998938
).finished();
template <typename Scalar>
Array<Scalar, 20, 1> QuadratureKronrod<Scalar>::abscissaeGauss81 =
(Array<Scalar, 20, 1>() <<
0.99823770971055920034962270242058649233577038159505,
0.99072623869945700645305435222137215496222208135109,
0.97725994998377426266337028371290380697866793203799,
0.95791681921379165580454099945275928509488349060275,
0.93281280827867653336085216684520571643475357528269,
0.90209880696887429672825333086849310358448808105767,
0.86595950321225950382078180835461996357054655301110,
0.82461223083331166319632023066609877390724038424299,
0.77830565142651938769497154550649484802069131612689,
0.72731825518992710328099645175493054855737867353333,
0.67195668461417954837935451496149410997032598138385,
0.61255388966798023795261245023069487738012378168315,
0.54946712509512820207593130552951797023397510159565,
0.48307580168617871290856657424482300459902239553311,
0.41377920437160500152487974580371368297409962405292,
0.34199409082575847300749248117919431006695362002733,
0.26815218500725368114118434480859618342480437323625,
0.19269758070137109971551685206514989481409202110521,
0.11608407067525520848345128440802411376872853085422,
0.03877241750605082193319344402462329467936463438314
).finished();
template <typename Scalar>
Array<Scalar, 20, 1> QuadratureKronrod<Scalar>::weightsGauss81 =
(Array<Scalar, 20, 1>() <<
0.00452127709853319125847173287818533272783111019971,
0.01049828453115281361474217106727965237679262131580,
0.01642105838190788871286348488236392729234229334696,
0.02224584919416695726150432418420857320703319667935,
0.02793700698002340109848915750772107730255086205077,
0.03346019528254784739267818308641084897724178665376,
0.03878216797447201763997203129044616225345921123203,
0.04387090818567327199167468604171549581100683717024,
0.04869580763507223206143416044814638806784302737712,
0.05322784698393682435499647977226050455532117182201,
0.05743976909939155136661773091042598560010483585445,
0.06130624249292893916653799640839859590259376351117,
0.06480401345660103807455452956675273003269296420849,
0.06791204581523390382569010823192398598419723837929,
0.07061164739128677969548363085528683235955910399559,
0.07288658239580405906051068344251783585755908098580,
0.07472316905796826420018933626132467319120293442036,
0.07611036190062624237155807592249482301255955384507,
0.07703981816424796558830753428381024852443975416394,
0.07750594797842481126372396295832632696366865278811
).finished();
// Nodes and Weights - Rule 91
template <typename Scalar>
Array<Scalar, 46, 1> QuadratureKronrod<Scalar>::abscissaeGaussKronrod91 =
(Array<Scalar, 46, 1>() <<
0.99976825819812844184050501808521584761778222189509,
0.99860364518193663815654767690082054809228852721448,
0.99623648400262042934511873220069067464977892521152,
0.99264999844720374174861712059773534497500164330364,
0.98788929755834198035845135712826787436469148879696,
0.98196871503454056823931847363434148153836524922046,
0.97487472859984626283011247237626660754074730116341,
0.96660831039689460473642516089247809711667903060942,
0.95719162401328142632709071186085980254412868260598,
0.94664169099562906178472059695383707845046834416714,
0.93496270170823072849720985525389099772434749839611,
0.92216393671900038809746736096052731926536287725066,
0.90826680768346396973605068382241808590437868337987,
0.89329167175324173846464905149305734636540124041379,
0.87725158719302201212972841148989357243612789849954,
0.86016247596066422533907887056713438778341442357702,
0.84204857691040695822975484057647336894431244265745,
0.82293422050208633703577526002650202999341525805627,
0.80283895116683677464879103408690607617541369942643,
0.78178431259390629131236318809860281766027351683906,
0.75979816452041089331911717455595240315934812011636,
0.73690884894549035262373884859489201987960568083762,
0.71314122427549174186073675931964340779561425372465,
0.68852168077120052523201982588043823595523300561853,
0.66308170860223379808352326376326032948162534249687,
0.63685339445322335927122384590338315069203124346568,
0.60986605449389570894381986808230584011038649491498,
0.58215021256935318668096733444417713100008882598323,
0.55374067215934876332312890149007883417942668039810,
0.52467282046291606709113410046016240260710827080028,
0.49497965749810183702993192041027660372919586028089,
0.46469512391963509857960150230974851083981266723719,
0.43385684374178294432166447428181289994755173351281,
0.40250294385854191407797450854834645597029935756940,
0.37066934235973061895322964085585254593836191642981,
0.33839265425060216164340410003187298953234144313742,
0.30571272186623304325858534427814131940857839940069,
0.27266976975237756060876539161564518189821175427664,
0.23930185320471224179064130404859896962578658675486,
0.20564748978326374571978722547154393070323638569943,
0.17174806594978090965313436541849815218083385269234,
0.13764520598325302875659004142306555310676555654153,
0.10337830283214540467276673615057417548074590315259,
0.06898698016314417249041461410381172969392066693097,
0.03451344875177766949491534277959965665394030693585,
0.00000000000000000000000000000000000000000000000000
).finished();
template <typename Scalar>
Array<Scalar, 46, 1> QuadratureKronrod<Scalar>::weightsGaussKronrod91 =
(Array<Scalar, 46, 1>() <<
0.00062429185718321579857663066059662918012169021712,
0.00174909651748535281285516187173085451060602294185,
0.00298417191546646641998503281623961554312494201463,
0.00417984253266509402229000496101711163585669889060,
0.00533932741808091123159062611497603209976404263396,
0.00650505471381696245904611633672298845714802006140,
0.00768283318432149209385511436140106223666562795019,
0.00884569677730051624106815234208503432536098150154,
0.00998469672095943272307954145417903769666978042970,
0.01111490008447190508121416984333781971203603824680,
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