quad.c

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/* Parallel Hierarchical Grid -- an adaptive finite element library.
 *
 * Copyright (C) 2005-2010 State Key Laboratory of Scientific and
 * Engineering Computing, Chinese Academy of Sciences. */
 
/* This library is free software; you can redistribute it and/or
 * modify it under the terms of the GNU Lesser General Public
 * License as published by the Free Software Foundation; either
 * version 2.1 of the License, or (at your option) any later version.
 *
 * This library is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
 * Lesser General Public License for more details.
 *
 * You should have received a copy of the GNU Lesser General Public
 * License along with this library; if not, write to the Free Software
 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
 * MA  02110-1301  USA */
 
/* $Id: quad.c,v 1.183 2011/06/08 06:31:34 zlb Exp $ */
 
/* Numerical Quadrature */
 
 
#include "quad.h"
 
#ifdef Length
# undef Length
#endif
#define Length(wts) (sizeof(wts) / (sizeof(wts[0])))
 
 
/*--------------------------- 1D quadrature rules ------------------------*/
 
static FLOAT QUAD_1D_P1_wts[] = {
    Dup2(1.)
};
static FLOAT QUAD_1D_P1_pts[Length(QUAD_1D_P1_wts) * 2] = {
    Perm2(.5)
};
QUAD QUAD_1D_P1_ = {
    "1D P1",            /* name */
    1,              /* dim */
    1,              /* order */
    Length(QUAD_1D_P1_wts), /* npoints */
    QUAD_1D_P1_pts,     /* points */
    QUAD_1D_P1_wts,     /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_1D_P3_wts[] = {
    Dup11(.5)
};
static FLOAT QUAD_1D_P3_pts[Length(QUAD_1D_P3_wts) * 2] = {
    /* (3 - sqrt(3)) / 6, (3 + sqrt(3)) / 6 */
    Perm11(.21132486540518711774542560974902127)
};
QUAD QUAD_1D_P3_ = {
    "1D P3",            /* name */
    1,              /* dim */
    3,              /* order */
    Length(QUAD_1D_P3_wts), /* npoints */
    QUAD_1D_P3_pts,     /* points */
    QUAD_1D_P3_wts,     /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_1D_P5_wts[] = {
    Dup11(5./18.),
    Dup2(4./9.)
};
static FLOAT QUAD_1D_P5_pts[Length(QUAD_1D_P5_wts) * 2] = {
    /* (5 - sqrt(15)) / 10, 1 / 2, (5 + sqrt(15)) / 10 */
    Perm11(.11270166537925831148207346002176004),
    Perm2(.5)
};
QUAD QUAD_1D_P5_ = {
    "1D P5",            /* name */
    1,              /* dim */
    5,              /* order */
    Length(QUAD_1D_P5_wts), /* npoints */
    QUAD_1D_P5_pts,     /* points */
    QUAD_1D_P5_wts,     /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_1D_P7_wts[] = {
    /* (18 - sqrt(30)) / 72, (18 + sqrt(30)) / 72 */
    Dup11(.17392742256872692868653197461099970),
    Dup11(.32607257743127307131346802538900030)
};
static FLOAT QUAD_1D_P7_pts[Length(QUAD_1D_P7_wts) * 2] = {
    /* (35 \pm sqrt(525 \pm 70 * sqrt(30))) / 70 */
    Perm11(.06943184420297371238802675555359525),
    Perm11(.33000947820757186759866712044837766)
};
QUAD QUAD_1D_P7_ = {
    "1D P7",            /* name */
    1,              /* dim */
    7,              /* order */
    Length(QUAD_1D_P7_wts), /* npoints */
    QUAD_1D_P7_pts,     /* points */
    QUAD_1D_P7_wts,     /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_1D_P9_wts[] = {
    /* (322 \pm 13 * sqrt(70)) / 1800 */
    Dup2(128./450.),
    Dup11(.11846344252809454375713202035995868),
    Dup11(.23931433524968323402064575741781910)
};
static FLOAT QUAD_1D_P9_pts[Length(QUAD_1D_P9_wts) * 2] = {
    /* (21 \pm sqrt(245 \pm 14 * sqrt(70))) / 42 */
    Perm2(0.5),
    Perm11(.04691007703066800360118656085030352),
    Perm11(.23076534494715845448184278964989560)
};
QUAD QUAD_1D_P9_ = {
    "1D P9",            /* name */
    1,              /* dim */
    9,              /* order */
    Length(QUAD_1D_P9_wts), /* npoints */
    QUAD_1D_P9_pts,     /* points */
    QUAD_1D_P9_wts,     /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_1D_P11_wts[] = {
    Dup11(.08566224618958517252014807108636645),
    Dup11(.18038078652406930378491675691885806),
    Dup11(.23395696728634552369493517199477550)
};
static FLOAT QUAD_1D_P11_pts[Length(QUAD_1D_P11_wts) * 2] = {
    Perm11(.96623475710157601390615077724699730),
    Perm11(.83060469323313225683069979750995267),
    Perm11(.61930959304159845431525086084035597)
};
QUAD QUAD_1D_P11_ = {
    "1D P11",           /* name */
    1,              /* dim */
    11,             /* order */
    Length(QUAD_1D_P11_wts),    /* npoints */
    QUAD_1D_P11_pts,        /* points */
    QUAD_1D_P11_wts,        /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_1D_P13_wts[] = {
    Dup11(.06474248308443484663530571633954101),
    Dup11(.13985269574463833395073388571188979),
    Dup11(.19091502525255947247518488774448757),
    Dup2(.20897959183673469387755102040816327)
};
static FLOAT QUAD_1D_P13_pts[Length(QUAD_1D_P13_wts) * 2] = {
    Perm11(.97455395617137926226309484202392563),
    Perm11(.87076559279969721993193238664039420),
    Perm11(.70292257568869858345330320603848073),
    Perm2(.5)
};
QUAD QUAD_1D_P13_ = {
    "1D P13",           /* name */
    1,              /* dim */
    13,             /* order */
    Length(QUAD_1D_P13_wts),    /* npoints */
    QUAD_1D_P13_pts,        /* points */
    QUAD_1D_P13_wts,        /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_1D_P15_wts[] = {
    Dup11(.05061426814518812957626567715498110),
    Dup11(.11119051722668723527217799721312044),
    Dup11(.15685332293894364366898110099330066),
    Dup11(.18134189168918099148257522463859781)
};
static FLOAT QUAD_1D_P15_pts[Length(QUAD_1D_P15_wts) * 2] = {
    Perm11(.98014492824876811584178043428473650),
    Perm11(.89833323870681336979577696823791522),
    Perm11(.76276620495816449290886952459462317),
    Perm11(.59171732124782490246973807118009199)
};
QUAD QUAD_1D_P15_ = {
    "1D P15",       /* name */
    1,          /* dim */
    15,         /* order */
    Length(QUAD_1D_P15_wts),    /* npoints */
    QUAD_1D_P15_pts,    /* points */
    QUAD_1D_P15_wts,    /* weights */
    -1          /* id */
};
 
static FLOAT QUAD_1D_P17_wts[] = {
    Dup11(.04063719418078720598594607905526183),
    Dup11(.09032408034742870202923601562145640),
    Dup11(.13030534820146773115937143470931642),
    Dup11(.15617353852000142003431520329222183),
    Dup2(.16511967750062988158226253464348702)
};
static FLOAT QUAD_1D_P17_pts[Length(QUAD_1D_P17_wts) * 2] = {
    Perm11(.98408011975381304491778810145183644),
    Perm11(.91801555366331789714971489403486744),
    Perm11(.80668571635029519865435101967073709),
    Perm11(.66212671170190446451926900732166830),
    Perm2(.5)
};
QUAD QUAD_1D_P17_ = {
    "1D P17",       /* name */
    1,          /* dim */
    17,         /* order */
    Length(QUAD_1D_P17_wts),    /* npoints */
    QUAD_1D_P17_pts,    /* points */
    QUAD_1D_P17_wts,    /* weights */
    -1          /* id */
};
 
static FLOAT QUAD_1D_P19_wts[] = {
    Dup11(.03333567215434406879678440494666590),
    Dup11(.07472567457529029657288816982884867),
    Dup11(.10954318125799102199776746711408160),
    Dup11(.13463335965499817754561346078473468),
    Dup11(.14776211235737643508694649732566916)
};
static FLOAT QUAD_1D_P19_pts[Length(QUAD_1D_P19_wts) * 2] = {
    Perm11(.98695326425858586003898200604222603),
    Perm11(.93253168334449225536604834421174652),
    Perm11(.83970478414951220311716368255743679),
    Perm11(.71669769706462359539963297158289208),
    Perm11(.57443716949081560544241300056485999)
};
QUAD QUAD_1D_P19_ = {
    "1D P19",       /* name */
    1,          /* dim */
    19,         /* order */
    Length(QUAD_1D_P19_wts),    /* npoints */
    QUAD_1D_P19_pts,    /* points */
    QUAD_1D_P19_wts,    /* weights */
    -1          /* id */
};
 
static FLOAT QUAD_1D_P21_wts[] = {
    Dup11(.02783428355808683324137686022127429),
    Dup11(.06279018473245231231734714961197005),
    Dup11(.09314510546386712571304882071582795),
    Dup11(.11659688229599523995926185242158757),
    Dup11(.13140227225512333109034443494525460),
    Dup2(.13646254338895031535724176416817109)
};
static FLOAT QUAD_1D_P21_pts[Length(QUAD_1D_P21_wts) * 2] = {
    Perm11(.98911432907302849640196900056142870),
    Perm11(.94353129988404764953757888465196363),
    Perm11(.86507600278702466204670812601557673),
    Perm11(.75954806460340590796286283472930478),
    Perm11(.63477157797617248616576599270043076),
    Perm2(.5)
};
QUAD QUAD_1D_P21_ = {
    "1D P21",       /* name */
    1,          /* dim */
    21,         /* order */
    Length(QUAD_1D_P21_wts),    /* npoints */
    QUAD_1D_P21_pts,    /* points */
    QUAD_1D_P21_wts,    /* weights */
    -1          /* id */
};
 
/*--------------------------- 2D cubature rules ------------------------*/
 
static FLOAT QUAD_2D_P1_wts[] = {
    Dup3(1.)
};
static FLOAT QUAD_2D_P1_pts[Length(QUAD_2D_P1_wts) * 3] = {
    Perm3(1./3.)
};
QUAD QUAD_2D_P1_ = {
    "2D P1",            /* name */
    2,              /* dim */
    1,              /* order */
    Length(QUAD_2D_P1_wts), /* npoints */
    QUAD_2D_P1_pts,     /* points */
    QUAD_2D_P1_wts,     /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_2D_P2_wts[] = {
    Dup21(1./3.)
};
static FLOAT QUAD_2D_P2_pts[Length(QUAD_2D_P2_wts) * 3] = {
    Perm21(1./6.)
};
QUAD QUAD_2D_P2_ = {
    "2D P2",            /* name */
    2,              /* dim */
    2,              /* order */
    Length(QUAD_2D_P2_wts), /* npoints = 3 */
    QUAD_2D_P2_pts,     /* points */
    QUAD_2D_P2_wts,     /* weights */
    -1              /* id */
};
 
#if 0
/* Note: this rule has points on the edges */
static FLOAT QUAD_2D_P3_wts[] = {
    Dup21(1./30.),
    Dup21(3./10.)
};
static FLOAT QUAD_2D_P3_pts[Length(QUAD_2D_P3_wts) * 3] = {
    Perm21(.5),
    /* 1/6 */
    Perm21(1./6.)
};
#else
static FLOAT QUAD_2D_P3_wts[] = {
    Dup21(.28114980244097964825351432270207695),
    Dup21(.05218353089235368507981901063125638)
};
static FLOAT QUAD_2D_P3_pts[Length(QUAD_2D_P3_wts) * 3] = {
    Perm21(.16288285039589191090016180418490635),
    Perm21(.47791988356756370000000000000000000)
};
#endif
QUAD QUAD_2D_P3_ = {
    "2D P3",            /* name */
    2,              /* dim */
    3,              /* order */
    Length(QUAD_2D_P3_wts), /* npoints = 6 */
    QUAD_2D_P3_pts,     /* points */
    QUAD_2D_P3_wts,     /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_2D_P4_wts[] = {
    /* (620 + sqrt(213125 - 53320 * sqrt(10))) / 3720 */
    Dup21(.22338158967801146569500700843312280),
    /* (620 - sqrt(213125 - 53320 * sqrt(10))) / 3720 */
    Dup21(.10995174365532186763832632490021053)
};
static FLOAT QUAD_2D_P4_pts[Length(QUAD_2D_P4_wts) * 3] = {
    /* (8 - sqrt(10) + sqrt(38 - 44 * sqrt(2 / 5))) / 18 */
    Perm21(.44594849091596488631832925388305199),
    /* (8 - sqrt(10) - sqrt(38 - 44 * sqrt(2 / 5))) / 18 */
    Perm21(.09157621350977074345957146340220151)
};
QUAD QUAD_2D_P4_ = {
    "2D P4",            /* name */
    2,              /* dim */
    4,              /* order */
    Length(QUAD_2D_P4_wts), /* npoints = 6 */
    QUAD_2D_P4_pts,     /* points */
    QUAD_2D_P4_wts,     /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_2D_P5_wts[] = {
    /* (155 - sqrt(15)) / 1200 */
    Dup21(.12593918054482715259568394550018133),
    /* (155 + sqrt(15)) / 1200 */
    Dup21(.13239415278850618073764938783315200),
    Dup3(9./40.)
};
static FLOAT QUAD_2D_P5_pts[Length(QUAD_2D_P5_wts) * 3] = {
    /* (6 - sqrt(15)) / 21 */
    Perm21(.10128650732345633880098736191512383),
    /* (6 + sqrt(15)) / 21 */
    Perm21(.47014206410511508977044120951344760),
    /* 1 / 3 */
    Perm3(1./3.)
};
QUAD QUAD_2D_P5_ = {
    "2D P5",            /* name */
    2,              /* dim */
    5,              /* order */
    Length(QUAD_2D_P5_wts), /* npoints = 7 */
    QUAD_2D_P5_pts,     /* points */
    QUAD_2D_P5_wts,     /* weights */
    -1              /* id */
};
 
/* Note: an 11-point rule has been found in:
 *   Day, David M., Mark A. Taylor, "A new 11 point degree 6 cubature
 *      formula for the triangle [Proceedings of the ICIAM 2007],"
 *   Journal Article, Proceedings of Applied Mathematics and Mechanics,
 *   Accepted/Published January 2008. */
static FLOAT QUAD_2D_P6_wts[] = {
    Dup21(.05084490637020681692093680910686898),
    Dup21(.11678627572637936602528961138557944),
    Dup111(.08285107561837357519355345642044245)
};
static FLOAT QUAD_2D_P6_pts[Length(QUAD_2D_P6_wts) * 3] = {
    Perm21(.06308901449150222834033160287081916),
    Perm21(.24928674517091042129163855310701908),
    Perm111(.05314504984481694735324967163139815,
        .31035245103378440541660773395655215)
};
QUAD QUAD_2D_P6_ = {
    "2D P6",            /* name */
    2,              /* dim */
    6,              /* order */
    Length(QUAD_2D_P6_wts), /* npoints = 12 */
    QUAD_2D_P6_pts,     /* points */
    QUAD_2D_P6_wts,     /* weights */
    -1              /* id */
};
 
#if 0
static FLOAT QUAD_2D_P7_wts[] = {
    Dup21(0.0102558174092),
    Dup111(0.1116047046647),
    Dup21(0.1679775595335),
    Dup21(0.2652238803946)
};
static FLOAT QUAD_2D_P7_pts[Length(QUAD_2D_P7_wts) * 3] = {
    Perm21(0.),
    Perm111(0.7839656651012, 0.1738960507345),
    Perm21(0.4743880861752),
    Perm21(0.2385615300181)
};
#else
static FLOAT QUAD_2D_P7_wts[] = {
    Dup21(.01353386251566556156682309245259393),
    Dup21(.07895125443201098137652145029770332),
    Dup21(.12860792781890607455665553308952344),
    Dup111(.05612014428337535791666662874675632)
};
static FLOAT QUAD_2D_P7_pts[Length(QUAD_2D_P7_wts) * 3] = {
    Perm21(.02826392415607634022359600691324002),
    Perm21(.47431132326722257527522522793181654),
    Perm21(.24114332584984881025414351267036207),
    Perm111(.76122274802452380000000000000000000,
        .04627087779880891064092559391702049)
};
#endif
QUAD QUAD_2D_P7_ = {
    "2D P7",            /* name */
    2,              /* dim */
    7,              /* order */
    Length(QUAD_2D_P7_wts), /* npoints = 15 */
    QUAD_2D_P7_pts,     /* points */
    QUAD_2D_P7_wts,     /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_2D_P8_wts[] = {
    Dup3(.14431560767778716825109111048906462),
    Dup21(.10321737053471825028179155029212903),
    Dup21(.03245849762319808031092592834178060),
    Dup21(.09509163426728462479389610438858432),
    Dup111(.02723031417443499426484469007390892)
};
static FLOAT QUAD_2D_P8_pts[Length(QUAD_2D_P8_wts) * 3] = {
    Perm3(.33333333333333333333333333333333333),
    Perm21(.17056930775176020662229350149146450),
    Perm21(.05054722831703097545842355059659895),
    Perm21(.45929258829272315602881551449416932),
    Perm111(.26311282963463811342178578628464359,
        .00839477740995760533721383453929445)
};
QUAD QUAD_2D_P8_ = {
    "2D P8",            /* name */
    2,              /* dim */
    8,              /* order */
    Length(QUAD_2D_P8_wts), /* npoints = 16 */
    QUAD_2D_P8_pts,     /* points */
    QUAD_2D_P8_wts,     /* weights */
    -1              /* id */
};
 
#if 0
static FLOAT QUAD_2D_P9_wts[] = {
    Dup21(0.0519871420646),
    Dup111(0.0707034101784),
    Dup111(0.0909390760952),
    Dup21(0.1032344051380),
    Dup21(0.1881601469167)
};
static FLOAT QUAD_2D_P9_pts[Length(QUAD_2D_P9_wts) * 3] = {
    Perm21(0.0451890097844),
    Perm111(0.7475124727339, 0.2220631655373),
    Perm111(0.1369912012649, 0.2182900709714),
    Perm21(0.4815198347833),
    Perm21(0.4036039798179)
};
QUAD QUAD_2D_P9_ = {
    "2D P9",            /* name */
    2,              /* dim */
    9,              /* order */
    21,             /* npoints */
    QUAD_2D_P9_pts,     /* points */
    QUAD_2D_P9_wts,     /* weights */
    -1              /* id */
};
#else
static FLOAT QUAD_2D_P9_wts[] = {
    Dup3(.09713579628279883381924198250728863),
    Dup21(.03133470022713907053685483128720932),
    Dup21(.02557767565869803126167879855899982),
    Dup21(.07782754100477427931673935629940396),
    Dup21(.07964773892721025303289177426404527),
    Dup111(.04328353937728937728937728937728938)
};
static FLOAT QUAD_2D_P9_pts[Length(QUAD_2D_P9_wts) * 3] = {
    Perm3(.33333333333333333333333333333333333),
    Perm21(.48968251919873762778370692483619280),
    Perm21(.04472951339445270986510658996627636),
    Perm21(.43708959149293663726993036443535497),
    Perm21(.18820353561903273024096128046733557),
    Perm111(.74119859878449802069007987352342383,
        .22196298916076569567510252769319107)
};
QUAD QUAD_2D_P9_ = {
    "2D P9",            /* name */
    2,              /* dim */
    9,              /* order */
    Length(QUAD_2D_P9_wts), /* npoints = 19 */
    QUAD_2D_P9_pts,     /* points */
    QUAD_2D_P9_wts,     /* weights */
    -1              /* id */
};
#endif
 
static FLOAT QUAD_2D_P10_wts[] = {
    Dup3(.08093742879762288025711312381650193),
    Dup21(.07729858800296312168250698238034344),
    Dup21(.07845763861237173136809392083439673),
    Dup21(.01746916799592948691760716329067815),
    Dup21(.00429237418483282803048040209013191),
    Dup111(.03746885821046764297902076548504452),
    Dup111(.02694935259187995964544947958109671)
};
static FLOAT QUAD_2D_P10_pts[Length(QUAD_2D_P10_wts) * 3] = {
    Perm3(.33333333333333333333333333333333333),
    Perm21(.42727317884677553809044271751544715),
    Perm21(.18309922244867502052157438485022004),
    Perm21(.49043401970113058745397122237684843),
    Perm21(.01257244555158053273132908502104126),
    Perm111(.65426866792006614066657009558762790,
        .30804600168524770000000000000000000),
    Perm111(.12280457706855927343012981748128116,
        .03337183373930478624081644177478038)
};
QUAD QUAD_2D_P10_ = {
    "2D P10",           /* name */
    2,              /* dim */
    10,             /* order */
    Length(QUAD_2D_P10_wts),    /* npoints = 25 */
    QUAD_2D_P10_pts,        /* points */
    QUAD_2D_P10_wts,        /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_2D_P11_wts[] = {
    Dup3(.08117796029686715951547596874982357),
    Dup21(.01232404350690949411847390101623284),
    Dup21(.06282800974441010728333942816029398),
    Dup21(.01222037904936452975521221500393789),
    Dup21(.06770134895281150992098886182322559),
    Dup21(.04021969362885169042356688960756866),
    Dup111(.01476227271771610133629306558778206),
    Dup111(.04072799645829903966033695848161786)
};
static FLOAT QUAD_2D_P11_pts[Length(QUAD_2D_P11_wts) * 3] = {
    Perm3(.33333333333333333333333333333333333),
    Perm21(.03093835524543078489519501499130475),
    Perm21(.43649818113412884191761527655997324),
    Perm21(.49898476370259326628798698383139087),
    Perm21(.21468819795859433660687581387825086),
    Perm21(.11368310404211339020529315622836178),
    Perm111(.82561876616486290435880620030835800,
        .15974230459185018980086078822500751),
    Perm111(.64047231013486526767703659081896681,
        .31178371570959900000000000000000000)
};
QUAD QUAD_2D_P11_ = {
    "2D P11",           /* name */
    2,              /* dim */
    11,             /* order */
    Length(QUAD_2D_P11_wts),    /* npoints = 28 */
    QUAD_2D_P11_pts,        /* points */
    QUAD_2D_P11_wts,        /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_2D_P12_wts[] = {
    Dup21(.00616626105155901723386648378523035),
    Dup21(.06285822421788510035427051309288255),
    Dup21(.03479611293070894298932839729499937),
    Dup21(.04369254453803840213545726255747497),
    Dup21(.02573106644045533541779092307156443),
    Dup111(.02235677320230344571183907670231999),
    Dup111(.01731623110865889237164210081103407),
    Dup111(.04037155776638092951782869925223677)
};
static FLOAT QUAD_2D_P12_pts[Length(QUAD_2D_P12_wts) * 3] = {
    Perm21(.02131735045321037024685697551572825),
    Perm21(.27121038501211592234595134039689474),
    Perm21(.12757614554158592467389632515428357),
    Perm21(.43972439229446027297973662348436108),
    Perm21(.48821738977380488256466206525881104),
    Perm111(.69583608678780342214163552323607254,
        .28132558098993954824813069297455275),
    Perm111(.85801403354407263059053661662617818,
        .11625191590759714124135414784260182),
    Perm111(.60894323577978780685619243776371007,
        .27571326968551419397479634607976398)
};
QUAD QUAD_2D_P12_ = {
    "2D P12",           /* name */
    2,              /* dim */
    12,             /* order */
    Length(QUAD_2D_P12_wts),    /* npoints = 33 */
    QUAD_2D_P12_pts,        /* points */
    QUAD_2D_P12_wts,        /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_2D_P13_wts[] = {
    Dup3(.06796003658683164428177442468088488),
    Dup21(.05560196753045332870725746601046147),
    Dup21(.05827848511919998140476708351333981),
    Dup21(.00605233710353917184179280003229082),
    Dup21(.02399440192889473077371079945095965),
    Dup111(.03464127614084837046598682851091822),
    Dup111(.01496540110516566726324585713290344),
    Dup111(.02417903981159381913744574557306076),
    Dup111(.00959068100354326272259509016611089)
};
static FLOAT QUAD_2D_P13_pts[Length(QUAD_2D_P13_wts) * 3] = {
    Perm3(.33333333333333333333333333333333333),
    Perm21(.42694141425980040602081253503137421),
    Perm21(.22137228629183290065481255470507908),
    Perm21(.02150968110884318386929131353405208),
    Perm21(.48907694645253934990068971909020439),
    Perm111(.62354599555367557081585435318623659,
        .30844176089211777465847185254124531),
    Perm111(.86470777029544277530254595089569318,
        .11092204280346339541286954522167452),
    Perm111(.74850711589995219517301859578870965,
        .16359740106785048023388790171095725),
    Perm111(.72235779312418796526062013230478405,
        .27251581777342966618005046435408685)
};
QUAD QUAD_2D_P13_ = {
    "2D P13",           /* name */
    2,              /* dim */
    13,             /* order */
    Length(QUAD_2D_P13_wts),    /* npoints = 37 */
    QUAD_2D_P13_pts,        /* points */
    QUAD_2D_P13_wts,        /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_2D_P14_wts[] = {
    Dup3(.05859628522602859412789380634775601),
    Dup21(.00173515122972526756806186388080941),
    Dup21(.02616378255861452177782885918197827),
    Dup21(.00391972924240182909652082757014540),
    Dup21(.01224735975694086609728698992625048),
    Dup21(.02819962850325796010736630715156571),
    Dup21(.05088708718595948529603482754545404),
    Dup21(.05045343990160359919102089713411889),
    Dup111(.01706364421223345129002539938494722),
    Dup111(.00968346642550660040752096309341938),
    Dup111(.03638575592848500562201132776427165),
    Dup111(.00696466337351841242539972250424131)
};
static FLOAT QUAD_2D_P14_pts[Length(QUAD_2D_P14_wts) * 3] = {
    Perm3(.33333333333333333333333333333333333),
    Perm21(.00997976080645843241529352958205243),
    Perm21(.47997789352118838981055286508838991),
    Perm21(.15381195917696690000000000000000000),
    Perm21(.07402347711698781000000000000000000),
    Perm21(.13035468250333000000000000000000000),
    Perm21(.23061722602665313429960537009838312),
    Perm21(.42233208341914782411440871379139388),
    Perm111(.78623738593466100332962211403309001,
        .19061636003190090424614328286530343),
    Perm111(.63055214366060744162240907556881292,
        .36232313774354714461832673435977294),
    Perm111(.62657732985630631423351231375342650,
        .29077120588366741502481681748167319),
    Perm111(.91420998492962541223996709938504695,
        .07116571087775076254759245029243364)
};
QUAD QUAD_2D_P14_ = {
    "2D P14",           /* name */
    2,              /* dim */
    14,             /* order */
    Length(QUAD_2D_P14_wts),    /* npoints = 46 */
    QUAD_2D_P14_pts,        /* points */
    QUAD_2D_P14_wts,        /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_2D_P15_wts[] = {
    Dup3(.04403871087843427985301732721493388),
    Dup21(.04618478718202697994871566760191669),
    Dup21(.00649890661733271652688280349281019),
    Dup21(.01799361425265840324466992416715655),
    Dup21(.04177310503914135411968606056414597),
    Dup21(.00305954760911646654843016992834484),
    Dup21(.00201243505255864734409031875654046),
    Dup21(.01677561093050912232611145688795876),
    Dup111(.01546074918971427486608803040924742),
    Dup111(.02849989033954742339273955875330195),
    Dup111(.03209435048348959564209923573709566),
    Dup111(.01150858163687071128402324377324186),
    Dup111(.00461430652896710314358717609185406)
};
static FLOAT QUAD_2D_P15_pts[Length(QUAD_2D_P15_wts) * 3] = {
    Perm3(.33333333333333333333333333333333333),
    Perm21(.22733221881914287420250436849229406),
    Perm21(.49716257743188742987380980001602329),
    Perm21(.47884973534895458333922920014385258),
    Perm21(.40498603909827199169724464234269204),
    Perm21(.01593121667174443211342773294126896),
    Perm21(.16558326242608140000000000000000000),
    Perm21(.07313360471922872772687381210732441),
    Perm111(.66526073307221393906236441338569119,
        .31635283934494723008633813095024529),
    Perm111(.71252198724254553304884901162338783,
        .09346075114991753000000000000000047),
    Perm111(.55964836223539321841224845401923000,
        .34422901758219320000000000000000160),
    Perm111(.81047659761907686304683273029057126,
        .17104724831425795154765033192558481),
    Perm111(.91607564403173118856460883877832000,
        .07305599647918648961294908192742498)
};
QUAD QUAD_2D_P15_ = {
    "2D P15",           /* name */
    2,              /* dim */
    15,             /* order */
    Length(QUAD_2D_P15_wts),    /* npoints = 52 */
    QUAD_2D_P15_pts,        /* points */
    QUAD_2D_P15_wts,        /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_2D_P16_wts[] = {
    Dup3(.04802218868037709055183940458051988),
    Dup21(.01470910030680192710340364286186919),
    Dup21(.02954458654931925599530972679646409),
    Dup21(.02612501735108837749859756549171557),
    Dup21(.00278038735239000697500301613866207),
    Dup21(.03182177300053664950342729005594961),
    Dup21(.00864583434950965990117373416984893),
    Dup111(.01430033290449536514661642536825213),
    Dup111(.02784977720360082995222987342395349),
    Dup111(.00704167340663609756237018808928069),
    Dup111(.01789983825993372860177020907581078),
    Dup111(.02745820038434976307247003810091720),
    Dup111(.00729979693943176208411254408777766)
};
static FLOAT QUAD_2D_P16_pts[Length(QUAD_2D_P16_wts) * 3] = {
    Perm3(.33333333333333333333333333333333333),
    Perm21(.08179498313137387264146559311886101),
    Perm21(.16530060196977965062676193293355656),
    Perm21(.46859210534946138669460289729660561),
    Perm21(.01443881344541668261410895669566020),
    Perm21(.24178428539178335340689445929320769),
    Perm21(.49531034298776996406549508687740551),
    Perm111(.65051340266135229943114468484168666,
        .33139974453708955658132316818259388),
    Perm111(.60401128149599703984940410303596702,
        .30324716274994218504155217807834692),
    Perm111(.80216825757474166361686194781166705,
        .18802805952123717344418211429398875),
    Perm111(.75650560644282839655115407575806082,
        .18350466852229686368238027743700035),
    Perm111(.46593843871411818488381073359154639,
        .35964594879750460000000000000001000),
    Perm111(.90639484399204150136249966186534000,
        .07719437129575543228251522505271386)
};
QUAD QUAD_2D_P16_ = {
    "2D P16",           /* name */
    2,              /* dim */
    16,             /* order */
    Length(QUAD_2D_P16_wts),    /* npoints = 55 */
    QUAD_2D_P16_pts,        /* points */
    QUAD_2D_P16_wts,        /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_2D_P17_wts[] = {
    Dup3(.04475687144434462937183647670425513),
    Dup21(.01736688502674779645049111764776038),
    Dup21(.03059934807610353272266564726895704),
    Dup21(.02858770857859978020704009121768920),
    Dup21(.00664743192975369323231849554546676),
    Dup21(.00747618940201851182224557347080098),
    Dup21(.02504998650383874531455895190550778),
    Dup111(.00147981089211964494480953682750477),
    Dup111(.00512113624674810606589435040573260),
    Dup111(.02731735936959280591853168984239567),
    Dup111(.01400572867590928159786633211401179),
    Dup111(.00780927569745836009810987323288001),
    Dup111(.01816572845979167217607207751722372),
    Dup111(.02744437399245832776208345541478450)
};
static FLOAT QUAD_2D_P17_pts[Length(QUAD_2D_P17_wts) * 3] = {
    Perm3(.33333333333333333333333333333333333),
    Perm21(.09569850886271093994316257860237634),
    Perm21(.17013863967877544672324723079568443),
    Perm21(.41802068586795497622263464239342782),
    Perm21(.49658148050662495497054035306227918),
    Perm21(.04166211482880764279207885159834192),
    Perm21(.46793290572942357826819008536171190),
    Perm111(.96953119890372205619454058305953242,
        .02892509162021824607152804771406821),
    Perm111(.75972438753862412955532719532266120,
        .23444175526356877454266053091297880),
    Perm111(.29549931696830150000000000000001000,
        .49591124666075357542303450004375516),
    Perm111(.62560638215769702707019209266693705,
        .35341769454149706762632499077099377),
    Perm111(.87217444723318479290318300141560742,
        .11272864181421976861888886768074196),
    Perm111(.74751231944000604006240678176087530,
        .19907027879785788131339143981558306),
    Perm111(.59886879088323805980616769726351095,
        .30358518307132607653202051204584937)
};
QUAD QUAD_2D_P17_ = {
    "2D P17",           /* name */
    2,              /* dim */
    17,             /* order */
    Length(QUAD_2D_P17_wts),    /* npoints = 61 */
    QUAD_2D_P17_pts,        /* points */
    QUAD_2D_P17_wts,        /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_2D_P18_wts[] = {
    Dup21(.01397786164528602097958400799055493),
    Dup21(.00055490697921321378506845551525090),
    Dup21(.02102681381970466902842986851624498),
    Dup21(.03401821217992769974722652741822109),
    Dup111(.02791016580477499514184347400781694),
    Dup111(.01821468612715086612673395662068577),
    Dup111(.01426702365810979307751982410955666),
    Dup111(.01423712309067505070431276377415596),
    Dup111(.01925758385467478779913738368202133),
    Dup111(.00970513228438064114878227633239023),
    Dup111(.00762978813433212899578245563385336),
    Dup111(.01061873913635034479446354367052828),
    Dup111(.00571066980327583881341421438268949),
    Dup111(.00432685746087641829452234473283270)
};
static FLOAT QUAD_2D_P18_pts[Length(QUAD_2D_P18_wts) * 3] = {
    Perm21(.07327088646438283157861967148768954),
    Perm21(.00391774898322823164278407441958060),
    Perm21(.46759731898871106165151299662296245),
    Perm21(.41791621096741131201212681051399350),
    Perm111(.16538169336028948005449026923917662,
        .56369670566087075380514589393807372),
    Perm111(.28750089440578398999619391313966060,
        .28604232613920474912095810748030295),
    Perm111(.12588931431982479601706483994903803,
        .69604321864246119579257486028195390),
    Perm111(.06322191594650261449357508011699804,
        .76054555188768243261459476379786871),
    Perm111(.07891022745402051775207221037548892,
        .59201963127175856332262057540222541),
    Perm111(.03805805350678571432611899159626213,
        .68368125963599985248012408745381310),
    Perm111(.01429035213045402564992411031307492,
        .85170403713705581502852165344276639),
    Perm111(.01296727234325317231234163433009032,
        .57473249288814902889945093868968971),
    Perm111(.00764859482084089933079262881822729,
        .73551044083072929870313522448164061),
    Perm111(.01271046057225546793114249181358220,
        .93934508764373178870740420268282255)
};
QUAD QUAD_2D_P18_ = {
    "2D P18",           /* name */
    2,              /* dim */
    18,             /* order */
    Length(QUAD_2D_P18_wts),    /* npoints = 72 */
    QUAD_2D_P18_pts,        /* points */
    QUAD_2D_P18_wts,        /* weights */
    -1              /* id */
};
 
/* Note: the rule QUAD_2D_P19 was taken from the book by
 *  P. Solin, K. Segeth, and I. Dolezel,
 *  "Higer-order Finite Element Methods",
 *  Chapman and Hall/CRC Press, 2003. */
static FLOAT QUAD_2D_P19_wts[] = {
    Dup3(.03290633138891865208361434484647497),
    Dup21(.01033073189127205336703996357174833),
    Dup21(.02238724726301639252918455603516271),
    Dup21(.03026612586946807086528019098259122),
    Dup21(.03049096780219778100003158657852042),
    Dup21(.02415921274164090491184803098664001),
    Dup21(.01605080358680087529162277027642948),
    Dup21(.00808458026178406048180567324219442),
    Dup21(.00207936202748478075134750167439841),
    Dup111(.00388487690498138975670499199277266),
    Dup111(.02557416061202190389292970195260027),
    Dup111(.00888090357333805774552592470351753),
    Dup111(.01612454676173139121978526932783766),
    Dup111(.00249194181749067544058464757594956),
    Dup111(.01824284011895057837766571320973615),
    Dup111(.01025856373619852130804807004235813),
    Dup111(.00379992885530191397907315371363970)
};
static FLOAT QUAD_2D_P19_pts[Length(QUAD_2D_P19_wts) * 3] = {
    Perm3(.33333333333333333333333333333333333),
    Perm21(.48960998707300633196613106574829817),
    Perm21(.45453689269789266204675939053572830),
    Perm21(.40141668064943118739399562381068860),
    Perm21(.25555165440309761132218176810926787),
    Perm21(.17707794215212955164267520651590115),
    Perm21(.11006105322795186130008495167737397),
    Perm21(.05552862425183967124867841247135571),
    Perm21(.01262186377722866849023476677870599),
    Perm111(.60063379479464500000000000000000000,
        .39575478735694286230479469406582787),
    Perm111(.13446675453077978561204319893264695,
        .55760326158878396836395324250118097),
    Perm111(.72098702581736505521665290233827892,
        .26456694840652020804030173490121494),
    Perm111(.59452706895587092461388928802650670,
        .35853935220595058842492699064590088),
    Perm111(.83933147368083857861749007714840520,
        .15780740596859474473767360335950651),
    Perm111(.22386142409791569130336938950653642,
        .70108797892617336732328833655951158),
    Perm111(.82293132406985663162747155916053316,
        .14242160111338343731557475687723745),
    Perm111(.92434425262078402945585913790156314,
        .06549462808293770339232652498592557)
};
QUAD QUAD_2D_P19_ = {
    "2D P19",           /* name */
    2,              /* dim */
    19,             /* order */
    Length(QUAD_2D_P19_wts),    /* npoints = 73 */
    QUAD_2D_P19_pts,        /* points */
    QUAD_2D_P19_wts,        /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_2D_P20_wts[] = {
    Dup3(.01253760799449665657358563677239480),
    Dup21(.02747186987642421374845354960735985),
    Dup21(.00976527227705142304136469142942368),
    Dup21(.00139841953539182352392336315978673),
    Dup21(.00929210262518518263042820340303303),
    Dup21(.01657787603236692532602362503518398),
    Dup111(.02066776234866507696142197001297288),
    Dup111(.02082223552115450730687855619932975),
    Dup111(.00956863841984906068887584504583203),
    Dup111(.02445277096897246388564392070240889),
    Dup111(.00315573063063053400382640032072957),
    Dup111(.01213679636532129693701330908075738),
    Dup111(.01496648014388644903652491185157070),
    Dup111(.00632759332177773956932403275043979),
    Dup111(.00134256031206369588497985129814333),
    Dup111(.00277607691634755406772935615580153),
    Dup111(.01073984447418494155517344744795167),
    Dup111(.00536780573818745320524741002126972)
};
static FLOAT QUAD_2D_P20_pts[Length(QUAD_2D_P20_wts) * 3] = {
    Perm3(.33333333333333333333333333333333333),
    Perm21(.21587430593299197319025454384018276),
    Perm21(.07537676652974727809728543094591628),
    Perm21(.01030082813722179211368621600969694),
    Perm21(.49360221129870016551192083214505357),
    Perm21(.46155093810692529674104871029151803),
    Perm111(.32862140642423699330349746095091325,
        .42934057025821037521395880046639840),
    Perm111(.26048036178656875641959301708115346,
        .10157753428096944616875500619617966),
    Perm111(.13707423584645530000000000000000000,
        .71006597300113015998790407454640790),
    Perm111(.14672694587229978430416098848745303,
        .49854547767841484938962269670761193),
    Perm111(.02699897774255329000000000000000000,
        .04918672267258200161970371257758717),
    Perm111(.06187178593361702684171247001223384,
        .77966014654056939536035061907681080),
    Perm111(.04772436742762199620835268010429344,
        .37049153914954763692014962025673877),
    Perm111(.12060051518636437996723378704007933,
        .86334694875475264849798799609252174),
    Perm111(.00269714779670978767164891450128273,
        .05619493818774550298789230198658868),
    Perm111(.00301563327794236265727625982347101,
        .20867500674842135095759446306135771),
    Perm111(.02990537578845701880692877386433865,
        .72115124091203409102810415020509411),
    Perm111(.00675665422246098853994581751922784,
        .64005544194054188990405366827216467)
};
QUAD QUAD_2D_P20_ = {
    "2D P20",                   /* name */
    2,                          /* dim */
    20,                         /* order */
    Length(QUAD_2D_P20_wts),    /* npoints = 88 */
    QUAD_2D_P20_pts,            /* points */
    QUAD_2D_P20_wts,            /* weights */
    -1                          /* id */
};
 
static FLOAT QUAD_2D_P21_wts[] = {
    Dup3(.02756225695287648096690704482451431),
    Dup21(.02206021541348850119135073403311636),
    Dup21(.02346001593867148849301344495230002),
    Dup21(.00032688959504719054621455750154654),
    Dup21(.00326531946293996823433530409586668),
    Dup21(.01175646291541279770430796921338205),
    Dup21(.01178076841991151684555757909867614),
    Dup111(.00226881081880114080533570433430431),
    Dup111(.00259601096443632006067378366548822),
    Dup111(.00463452978587186021234789056159687),
    Dup111(.00479433605454885793485744871991192),
    Dup111(.00571247883672361156725063834296336),
    Dup111(.00586582760432212163695579870000227),
    Dup111(.00941376305909158758981826852034708),
    Dup111(.01341494379665642491002202661089309),
    Dup111(.01571691809208324594350000113784617),
    Dup111(.01686368301443690459165096388619991),
    Dup111(.02139002708532009837783229808035898),
    Dup111(.02307679218949268136788087552189154)
};
static FLOAT QUAD_2D_P21_pts[Length(QUAD_2D_P21_wts) * 3] = {
    Perm3(.33333333333333333333333333333333333),
    Perm21(.20093527706508527987296185156416367),
    Perm21(.43765916596192717973183384418805413),
    Perm21(.00343395649059617685095991220960492),
    Perm21(.04664348477530675349517624043214192),
    Perm21(.38642225176307149094035202416772642),
    Perm21(.09543547110853091010857168104147605),
    Perm111(.95551380335045636050131472514677118,
        .03571862787316335823804160897543867),
    Perm111(.88663881342886822612490057469143760,
        .10814322491564621152738861104631270),
    Perm111(.78426284588043415429664399039819537,
        .20746444959987645682438042951572740),
    Perm111(.88292395505020003271134898731688967,
        .08568470872031694000000000000001000),
    Perm111(.66899196444107724049132248320989459,
        .32149400301428881688168321268348603),
    Perm111(.55207212103556096415716096525277878,
        .43794221879334138355236807696291701),
    Perm111(.79759296559656856762931422329572582,
        .16191645306357785675100677020385905),
    Perm111(.67751471511977148463499116634413256,
        .27450476740199490385900297290733317),
    Perm111(.54299741558909160533113611683919342,
        .40533599807500692794989089537632556),
    Perm111(.70545990556996856165885634154060172,
        .18773768065643534277281674394512005),
    Perm111(.57480057306650846221598245054985001,
        .30569683476605516651279255664984316),
    Perm111(.47177880850461481660397704013492420,
        .31214446687089088167080460581557645)
};
QUAD QUAD_2D_P21_ = {
    "2D P21",           /* name */
    2,              /* dim */
    21,             /* order */
    Length(QUAD_2D_P21_wts),    /* npoints = 91 */
    QUAD_2D_P21_pts,        /* points */
    QUAD_2D_P21_wts,        /* weights */
    -1              /* id */
};
 
/*---------------------------- 3D cubature rules ---------------------------*/
 
static FLOAT QUAD_3D_P1_wts[] = {
    Dup4(1.)
};
static FLOAT QUAD_3D_P1_pts[Length(QUAD_3D_P1_wts) * 4] = {
    Perm4(.25)
};
QUAD QUAD_3D_P1_ = {
    "3D P1",            /* name */
    3,              /* dim */
    1,              /* order */
    Length(QUAD_3D_P1_wts), /* npoints = 1 */
    QUAD_3D_P1_pts,     /* points */
    QUAD_3D_P1_wts,     /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_3D_P2_wts[] = {
    Dup31(.25)
};
static FLOAT QUAD_3D_P2_pts[Length(QUAD_3D_P2_wts) * 4] = {
    /* (5. - sqrt(5.)) / 20, (5. + 3 * sqrt(5.)) / 20 */
    Perm31(.13819660112501051517954131656343619)
};
QUAD QUAD_3D_P2_ = {
    "3D P2",            /* name */
    3,              /* dim */
    2,              /* order */
    Length(QUAD_3D_P2_wts), /* npoints = 4 */
    QUAD_3D_P2_pts,     /* points */
    QUAD_3D_P2_wts,     /* weights */
    -1              /* id */
};
 
#if 0
static FLOAT QUAD_3D_P3_wts[] = {
    Dup31(1./40.),
    Dup31(9./40.)
};
static FLOAT QUAD_3D_P3_pts[Length(QUAD_3D_P3_wts) * 4] = {
    Perm31(0., 1.),
    Perm31(1./3., 0.)
};
#else
static FLOAT QUAD_3D_P3_wts[] = {
    /* 1 / 8 + sqrt((1715161837 - 406006699 * sqrt(17)) / 23101) / 3120 */
    Dup31(.13852796651186214232361769837564129),
    /* 1 / 8 - sqrt((1715161837 - 406006699 * sqrt(17)) / 23101) / 3120 */
    Dup31(.11147203348813785767638230162435871)
};
static FLOAT QUAD_3D_P3_pts[Length(QUAD_3D_P3_wts) * 4] = {
    /* (55 - 3 * sqrt(17) + sqrt(1022 - 134 * sqrt(17))) / 196 */
    Perm31(.32805469671142664733580581998119743),
    /* (55 - 3 * sqrt(17) - sqrt(1022 - 134 * sqrt(17))) / 196 */
    Perm31(.10695227393293068277170204157061650)
};
#endif
QUAD QUAD_3D_P3_ = {
    "3D P3",            /* name */
    3,              /* dim */
    3,              /* order */
    Length(QUAD_3D_P3_wts), /* npoints = 8 */
    QUAD_3D_P3_pts,     /* points */
    QUAD_3D_P3_wts,     /* weights */
    -1              /* id */
};
 
#if 0
static FLOAT QUAD_3D_P4_wts[] = {
    Dup4(-148./1875.),  /* negative weight */
    Dup31(343./7500.),
    Dup22(56./375.)
};
static FLOAT QUAD_3D_P4_pts[Length(QUAD_3D_P4_wts) * 4] = {
    Perm4(0.25),
    Perm31(1./14.),
    Perm22(0.1005964238332008)
};
QUAD QUAD_3D_P4_ = {
    "3D P4",            /* name */
    3,              /* dim */
    4,              /* order */
    11,             /* npoints */
    QUAD_3D_P4_pts,     /* points */
    QUAD_3D_P4_wts,     /* weights */
    -1              /* id */
};
#else
static FLOAT QUAD_3D_P4_wts[] = {
    Dup31(.07349304311636194934358694586367885),
    Dup31(.11268792571801585036501492847638892),
    Dup22(.04254602077708146686093208377328816)
};
static FLOAT QUAD_3D_P4_pts[Length(QUAD_3D_P4_wts) * 4] = {
    Perm31(.09273525031089122628655892066032137),
    Perm31(.31088591926330060975814749494040332),
    Perm22(.04550370412564965000000000000000000)
};
QUAD QUAD_3D_P4_ = {
    "3D P4",            /* name */
    3,              /* dim */
    4,              /* order */
    Length(QUAD_3D_P4_wts), /* npoints = 14 */
    QUAD_3D_P4_pts,     /* points */
    QUAD_3D_P4_wts,     /* weights */
    -1              /* id */
};
#endif
 
#if 0
/* Stroud T3:5-1 p315 */
static FLOAT QUAD_3D_P5_wts[] = {
    Dup4(16./135.),
    /* (2665 + 14 * sqrt(15)) / 37800 */
    Dup31(.07193708377901862),
    /* (2665 - 14 * sqrt(15)) / 37800 */
    Dup31(.06906820722627239),
    Dup22(20./378.)
};
static FLOAT QUAD_3D_P5_pts[Length(QUAD_3D_P5_wts) * 4] = {
    Perm4(.25),
    /* (7 - sqrt(15)) / 34, (13 + 3 * sqrt(15)) / 34 */
    Perm31(.09197107805272303),
    /* (7 + sqrt(15)) / 34, (13 - 3 * sqrt(15)) / 34 */
    Perm31(.31979362782962991),
    /* (10 - 2 * sqrt(15)) / 40, (10 + 2 * sqrt(15)) / 40 */
    Perm22(.05635083268962916)
};
QUAD QUAD_3D_P5_ = {
    "3D P5",            /* name */
    3,              /* dim */
    5,              /* order */
    15,             /* npoints */
    QUAD_3D_P5_pts,     /* points */
    QUAD_3D_P5_wts,     /* weights */
    -1              /* id */
};
#else
static FLOAT QUAD_3D_P5_wts[] = {
    Dup31(.11268792571801585079918565233328633),
    Dup31(.07349304311636194954371020548632750),
    Dup22(.04254602077708146643806942812025744)
 
};
static FLOAT QUAD_3D_P5_pts[Length(QUAD_3D_P5_wts) * 4] = {
    Perm31(.31088591926330060979734573376345783),
    Perm31(.09273525031089122640232391373703061),
    Perm22(.04550370412564964949188052627933943)
};
QUAD QUAD_3D_P5_ = {
    "3D P5",            /* name */
    3,              /* dim */
    5,              /* order */
    Length(QUAD_3D_P5_wts), /* npoints = 14 */
    QUAD_3D_P5_pts,     /* points */
    QUAD_3D_P5_wts,     /* weights */
    -1              /* id */
};
#endif
 
static FLOAT QUAD_3D_P6_wts[] = {
    Dup31(.03992275025816749209969062755747998),
    Dup31(.01007721105532064294801323744593686),
    Dup31(.05535718154365472209515327785372602),
    Dup211(27./560.)
};
static FLOAT QUAD_3D_P6_pts[Length(QUAD_3D_P6_wts) * 4] = {
    Perm31(.21460287125915202928883921938628499),
    Perm31(.04067395853461135311557944895641006),
    Perm31(.32233789014227551034399447076249213),
    /* (3 - sqrt(5)) / 12, (5 + sqrt(5)) / 12, (1 + sqrt(5)) / 12 */
    Perm211(.06366100187501752529923552760572698,
        .60300566479164914136743113906093969)
};
QUAD QUAD_3D_P6_ = {
    "3D P6",            /* name */
    3,              /* dim */
    6,              /* order */
    Length(QUAD_3D_P6_wts), /* npoints = 24 */
    QUAD_3D_P6_pts,     /* points */
    QUAD_3D_P6_wts,     /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_3D_P7_wts[] = {
    Dup4(.09548528946413084886057843611722638),
    Dup31(.04232958120996702907628617079854674),
    Dup22(.03189692783285757993427482408294246),
    Dup211(.03720713072833462136961556119148112),
    Dup211(.00811077082990334156610343349109654)
};
static FLOAT QUAD_3D_P7_pts[Length(QUAD_3D_P7_wts) * 4] = {
    Perm4(.25),
    Perm31(.31570114977820279942342999959331149),
    Perm22(.05048982259839636876305382298656247),
    Perm211(.18883383102600104773643110385458576,
        .57517163758700002348324157702230752),
    Perm211(.02126547254148324598883610149981994,
        .81083024109854856111810537984823239)
};
QUAD QUAD_3D_P7_ = {
    "3D P7",                    /* name */
    3,                          /* dim */
    7,                          /* order */
    Length(QUAD_3D_P7_wts),     /* npoints = 35 */
    QUAD_3D_P7_pts,             /* points */
    QUAD_3D_P7_wts,             /* weights */
    -1                          /* id */
};
 
static FLOAT QUAD_3D_P8_wts[] = {
    Dup31(.00639714777990232132145142033517302),
    Dup31(.04019044802096617248816115847981783),
    Dup31(.02430797550477032117486910877192260),
    Dup31(.05485889241369744046692412399039144),
    Dup22(.03571961223409918246495096899661762),
    Dup211(.00718319069785253940945110521980376),
    Dup211(.01637218194531911754093813975611913)
};
static FLOAT QUAD_3D_P8_pts[Length(QUAD_3D_P8_wts) * 4] = {
    Perm31(.03967542307038990126507132953938949),
    Perm31(.31448780069809631378416056269714830),
    Perm31(.10198669306270330000000000000000000),
    Perm31(.18420369694919151227594641734890918),
    Perm22(.06343628775453989240514123870189827),
    Perm211(.02169016206772800480266248262493018,
        .71993192203946593588943495335273478),
    Perm211(.20448008063679571424133557487274534,
        .58057719012880922417539817139062041)
};
QUAD QUAD_3D_P8_ = {
    "3D P8",            /* name */
    3,              /* dim */
    8,              /* order */
    Length(QUAD_3D_P8_wts), /* npoints = 46 */
    QUAD_3D_P8_pts,     /* points */
    QUAD_3D_P8_wts,     /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_3D_P9_wts[] = {
    Dup4(.05642669317950620658871504327612541),
    Dup31(.00334109507471348040299974430471765),
    Dup31(.03011375476877376390731423843157491),
    Dup31(.00649096092006153463576211689456861),
    Dup211(.00980928586825458643196874259255500),
    Dup211(.02811915382336547255163261742529262),
    Dup211(.00789458690833150076834149200960885),
    Dup211(.01949281204723999671697219448924602)
};
static FLOAT QUAD_3D_P9_pts[Length(QUAD_3D_P9_wts) * 4] = {
    Perm4(.25),
    Perm31(.03402217700104486646540370887876764),
    Perm31(.32277033353380052539137668325496398),
    Perm31(.06045707742577493000000000000000000),
    Perm211(.45536299094720821180030815044164301,
        .00568317736533017990610016014574474),
    Perm211(.11950225539382580097797370469611438,
        .46311683247848994097622449365772955),
    Perm211(.02802195578340115815505750665412373,
        .72520607683986748873856595428480993),
    Perm211(.17483303201157461578532464597224522,
        .61668257178125640457068309097954073)
};
QUAD QUAD_3D_P9_ = {
    "3D P9",            /* name */
    3,              /* dim */
    9,              /* order */
    Length(QUAD_3D_P9_wts), /* npoints = 61 */
    QUAD_3D_P9_pts,     /* points */
    QUAD_3D_P9_wts,     /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_3D_P10_wts[] = {
    Dup4(.04739977355602073838473882117805110),
    Dup31(.02693705999226869980276416100488208),
    Dup31(.00986915971679338323455773543017308),
    Dup211(.00036194434433925362423987838480851),
    Dup211(.01013587167975579278851647011501678),
    Dup211(.01139388122019523162362093488071434),
    Dup211(.00657614727703590416745574020045070),
    Dup211(.02573973198045607127903601225965471),
    Dup211(.01290703579886199063929543024949899)
};
static FLOAT QUAD_3D_P10_pts[Length(QUAD_3D_P10_wts) * 4] = {
    Perm4(.25),
    Perm31(.31225006869518864772980831868682746),
    Perm31(.11430965385734615058737119765365045),
    Perm211(.00613800882479074784759371324841535,
        .94298876734520486619763058691825076),
    Perm211(.03277946821644267077472102033232419,
        .34018479408710763278898792494967132),
    Perm211(.41043073921896549428789784425151169,
        .16548602561961105160449012444452641),
    Perm211(.03248528156482304783551493997842620,
        .13385215221200951309782843596456662),
    Perm211(.12105018114558942599389500159505053,
        .47719037990428035054410640829690722),
    Perm211(.17497934218393902428494922652831040,
        .62807184547536601069327607221790967)
};
QUAD QUAD_3D_P10_ = {
    "3D P10",           /* name */
    3,              /* dim */
    10,             /* order */
    Length(QUAD_3D_P10_wts),    /* npoints = 81 */
    QUAD_3D_P10_pts,        /* points */
    QUAD_3D_P10_wts,        /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_3D_P11_wts[] = {
    Dup4(.03943210802865886350733033449120443),
    Dup31(.01566212622727911315008856276876506),
    Dup31(.00333217237490140814440923615401491),
    Dup31(.01402607740748974743749136099769235),
    Dup211(.00108590752933246630682209837723547),
    Dup211(.02023596043066317891111657316540838),
    Dup211(.01179021487212586353684938046770181),
    Dup211(.00769031498252129590113157802073890),
    Dup211(.00443730570345920390473072602143959),
    Dup211(.01142954846718404041077055259859402),
    Dup1111(.00618564017121781141281925508389534)
};
static FLOAT QUAD_3D_P11_pts[Length(QUAD_3D_P11_wts) * 4] = {
    Perm4(.25),
    Perm31(.12149136777653379449770230990807224),
    Perm31(.03231625915107289635395445208958103),
    Perm31(.32492614978860679781284190241442197),
    Perm211(.00414835697166001200000000000001000,
        .59826599679018635020545384277617780),
    Perm211(.22462461067637714141447515116498644,
        .47366228783234957140836966920205236),
    Perm211(.05190508777256569674422721644265892,
        .56314477790827989873710197630305713),
    Perm211(.13493013121624020422375917234299303,
        .70835883078581895385699500512712996),
    Perm211(.02519119210825247292005118506530550,
        .78371950734007737543057403429990901),
    Perm211(.36531877978173361396933198009886720,
        .13460390831686580000000000000001000),
    Perm1111(.52290753950993847296521692758602923,
         .14075363054369590184253913949127849,
         .00976243819645261550829228038997777)
};
QUAD QUAD_3D_P11_ = {
    "3D P11",           /* name */
    3,              /* dim */
    11,             /* order */
    Length(QUAD_3D_P11_wts),    /* npoints = 109 */
    QUAD_3D_P11_pts,        /* points */
    QUAD_3D_P11_wts,        /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_3D_P12_wts[] = {
    Dup31(.01276763770097074150203778596512505),
    Dup31(.01612110423790926821858154489575762),
    Dup31(.00037161269857844220004255818986081),
    Dup31(.01971744178668545763955330903818868),
    Dup31(.00257139093086271836218234759448548),
    Dup22(.00381724787051057590575318412783326),
    Dup22(.01208722707766311317860318419314605),
    Dup211(.00310586115843473343431688149929620),
    Dup211(.00545953133647103066912742126769441),
    Dup211(.00214289974849699750666852093655947),
    Dup211(.00552467146725782962244930098165075),
    Dup211(.00853695669449918042985177836672201),
    Dup1111(.01151017784832330697333644123403294),
    Dup1111(.00520387865288561360396792421252454)
};
static FLOAT QUAD_3D_P12_pts[Length(QUAD_3D_P12_wts) * 4] = {
    Perm31(.11529974435148014530455720738915911),
    Perm31(.20233628224059090000000000000001000),
    Perm31(.01171759795761995151247906754831398),
    Perm31(.31330644136780106727760279964458934),
    Perm31(.25000573011558370000000000000001000),
    Perm22(.02099547435075800669020182527059018),
    Perm22(.15177401824745010000000000000001000),
    Perm211(.02441977874343536478314000904761661,
        .84832928469787285064520886743481574),
    Perm211(.25620709853201830896382010708562210,
        .48248737387384884780289289672973542),
    Perm211(.01679032097960299061471796028857942,
        .69477194236575592695949850988417719),
    Perm211(.12616082113987204239970703846895919,
        .72541048930294811897485950521263380),
    Perm211(.43143517452637984721670695066371957,
        .11272193989285241520959977211007542),
    Perm1111(.50167006246250569747515507168476130,
         .27247180286952239178351046753060445,
         .07207432880729891465015948456335820),
    Perm1111(.26164485453781874566945505006396799,
         .08629229194706173191742351944352488,
         .02056541065587613830062489762710900)
};
QUAD QUAD_3D_P12_ = {
    "3D P12",           /* name */
    3,              /* dim */
    12,             /* order */
    Length(QUAD_3D_P12_wts),    /* npoints = 140 */
    QUAD_3D_P12_pts,        /* points */
    QUAD_3D_P12_wts,        /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_3D_P13_wts[] = {
    Dup4(.01501368777308314675062970631615983),
    Dup31(.01822520928017342532379068941490097),
    Dup31(.00700610921774146424038518693926311),
    Dup22(.01642354974394954829540573107905531),
    Dup22(.00512061009636059707262596949702171),
    Dup22(.01119669865290491634382032086351956),
    Dup211(.01561914973337995400953811302431969),
    Dup211(.00248442301331647441904056776338473),
    Dup211(.00163859853481823893844525309440751),
    Dup211(.00590303044012492197171914655535865),
    Dup211(.01102208245821805240445097989201525),
    Dup211(.00040645183996417822585155512755848),
    Dup1111(.00268796997296854209745781926651729),
    Dup1111(.00197950480552671190531894675510740),
    Dup1111(.00544631918142579120943187040108667)
};
static FLOAT QUAD_3D_P13_pts[Length(QUAD_3D_P13_wts) * 4] = {
    Perm4(.25),
    Perm31(.15521609351908950314115784335704739),
    Perm31(.33012266333967360024433192595196779),
    Perm22(.16680640389386249928937782601144234),
    Perm22(.02492378854777361779701400374860089),
    Perm22(.09719762991575100143072243716240818),
    Perm211(.24785929015736256692746910620827934,
        .43365324235685144718726061434767377),
    Perm211(.02223159608186700290879521860892929,
        .83690032040373400514509486595698594),
    Perm211(.10727869331305341049150459639584801,
        .77498030597500180756587877274179289),
    Perm211(.19817684388398981142331840582142759,
        .58756930578220530259172017903595920),
    Perm211(.06917924347737931647732534347465502,
        .60420006666006644707935264871115302),
    Perm211(.02311471947193316000000000000001000,
        .93087579279244424864920228882888307),
    Perm1111(.11788928751019608922290117470644250,
         .11651536422540720000000000000001000,
         .04202400112551542095676634303719997),
    Perm1111(.67703279860228426355032221326746594,
         .04616537602461971083458041122176081,
         .00084434031890503975729899692135905),
    Perm1111(.48489008867363312201080094154790828,
         .35888294295520201572423646909421086,
         .13818283491762872996955080907912355)
};
QUAD QUAD_3D_P13_ = {
    "3D P13",           /* name */
    3,              /* dim */
    13,             /* order */
    Length(QUAD_3D_P13_wts),    /* npoints = 171 */
    QUAD_3D_P13_pts,        /* points */
    QUAD_3D_P13_wts,        /* weights */
    -1              /* id */
};
 
static FLOAT QUAD_3D_P14_wts[] = {
    Dup31(.00406511366527076704362088368356360),
    Dup31(.00221453853344557814375995695000715),
    Dup31(.00581343826788845054953733388214554),
    Dup31(.01962554338583572159756233339617148),
    Dup31(.00038757379059082143645387212483937),
    Dup211(.01164297197217703698552134010055516),
    Dup211(.00528904298828171313177368830528561),
    Dup211(.00183108541636005593766978234880692),
    Dup211(.00824964737721464520674496691736603),
    Dup1111(.00300992453470824513768887482089866),
    Dup1111(.00080471656173675346362618087603116),
    Dup1111(.00298504125884930711876556928839215),
    Dup1111(.00568960024187607669633614778119730),
    Dup1111(.00415908658785457156700139801826135),
    Dup1111(.00072823892045727243561364297456536),
    Dup1111(.00543265007699582482162423406519264)
};
static FLOAT QUAD_3D_P14_pts[Length(QUAD_3D_P14_wts) * 4] = {
    Perm31(.32725336252384856390930966926852893),
    Perm31(.04476130446668508088379420964788419),
    Perm31(.08614033110243635365372087402988575),
    Perm31(.20876264250043229682653570839761758),
    Perm31(.01410497380292096006358791521029282),
    Perm211(.10216532418077681234766925269825839,
        .57394636759433382028140028934601068),
    Perm211(.40757005166001071572132956513017833,
        .09222787013902013000000000000000000),
    Perm211(.01566400074028035855575867095780840,
        .70128109595894403271399676732084261),
    Perm211(.22549635625250290537807241542011034,
        .47690639744208871158605833541070112),
    Perm1111(.39059842812814580000000000000000000,
         .20135905441239221681230773272350923,
         .01611228807103002985780269315483708),
    Perm1111(.10613506799890214555561390298480794,
         .03273581868172692849440040779126601,
         .00359790765372716669079715233859245),
    Perm1111(.56363837316977438968968166306485017,
         .23029207223006574545025268741356515,
         .19071993417435518627124877906378985),
    Perm1111(.36762550953258608440922067759911669,
         .20788513802300449507171021252507348,
         .33121048851934490000000000000000000),
    Perm1111(.71923236898172952950234018407969909,
         .17632791180193297621579930336369727,
         .02076023625713100907549734406116442),
    Perm1111(.52782499521529872984092400758172763,
         .43728908922034181655262387608419181,
         .00922016518566419494631775549492202),
    Perm1111(.54836745449481907289949105056077457,
         .34478155061716412287036718709203314,
         .08672172833222153946294387400858277)
};
QUAD QUAD_3D_P14_ = {
    "3D P14",           /* name */
    3,              /* dim */
    14,             /* order */
    Length(QUAD_3D_P14_wts),    /* npoints = 236 */
    QUAD_3D_P14_pts,        /* points */
    QUAD_3D_P14_wts,        /* weights */
    -1              /* id */
};