Package sfc :: Package quadrature :: Module quad_tables
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Source Code for Module sfc.quadrature.quad_tables

   1  #!/usr/bin/env python 
   2  # -*- coding: utf-8 -*- 
   3  """ 
   4  This module contains quadrature rule tables copied from 
   5      P. Solin, K Segeth, and I Dolezel. Higher–Order Finite Element Methods. 
   6      Studies in Advanced Mathematics. Chapman and Hall/CRC, 2004. 
   7   
   8  Use the function load_table(polygon, family) to get a list of QuadRule objects. 
   9  The family can be one of {"composite_gauss", "economical_gauss"}, the polygon name one of the UFC polygon names. 
  10  """ 
  11   
  12  # Copyright (C) 2008 Martin Sandve Alnes and Simula Resarch Laboratory 
  13  # 
  14  # This file is part of SyFi. 
  15  # 
  16  # SyFi is free software: you can redistribute it and/or modify 
  17  # it under the terms of the GNU General Public License as published by 
  18  # the Free Software Foundation, either version 2 of the License, or 
  19  # (at your option) any later version. 
  20  # 
  21  # SyFi is distributed in the hope that it will be useful, 
  22  # but WITHOUT ANY WARRANTY; without even the implied warranty of 
  23  # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 
  24  # GNU General Public License for more details. 
  25  # 
  26  # You should have received a copy of the GNU General Public License 
  27  # along with SyFi. If not, see <http://www.gnu.org/licenses/>. 
  28  # 
  29  # First added:  2008-09-01 
  30  # Last changed: 2008-09-01 
  31   
  32  from sfc.quadrature.QuadRule import QuadRule 
  33   
  34   
  35  # TODO: To add non-Gauss rules here, just write them in the same format 
  36  #       as the gauss strings, and call them myrule_polygonname_table, 
  37  #       just like economical_gauss_triangle_table etc. 
  38   
  39  from ufl.geometry import domain2dim 
  40   
41 -def load_rule(polygon, order, family):
42 43 rules = load_table(polygon, family) 44 45 for repeat in range(10): 46 # find the first rule with this order 47 for r in rules: 48 if r.order == order: 49 return r 50 # if this failed, find the first rule order+1 51 order += 1 52 53 # If that fails, pick the largest order we have 54 maxr = -1 55 rule = rules[0] 56 for r in rules: 57 if r.order > rule.order: 58 maxr = r.order 59 rule = r 60 return rule 61 62 raise RuntimeError("Found no rule for a %s with order %d in family %s" % (polygon, order, family))
63 64
65 -def load_table(polygon, family):
66 """Return list of quadrature rules matching polygon and family.""" 67 rules = parse_table_string(polygon, family) 68 rules = scale_table(rules) 69 return rules
70 71
72 -def parse_table_string(polygon, family):
73 # Fetch a table string with quadrature points and weights for different orders. 74 name = "%s_%s_table" % (family, polygon) 75 tablestring = eval(name) 76 77 nsd = domain2dim[polygon] 78 lines = tablestring.split('\n') 79 rules = [] 80 i = 0 81 while i<len(lines): 82 n = int(lines[i]) 83 i += 1 84 points = [] 85 weights = [] 86 for j in range(n): 87 values = lines[i].split() 88 points += [ map(float, values[:nsd]) ] 89 weights += [ float(values[nsd]) ] 90 i += 1 91 comment = lines[i].strip() 92 i += 2 93 order = int(comment.split('=')[-1].split()[-1]) 94 rules.append( QuadRule(polygon, nsd, order, points, weights, comment) ) 95 return rules
96 97
98 -def scale_table(rules):
99 """Scales a set of quadrature rules from [-1,+1]^nsd to [0,1]^nsd.""" 100 point_offset = 0.5 101 point_scale = 0.5 102 for rule in rules: 103 ws = point_scale**rule.nsd 104 rule.weights = [ w*ws for w in rule.weights ] 105 rule.points = [ [x*point_scale+point_offset for x in p] for p in rule.points ] 106 return rules
107 108 109 # Gauss rules 110 gauss_interval_table = """\ 111 1 112 0.0000000000000000000000000 2.0000000000000000000000000 113 Gauss quadrature points and weights order 2n-1=1 114 115 2 116 -0.5773502691896257645091488 1.0000000000000000000000000 117 0.5773502691896257645091488 1.0000000000000000000000000 118 Gauss quadrature points and weights order 2n-1=3 119 120 3 121 -0.7745966692414833770358531 0.5555555555555555555555556 122 0.0000000000000000000000000 0.8888888888888888888888889 123 0.7745966692414833770358531 0.5555555555555555555555556 124 Gauss quadrature points and weights order 2n-1=5 125 126 4 127 -0.8611363115940525752239465 0.3478548451374538573730639 128 -0.3399810435848562648026658 0.6521451548625461426269361 129 0.3399810435848562648026658 0.6521451548625461426269361 130 0.8611363115940525752239465 0.3478548451374538573730639 131 Gauss quadrature points and weights order 2n-1=7 132 133 5 134 -0.9061798459386639927976269 0.2369268850561890875142640 135 -0.5384693101056830910363144 0.4786286704993664680412915 136 0.0000000000000000000000000 0.5688888888888888888888889 137 0.5384693101056830910363144 0.4786286704993664680412915 138 0.9061798459386639927976269 0.2369268850561890875142640 139 Gauss quadrature points and weights order 2n-1=9 140 141 6 142 -0.9324695142031520278123016 0.1713244923791703450402961 143 -0.6612093864662645136613996 0.3607615730481386075698335 144 -0.2386191860831969086305017 0.4679139345726910473898703 145 0.2386191860831969086305017 0.4679139345726910473898703 146 0.6612093864662645136613996 0.3607615730481386075698335 147 0.9324695142031520278123016 0.1713244923791703450402961 148 Gauss quadrature points and weights order 2n-1=11 149 150 7 151 -0.9491079123427585245261897 0.1294849661688696932706114 152 -0.7415311855993944398638648 0.2797053914892766679014678 153 -0.4058451513773971669066064 0.3818300505051189449503698 154 0.0000000000000000000000000 0.4179591836734693877551020 155 0.4058451513773971669066064 0.3818300505051189449503698 156 0.7415311855993944398638648 0.2797053914892766679014678 157 0.9491079123427585245261897 0.1294849661688696932706114 158 Gauss quadrature points and weights order 2n-1=13 159 160 8 161 -0.9602898564975362316835609 0.1012285362903762591525314 162 -0.7966664774136267395915539 0.2223810344533744705443560 163 -0.5255324099163289858177390 0.3137066458778872873379622 164 -0.1834346424956498049394761 0.3626837833783619829651504 165 0.1834346424956498049394761 0.3626837833783619829651504 166 0.5255324099163289858177390 0.3137066458778872873379622 167 0.7966664774136267395915539 0.2223810344533744705443560 168 0.9602898564975362316835609 0.1012285362903762591525314 169 Gauss quadrature points and weights order 2n-1=15 170 171 9 172 -0.9681602395076260898355762 0.0812743883615744119718922 173 -0.8360311073266357942994297 0.1806481606948574040584720 174 -0.6133714327005903973087020 0.2606106964029354623187429 175 -0.3242534234038089290385380 0.3123470770400028400686304 176 0.0000000000000000000000000 0.3302393550012597631645251 177 0.3242534234038089290385380 0.3123470770400028400686304 178 0.6133714327005903973087020 0.2606106964029354623187429 179 0.8360311073266357942994298 0.1806481606948574040584720 180 0.9681602395076260898355762 0.0812743883615744119718922 181 Gauss quadrature points and weights order 2n-1=17 182 183 10 184 -0.9739065285171717200779640 0.0666713443086881375935688 185 -0.8650633666889845107320967 0.1494513491505805931457763 186 -0.6794095682990244062343274 0.2190863625159820439955349 187 -0.4333953941292471907992659 0.2692667193099963550912269 188 -0.1488743389816312108848260 0.2955242247147528701738930 189 0.1488743389816312108848260 0.2955242247147528701738930 190 0.4333953941292471907992659 0.2692667193099963550912269 191 0.6794095682990244062343274 0.2190863625159820439955349 192 0.8650633666889845107320967 0.1494513491505805931457763 193 0.9739065285171717200779640 0.0666713443086881375935688 194 Gauss quadrature points and weights order 2n-1=19 195 196 11 197 -0.9782286581460569928039380 0.0556685671161736664827537 198 -0.8870625997680952990751578 0.1255803694649046246346943 199 -0.7301520055740493240934162 0.1862902109277342514260976 200 -0.5190961292068118159257257 0.2331937645919904799185237 201 -0.2695431559523449723315320 0.2628045445102466621806889 202 0.0000000000000000000000000 0.2729250867779006307144835 203 0.2695431559523449723315320 0.2628045445102466621806889 204 0.5190961292068118159257257 0.2331937645919904799185237 205 0.7301520055740493240934163 0.1862902109277342514260980 206 0.8870625997680952990751578 0.1255803694649046246346940 207 0.9782286581460569928039380 0.0556685671161736664827537 208 Gauss quadrature points and weights order 2n-1=21 209 210 12 211 -0.9815606342467192506905491 0.0471753363865118271946160 212 -0.9041172563704748566784659 0.1069393259953184309602547 213 -0.7699026741943046870368938 0.1600783285433462263346525 214 -0.5873179542866174472967024 0.2031674267230659217490645 215 -0.3678314989981801937526915 0.2334925365383548087608499 216 -0.1252334085114689154724414 0.2491470458134027850005624 217 0.1252334085114689154724414 0.2491470458134027850005624 218 0.3678314989981801937526915 0.2334925365383548087608499 219 0.5873179542866174472967024 0.2031674267230659217490645 220 0.7699026741943046870368938 0.1600783285433462263346525 221 0.9041172563704748566784659 0.1069393259953184309602547 222 0.9815606342467192506905491 0.0471753363865118271946160 223 Gauss quadrature points and weights order 2n-1=23 224 225 16 226 -0.9894009349916499325961542 0.0271524594117540948517806 227 -0.9445750230732325760779884 0.0622535239386478928628438 228 -0.8656312023878317438804679 0.0951585116824927848099251 229 -0.7554044083550030338951012 0.1246289712555338720524763 230 -0.6178762444026437484466718 0.1495959888165767320815017 231 -0.4580167776572273863424194 0.1691565193950025381893121 232 -0.2816035507792589132304605 0.1826034150449235888667637 233 -0.0950125098376374401853193 0.1894506104550684962853967 234 0.0950125098376374401853193 0.1894506104550684962853967 235 0.2816035507792589132304605 0.1826034150449235888667637 236 0.4580167776572273863424194 0.1691565193950025381893121 237 0.6178762444026437484466718 0.1495959888165767320815017 238 0.7554044083550030338951012 0.1246289712555338720524763 239 0.8656312023878317438804679 0.0951585116824927848099251 240 0.9445750230732325760779884 0.0622535239386478928628438 241 0.9894009349916499325961542 0.0271524594117540948517806 242 Gauss quadrature points and weights order 2n-1=31 243 244 20 245 -0.9931285991850949247861224 0.0176140071391521183118620 246 -0.9639719272779137912676661 0.0406014298003869413310400 247 -0.9122344282513259058677524 0.0626720483341090635695065 248 -0.8391169718222188233945291 0.0832767415767047487247581 249 -0.7463319064601507926143051 0.1019301198172404350367501 250 -0.6360536807265150254528367 0.1181945319615184173123774 251 -0.5108670019508270980043641 0.1316886384491766268984945 252 -0.3737060887154195606725482 0.1420961093183820513292983 253 -0.2277858511416450780804962 0.1491729864726037467878287 254 -0.0765265211334973337546404 0.1527533871307258506980843 255 0.0765265211334973337546404 0.1527533871307258506980843 256 0.2277858511416450780804962 0.1491729864726037467878287 257 0.3737060887154195606725482 0.1420961093183820513292983 258 0.5108670019508270980043641 0.1316886384491766268984945 259 0.6360536807265150254528367 0.1181945319615184173123774 260 0.7463319064601507926143051 0.1019301198172404350367501 261 0.8391169718222188233945291 0.0832767415767047487247581 262 0.9122344282513259058677524 0.0626720483341090635695065 263 0.9639719272779137912676661 0.0406014298003869413310400 264 0.9931285991850949247861224 0.0176140071391521183118620 265 Gauss quadrature points and weights order 2n-1=39 266 267 24 268 -0.9951872199970213601799974 0.0123412297999871995468057 269 -0.9747285559713094981983919 0.0285313886289336631813078 270 -0.9382745520027327585236490 0.0442774388174198061686027 271 -0.8864155270044010342131543 0.0592985849154367807463678 272 -0.8200019859739029219539499 0.0733464814110803057340336 273 -0.7401241915785543642438281 0.0861901615319532759171852 274 -0.6480936519369755692524958 0.0976186521041138882698807 275 -0.5454214713888395356583756 0.1074442701159656347825773 276 -0.4337935076260451384870842 0.1155056680537256013533445 277 -0.3150426796961633743867933 0.1216704729278033912044632 278 -0.1911188674736163091586398 0.1258374563468282961213754 279 -0.0640568928626056260850431 0.1279381953467521569740562 280 0.0640568928626056260850431 0.1279381953467521569740562 281 0.1911188674736163091586398 0.1258374563468282961213754 282 0.3150426796961633743867933 0.1216704729278033912044632 283 0.4337935076260451384870842 0.1155056680537256013533445 284 0.5454214713888395356583756 0.1074442701159656347825773 285 0.6480936519369755692524958 0.0976186521041138882698807 286 0.7401241915785543642438281 0.0861901615319532759171852 287 0.8200019859739029219539499 0.0733464814110803057340336 288 0.8864155270044010342131543 0.0592985849154367807463678 289 0.9382745520027327585236490 0.0442774388174198061686027 290 0.9747285559130949819839199 0.0285313886289336631813078 291 0.9951872199970213601799974 0.0123412297999871995468057 292 Gauss quadrature points and weights order 2n-1=47 293 294 32 295 -0.9972638618494815635449811 0.0070186100094700966004071 296 -0.9856115115452683354001750 0.0162743947309056706051706 297 -0.9647622555875064307738119 0.0253920653092620594557526 298 -0.9349060759377396891709191 0.0342738629130214331026877 299 -0.8963211557660521239653072 0.0428358980222266806568786 300 -0.8493676137325699701336930 0.0509980592623761761961632 301 -0.7944837959679424069630973 0.0586840934785355471452836 302 -0.7321821187402896803874267 0.0658222227763618468376501 303 -0.6630442669302152009751152 0.0723457941088485062253994 304 -0.5877157572407623290407455 0.0781938957870703064717409 305 -0.5068999089322293900237475 0.0833119242269467552221991 306 -0.4213512761306353453641194 0.0876520930044038111427715 307 -0.3318686022821276497799168 0.0911738786957638847128686 308 -0.2392873622521370745446032 0.0938443990808045656391802 309 -0.1444719615827964934851864 0.0956387200792748594190820 310 -0.0483076656877383162348126 0.0965400885147278005667648 311 0.0483076656877383162348126 0.0965400885147278005667648 312 0.1444719615827964934851864 0.0956387200792748594190820 313 0.2392873622521370745446032 0.0938443990808045656391802 314 0.3318686022821276497799168 0.0911738786957638847128686 315 0.4213512761306353453641194 0.0876520930044038111427715 316 0.5068999089322293900237475 0.0833119242269467552221991 317 0.5877157572407623290407455 0.0781938957870703064717409 318 0.6630442669302152009751152 0.0723457941088485062253994 319 0.7321821187402896803874267 0.0658222227763618468376501 320 0.7944837959679424069630973 0.0586840934785355471452836 321 0.8493676137325699701336930 0.0509980592623761761961632 322 0.8963211557660521239653072 0.0428358980222266806568787 323 0.9349060759377396891709191 0.0342738629130214331026877 324 0.9647622555875064307738119 0.0253920653092620594557526 325 0.9856115115452683354001750 0.0162743947309056706051706 326 0.9972638618494815635449811 0.0070186100094700966004071 327 Gauss quadrature points and weights order 2n-1=63 328 329 40 330 -0.9982377097105592003496227 0.0045212770985331912584717 331 -0.9907262386994570064530544 0.0104982845311528136147422 332 -0.9772599499837742626633703 0.0164210583819078887128635 333 -0.9579168192137916558045410 0.0222458491941669572615043 334 -0.9328128082786765333608522 0.0279370069800234010984892 335 -0.9020988069688742967282533 0.0334601952825478473926782 336 -0.8659595032122595038207818 0.0387821679744720176399720 337 -0.8246122308333116631963202 0.0438709081856732719916747 338 -0.7783056514265193876949715 0.0486958076350722320614342 339 -0.7273182551899271032809965 0.0532278469839368243549965 340 -0.6719566846141795483793545 0.0574397690993915513666177 341 -0.6125538896679802379526125 0.0613062424929289391665380 342 -0.5494671250951282020759313 0.0648040134566010380745545 343 -0.4830758016861787129085666 0.0679120458152339038256901 344 -0.4137792043716050015248797 0.0706116473912867796954836 345 -0.3419940908257584730074925 0.0728865823958040590605107 346 -0.2681521850072536811411843 0.0747231690579682642001893 347 -0.1926975807013710997155169 0.0761103619006262423715581 348 -0.1160840706752552084834513 0.0770398181642479655883075 349 -0.0387724175060508219331934 0.0775059479784248112637240 350 0.0387724175060508219331934 0.0775059479784248112637240 351 0.1160840706752552084834513 0.0770398181642479655883075 352 0.1926975807013710997155169 0.0761103619006262423715581 353 0.2681521850072536811411843 0.0747231690579682642001893 354 0.3419940908257584730074925 0.0728865823958040590605107 355 0.4137792043716050015248797 0.0706116473912867796954836 356 0.4830758016861787129085666 0.0679120458152339038256901 357 0.5494671250951282020759313 0.0648040134566010380745545 358 0.6125538896679802379526125 0.0613062424929289391665380 359 0.6719566846141795483793545 0.0574397690993915513666177 360 0.7273182551899271032809965 0.0532278469839368243549965 361 0.7783056514265193876949715 0.0486958076350722320614342 362 0.8246122308333116631963202 0.0438709081856732719916747 363 0.8659595032122595038207818 0.0387821679744720176399720 364 0.9020988069688742967282533 0.0334601952825478473926782 365 0.9328128082786765333608522 0.0279370069800234010984892 366 0.9579168192137916558045410 0.0222458491941669572615040 367 0.9772599499837742626633703 0.0164210583819078887128635 368 0.9907262386994570064530544 0.0104982845311528136147422 369 0.9982377097105592003496227 0.0045212770985331912584717 370 Gauss quadrature points and weights order 2n-1=79 371 372 48 373 -0.9987710072524261186005415 0.0031533460523058386326773 374 -0.9935301722663507575479288 0.0073275539012762621023840 375 -0.9841245837228268577445836 0.0114772345792345394895927 376 -0.9705915925462472504614120 0.0155793157229438487281770 377 -0.9529877031604308607229607 0.0196161604573555278144607 378 -0.9313866907065543331141744 0.0235707608393243791405193 379 -0.9058791367155696728220748 0.0274265097083569482000738 380 -0.8765720202742478859056936 0.0311672278327980889020658 381 -0.8435882616243935307110898 0.0347772225647704388925486 382 -0.8070662040294426270825530 0.0382413510658307063172173 383 -0.7671590325157403392538554 0.0415450829434647492140588 384 -0.7240341309238146546744822 0.0446745608566942804194486 385 -0.6778723796326639052118513 0.0476166584924904748259066 386 -0.6288673967765136239951649 0.0503590355538544749578076 387 -0.5772247260839727038178092 0.0528901894851936670955051 388 -0.5231609747222330336782259 0.0551995036999841628682035 389 -0.4669029047509584045449289 0.0572772921004032157051502 390 -0.4086864819907167299162255 0.0591148396983956357464748 391 -0.3487558862921607381598179 0.0607044391658938800529692 392 -0.2873624873554555767358865 0.0620394231598926639041978 393 -0.2247637903946890612248654 0.0631141922862540256571260 394 -0.1612223560688917180564374 0.0639242385846481866239062 395 -0.0970046992094626989300540 0.0644661644359500822065042 396 -0.0323801709628693620333222 0.0647376968126839225030249 397 0.0323801709628693620333222 0.0647376968126839225030249 398 0.0970046992094626989300540 0.0644661644359500822065042 399 0.1612223560688917180564374 0.0639242385846481866239062 400 0.2247637903946890612248654 0.0631141922862540256571260 401 0.2873624873554555767358865 0.0620394231598926639041978 402 0.3487558862921607381598179 0.0607044391658938800529692 403 0.4086864819907167299162255 0.0591148396983956357464748 404 0.4669029047509584045449289 0.0572772921004032157051502 405 0.5231609747222330336782259 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492 economical_gauss_quadrilateral_table = """\ 493 1 494 0.000000000000000 0.000000000000000 4.000000000000000 495 Gauss quadrature points and weights on the reference quadrilateral order p=0 1 496 497 4 498 0.577350269189626 0.577350269189626 1.000000000000000 499 0.577350269189626 -0.577350269189626 1.000000000000000 500 -0.577350269189626 0.577350269189626 1.000000000000000 501 -0.577350269189626 -0.577350269189626 1.000000000000000 502 Gauss quadrature points and weights on the reference quadrilateral order p=2 3 503 504 8 505 0.683130051063973 0.000000000000000 0.816326530612245 506 -0.683130051063973 0.000000000000000 0.816326530612245 507 0.000000000000000 0.683130051063973 0.816326530612245 508 0.000000000000000 -0.683130051063973 0.816326530612245 509 0.881917103688197 0.881917103688197 0.183673469387755 510 0.881917103688197 -0.881917103688197 0.183673469387755 511 -0.881917103688197 0.881917103688197 0.183673469387755 512 -0.881917103688197 -0.881917103688197 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2222 -0.538990774213829 0.754378926224437 0.258107746839302 0.039008180977840 2223 -0.538990774213829 0.754378926224437 -0.258107746839302 0.039008180977840 2224 -0.538990774213829 0.258107746839302 0.754378926224437 0.039008180977840 2225 -0.538990774213829 -0.258107746839302 0.754378926224437 0.039008180977840 2226 -0.538990774213829 -0.754378926224437 0.258107746839302 0.039008180977840 2227 -0.538990774213829 -0.754378926224437 -0.258107746839302 0.039008180977840 2228 -0.538990774213829 0.258107746839302 -0.754378926224437 0.039008180977840 2229 -0.538990774213829 -0.258107746839302 -0.754378926224437 0.039008180977840 2230 0.258107746839302 0.754378926224437 0.538990774213829 0.039008180977840 2231 0.258107746839302 0.754378926224437 -0.538990774213829 0.039008180977840 2232 0.258107746839302 0.538990774213829 0.754378926224437 0.039008180977840 2233 0.258107746839302 -0.538990774213829 0.754378926224437 0.039008180977840 2234 0.258107746839302 -0.754378926224437 0.538990774213829 0.039008180977840 2235 0.258107746839302 -0.754378926224437 -0.538990774213829 0.039008180977840 2236 0.258107746839302 0.538990774213829 -0.754378926224437 0.039008180977840 2237 0.258107746839302 -0.538990774213829 -0.754378926224437 0.039008180977840 2238 -0.258107746839302 0.754378926224437 0.538990774213829 0.039008180977840 2239 -0.258107746839302 0.754378926224437 -0.538990774213829 0.039008180977840 2240 -0.258107746839302 0.538990774213829 0.754378926224437 0.039008180977840 2241 -0.258107746839302 -0.538990774213829 0.754378926224437 0.039008180977840 2242 -0.258107746839302 -0.754378926224437 0.538990774213829 0.039008180977840 2243 -0.258107746839302 -0.754378926224437 -0.538990774213829 0.039008180977840 2244 -0.258107746839302 0.538990774213829 -0.754378926224437 0.039008180977840 2245 -0.258107746839302 -0.538990774213829 -0.754378926224437 0.039008180977840 2246 Gauss quadrature points and weights on the reference cube order p=18 19 2247 """ 2248 2249 2250 economical_gauss_triangle_table = """\ 2251 1 2252 -0.333333333333333 -0.333333333333333 2.000000000000000 2253 Gauss quadrature points and weights on the reference triangle order p=1 2254 2255 3 2256 -0.666666666666667 -0.666666666666667 0.666666666666667 2257 -0.666666666666667 0.333333333333333 0.666666666666667 2258 0.333333333333333 -0.666666666666667 0.666666666666667 2259 Gauss quadrature points and weights on the reference triangle order p=2 2260 2261 4 2262 -0.333333333333333 -0.333333333333333 -1.125000000000000 2263 -0.600000000000000 -0.600000000000000 1.041666666666667 2264 -0.600000000000000 0.200000000000000 1.041666666666667 2265 0.200000000000000 -0.600000000000000 1.041666666666667 2266 Gauss quadrature points and weights on the reference triangle order p=3 2267 2268 6 2269 -0.108103018168070 -0.108103018168070 0.446763179356022 2270 -0.108103018168070 -0.783793963663860 0.446763179356022 2271 -0.783793963663860 -0.108103018168070 0.446763179356022 2272 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0.797811086731876 0.064916995246395 2328 0.797811086731876 -0.898905543365938 0.064916995246396 2329 -0.473774340730724 0.456984785910808 0.054460628348870 2330 0.456984785910808 -0.983210445180084 0.054460628348870 2331 -0.983210445180084 -0.473774340730724 0.054460628348870 2332 -0.473774340730724 -0.983210445180084 0.054460628348870 2333 0.456984785910808 -0.473774340730724 0.054460628348870 2334 -0.983210445180084 0.456984785910808 0.054460628348870 2335 Gauss quadrature points and weights on the reference triangle order p=8 2336 2337 19 2338 -0.333333333333333 -0.333333333333333 0.194271592565598 2339 -0.020634961602524 -0.020634961602524 0.062669400454278 2340 -0.020634961602524 -0.958730076794950 0.062669400454278 2341 -0.958730076794950 -0.020634961602524 0.062669400454278 2342 -0.125820817014126 -0.125820817014126 0.155655082009548 2343 -0.125820817014126 -0.748358365971746 0.155655082009548 2344 -0.748358365971746 -0.125820817014126 0.155655082009548 2345 -0.623592928761934 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0.018843333927466 2380 0.847311867175000 -0.980918369199402 0.018843333927466 2381 -0.980918369199402 -0.866393497975600 0.018843333927466 2382 -0.866393497975600 -0.980918369199402 0.018843333927466 2383 0.847311867175000 -0.866393497975600 0.018843333927466 2384 -0.980918369199402 0.847311867175000 0.018843333927466 2385 Gauss quadrature points and weights on the reference triangle order p=10 2386 2387 27 2388 0.069222096541516 0.069222096541516 0.001854012657922 2389 0.069222096541516 -1.138444193083034 0.001854012657922 2390 -1.138444193083034 0.069222096541516 0.001854012657922 2391 -0.202061394068290 -0.202061394068290 0.154299069829626 2392 -0.202061394068290 -0.595877211863420 0.154299069829626 2393 -0.595877211863420 -0.202061394068290 0.154299069829626 2394 -0.593380199137436 -0.593380199137436 0.118645954761548 2395 -0.593380199137436 0.186760398274870 0.118645954761548 2396 0.186760398274870 -0.593380199137436 0.118645954761548 2397 -0.761298175434838 -0.761298175434838 0.072369081006836 2398 -0.761298175434838 0.522596350869674 0.072369081006836 2399 0.522596350869674 -0.761298175434838 0.072369081006836 2400 -0.935270103777448 -0.935270103777448 0.027319462005356 2401 -0.935270103777448 0.870540207554896 0.027319462005356 2402 0.870540207554896 -0.935270103777448 0.027319462005356 2403 -0.286758703477414 0.186402426856426 0.104674223924408 2404 0.186402426856426 -0.899643723379010 0.104674223924408 2405 -0.899643723379010 -0.286758703477414 0.104674223924408 2406 -0.286758703477414 -0.899643723379010 0.104674223924408 2407 0.186402426856426 -0.286758703477414 0.104674223924408 2408 -0.899643723379010 0.186402426856426 0.104674223924408 2409 -0.657022039391916 0.614978006319584 0.041415319278282 2410 0.614978006319584 -0.957955966927668 0.041415319278282 2411 -0.957955966927668 -0.657022039391916 0.041415319278282 2412 -0.657022039391916 -0.957955966927668 0.041415319278282 2413 0.614978006319584 -0.657022039391916 0.041415319278282 2414 -0.957955966927668 0.614978006319584 0.041415319278282 2415 Gauss quadrature points and weights on the reference triangle order p=11 2416 2417 33 2418 -0.023565220452390 -0.023565220452390 0.051462132880910 2419 -0.023565220452390 -0.952869559095220 0.051462132880910 2420 -0.952869559095220 -0.023565220452390 0.051462132880910 2421 -0.120551215411080 -0.120551215411080 0.087385089076076 2422 -0.120551215411080 -0.758897569177842 0.087385089076076 2423 -0.758897569177842 -0.120551215411080 0.087385089076076 2424 -0.457579229975768 -0.457579229975768 0.125716448435770 2425 -0.457579229975768 -0.084841540048464 0.125716448435770 2426 -0.084841540048464 -0.457579229975768 0.125716448435770 2427 -0.744847708916828 -0.744847708916828 0.069592225861418 2428 -0.744847708916828 0.489695417833656 0.069592225861418 2429 0.489695417833656 -0.744847708916828 0.069592225861418 2430 -0.957365299093580 -0.957365299093580 0.012332522103118 2431 -0.957365299093580 0.914730598187158 0.012332522103118 2432 0.914730598187158 -0.957365299093580 0.012332522103118 2433 -0.448573460628972 0.217886471559576 0.080743115532762 2434 0.217886471559576 -0.769313010930604 0.080743115532762 2435 -0.769313010930604 -0.448573460628972 0.080743115532762 2436 -0.448573460628972 -0.769313010930604 0.080743115532762 2437 0.217886471559576 -0.448573460628972 0.080743115532762 2438 -0.769313010930604 0.217886471559576 0.080743115532762 2439 -0.437348838020120 0.391672173575606 0.044713546404606 2440 0.391672173575606 -0.954323335555486 0.044713546404606 2441 -0.954323335555486 -0.437348838020120 0.044713546404606 2442 -0.437348838020120 -0.954323335555486 0.044713546404606 2443 0.391672173575606 -0.437348838020120 0.044713546404606 2444 -0.954323335555486 0.391672173575606 0.044713546404606 2445 -0.767496168184806 0.716028067088146 0.034632462217318 2446 0.716028067088146 -0.948531898903340 0.034632462217318 2447 -0.948531898903340 -0.767496168184806 0.034632462217318 2448 -0.767496168184806 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0.036709828212560 2919 -0.706069127893522 0.722805434309974 0.001408809355816 2920 0.722805434309974 -1.016736306416454 0.001408809355816 2921 -1.016736306416454 -0.706069127893522 0.001408809355816 2922 -0.706069127893522 -1.016736306416454 0.001408809355816 2923 0.722805434309974 -0.706069127893522 0.001408809355816 2924 -1.016736306416454 0.722805434309974 0.001408809355816 2925 -0.724546042342154 0.671173915824726 0.020225369854924 2926 0.671173915824726 -0.946627873482572 0.020225369854924 2927 -0.946627873482572 -0.724546042342154 0.020225369854924 2928 -0.724546042342154 -0.946627873482572 0.020225369854924 2929 0.671173915824726 -0.724546042342154 0.020225369854924 2930 -0.946627873482572 0.671173915824726 0.020225369854924 2931 -0.880607781701986 0.859512343113706 0.007147818771900 2932 0.859512343113706 -0.978904561411718 0.007147818771900 2933 -0.978904561411718 -0.880607781701986 0.007147818771900 2934 -0.880607781701986 -0.978904561411718 0.007147818771900 2935 0.859512343113706 -0.880607781701986 0.007147818771900 2936 -0.978904561411718 0.859512343113706 0.007147818771900 2937 Gauss quadrature points and weights on the reference triangle order p=20 2938 """ 2939 2940 2941 economical_gauss_tetrahedron_table = """\ 2942 1 2943 -0.500000000000000 -0.500000000000000 -0.500000000000000 1.333333333333333 2944 Gauss quadrature constants for the reference tetrahedron order p=1 2945 2946 4 2947 -0.723606797749979 -0.723606797749979 -0.723606797749979 0.333333333333333 2948 0.170820393249937 -0.723606797749979 -0.723606797749979 0.333333333333333 2949 -0.723606797749979 0.170820393249937 -0.723606797749979 0.333333333333333 2950 -0.723606797749979 -0.723606797749979 0.170820393249937 0.333333333333333 2951 Gauss quadrature constants for the reference tetrahedron order p=2 2952 2953 5 2954 -0.500000000000000 -0.500000000000000 -0.500000000000000 -1.066666666666667 2955 -0.666666666666667 -0.666666666666667 -0.666666666666667 0.600000000000000 2956 -0.666666666666667 -0.666666666666667 0.000000000000000 0.600000000000000 2957 -0.666666666666667 0.000000000000000 -0.666666666666667 0.600000000000000 2958 0.000000000000000 -0.666666666666667 -0.666666666666667 0.600000000000000 2959 Gauss quadrature constants for the reference tetrahedron order p=3 2960 2961 11 2962 -0.500000000000000 -0.500000000000000 -0.500000000000000 -0.105244444444444 2963 -0.857142857142857 -0.857142857142857 -0.857142857142857 0.060977777777778 2964 -0.857142857142857 -0.857142857142857 0.571428571428571 0.060977777777778 2965 -0.857142857142857 0.571428571428571 -0.857142857142857 0.060977777777778 2966 0.571428571428571 -0.857142857142857 -0.857142857142857 0.060977777777778 2967 -0.201192847666402 -0.201192847666402 -0.798807152333598 0.199111111111111 2968 -0.201192847666402 -0.798807152333598 -0.201192847666402 0.199111111111111 2969 -0.798807152333598 -0.201192847666402 -0.201192847666402 0.199111111111111 2970 -0.201192847666402 -0.798807152333598 -0.798807152333598 0.199111111111111 2971 -0.798807152333598 -0.201192847666402 -0.798807152333598 0.199111111111111 2972 -0.798807152333598 -0.798807152333598 -0.201192847666402 0.199111111111111 2973 Gauss quadrature constants for the reference tetrahedron order p=4 2974 2975 14 2976 -0.814529499378218 -0.814529499378218 -0.814529499378218 0.097990724155149 2977 0.443588498134653 -0.814529499378218 -0.814529499378218 0.097990724155149 2978 -0.814529499378218 0.443588498134653 -0.814529499378218 0.097990724155149 2979 -0.814529499378218 -0.814529499378218 0.443588498134653 0.097990724155149 2980 -0.378228161473399 -0.378228161473399 -0.378228161473399 0.150250567624021 2981 -0.865315515579804 -0.378228161473399 -0.378228161473399 0.150250567624021 2982 -0.378228161473399 -0.865315515579804 -0.378228161473399 0.150250567624021 2983 -0.378228161473399 -0.378228161473399 -0.865315515579804 0.150250567624021 2984 -0.091007408251299 -0.091007408251299 -0.908992591748701 0.056728027702775 2985 -0.091007408251299 -0.908992591748701 -0.091007408251299 0.056728027702775 2986 -0.908992591748701 -0.091007408251299 -0.091007408251299 0.056728027702775 2987 -0.091007408251299 -0.908992591748701 -0.908992591748701 0.056728027702775 2988 -0.908992591748701 -0.091007408251299 -0.908992591748701 0.056728027702775 2989 -0.908992591748701 -0.908992591748701 -0.091007408251299 0.056728027702775 2990 Gauss quadrature constants for the reference tetrahedron order p=5 2991 2992 24 2993 -0.570794257481696 -0.570794257481696 -0.570794257481696 0.053230333677557 2994 -0.287617227554912 -0.570794257481696 -0.570794257481696 0.053230333677557 2995 -0.570794257481696 -0.287617227554912 -0.570794257481696 0.053230333677557 2996 -0.570794257481696 -0.570794257481696 -0.287617227554912 0.053230333677557 2997 -0.918652082930777 -0.918652082930777 -0.918652082930777 0.013436281407094 2998 0.755956248792332 -0.918652082930777 -0.918652082930777 0.013436281407094 2999 -0.918652082930777 0.755956248792332 -0.918652082930777 0.013436281407094 3000 -0.918652082930777 -0.918652082930777 0.755956248792332 0.013436281407094 3001 -0.355324219715449 -0.355324219715449 -0.355324219715449 0.073809575391540 3002 -0.934027340853653 -0.355324219715449 -0.355324219715449 0.073809575391540 3003 -0.355324219715449 -0.934027340853653 -0.355324219715449 0.073809575391540 3004 -0.355324219715449 -0.355324219715449 -0.934027340853653 0.073809575391540 3005 -0.872677996249965 -0.872677996249965 -0.460655337083368 0.064285714285714 3006 -0.872677996249965 -0.460655337083368 -0.872677996249965 0.064285714285714 3007 -0.872677996249965 -0.872677996249965 0.206011329583298 0.064285714285714 3008 -0.872677996249965 0.206011329583298 -0.872677996249965 0.064285714285714 3009 -0.872677996249965 -0.460655337083368 0.206011329583298 0.064285714285714 3010 -0.872677996249965 0.206011329583298 -0.460655337083368 0.064285714285714 3011 -0.460655337083368 -0.872677996249965 -0.872677996249965 0.064285714285714 3012 -0.460655337083368 -0.872677996249965 0.206011329583298 0.064285714285714 3013 -0.460655337083368 0.206011329583298 -0.872677996249965 0.064285714285714 3014 0.206011329583298 -0.872677996249965 -0.460655337083368 0.064285714285714 3015 0.206011329583298 -0.872677996249965 -0.872677996249965 0.064285714285714 3016 0.206011329583298 -0.460655337083368 -0.872677996249965 0.064285714285714 3017 Gauss quadrature constants for the reference tetrahedron order p=6 3018 3019 31 3020 0.000000000000000 0.000000000000000 -1.000000000000000 0.007760141093474 3021 0.000000000000000 -1.000000000000000 0.000000000000000 0.007760141093474 3022 -1.000000000000000 0.000000000000000 0.000000000000000 0.007760141093474 3023 -1.000000000000000 -1.000000000000000 0.000000000000000 0.007760141093474 3024 -1.000000000000000 0.000000000000000 -1.000000000000000 0.007760141093474 3025 0.000000000000000 -1.000000000000000 -1.000000000000000 0.007760141093474 3026 -0.500000000000000 -0.500000000000000 -0.500000000000000 0.146113787728871 3027 -0.843573615339364 -0.843573615339364 -0.843573615339364 0.084799532195309 3028 -0.843573615339364 -0.843573615339364 0.530720846018092 0.084799532195309 3029 -0.843573615339364 0.530720846018092 -0.843573615339364 0.084799532195309 3030 0.530720846018092 -0.843573615339364 -0.843573615339364 0.084799532195309 3031 -0.756313566672190 -0.756313566672190 -0.756313566672190 -0.500141920914655 3032 -0.756313566672190 -0.756313566672190 0.268940700016569 -0.500141920914655 3033 -0.756313566672190 0.268940700016569 -0.756313566672190 -0.500141920914655 3034 0.268940700016569 -0.756313566672190 -0.756313566672190 -0.500141920914655 3035 -0.334921671107159 -0.334921671107159 -0.334921671107159 0.039131402104588 3036 -0.334921671107159 -0.334921671107159 -0.995234986678524 0.039131402104588 3037 -0.334921671107159 -0.995234986678524 -0.334921671107159 0.039131402104588 3038 -0.995234986678524 -0.334921671107159 -0.334921671107159 0.039131402104588 3039 -0.800000000000000 -0.800000000000000 -0.600000000000000 0.220458553791887 3040 -0.800000000000000 -0.600000000000000 -0.800000000000000 0.220458553791887 3041 -0.800000000000000 -0.800000000000000 0.200000000000000 0.220458553791887 3042 -0.800000000000000 0.200000000000000 -0.800000000000000 0.220458553791887 3043 -0.800000000000000 -0.600000000000000 0.200000000000000 0.220458553791887 3044 -0.800000000000000 0.200000000000000 -0.600000000000000 0.220458553791887 3045 -0.600000000000000 -0.800000000000000 -0.800000000000000 0.220458553791887 3046 -0.600000000000000 -0.800000000000000 0.200000000000000 0.220458553791887 3047 -0.600000000000000 0.200000000000000 -0.800000000000000 0.220458553791887 3048 0.200000000000000 -0.800000000000000 -0.600000000000000 0.220458553791887 3049 0.200000000000000 -0.800000000000000 -0.800000000000000 0.220458553791887 3050 0.200000000000000 -0.600000000000000 -0.800000000000000 0.220458553791887 3051 Gauss quadrature constants for the reference tetrahedron order p=7 3052 3053 43 3054 -0.500000000000000 -0.500000000000000 -0.500000000000000 -0.164001509269119 3055 -0.586340136778654 -0.586340136778654 -0.586340136778654 0.114002446582935 3056 -0.586340136778654 -0.586340136778654 -0.240979589664039 0.114002446582935 3057 -0.586340136778654 -0.240979589664039 -0.586340136778654 0.114002446582935 3058 -0.240979589664039 -0.586340136778654 -0.586340136778654 0.114002446582935 3059 -0.835792823378907 -0.835792823378907 -0.835792823378907 0.015736266505071 3060 -0.835792823378907 -0.835792823378907 0.507378470136720 0.015736266505071 3061 -0.835792823378907 0.507378470136720 -0.835792823378907 0.015736266505071 3062 0.507378470136720 -0.835792823378907 -0.835792823378907 0.015736266505071 3063 -0.988436098989604 -0.988436098989604 -0.988436098989604 0.001358672872743 3064 -0.988436098989604 -0.988436098989604 0.965308296968812 0.001358672872743 3065 -0.988436098989604 0.965308296968812 -0.988436098989604 0.001358672872743 3066 0.965308296968812 -0.988436098989604 -0.988436098989604 0.001358672872743 3067 -0.898934519962212 -0.898934519962212 -0.101065480037788 0.036637470595738 3068 -0.898934519962212 -0.101065480037788 -0.898934519962212 0.036637470595738 3069 -0.101065480037788 -0.898934519962212 -0.898934519962212 0.036637470595738 3070 -0.898934519962212 -0.101065480037788 -0.101065480037788 0.036637470595738 3071 -0.101065480037788 -0.898934519962212 -0.101065480037788 0.036637470595738 3072 -0.101065480037788 -0.101065480037788 -0.898934519962212 0.036637470595738 3073 -0.541866927766378 -0.541866927766378 -0.928720834422932 0.045635886469455 3074 -0.541866927766378 -0.928720834422932 -0.541866927766378 0.045635886469455 3075 -0.541866927766378 -0.541866927766378 0.012454689955687 0.045635886469455 3076 -0.541866927766378 0.012454689955687 -0.541866927766378 0.045635886469455 3077 -0.541866927766378 -0.928720834422932 0.012454689955687 0.045635886469455 3078 -0.541866927766378 0.012454689955687 -0.928720834422932 0.045635886469455 3079 -0.928720834422932 -0.541866927766378 -0.541866927766378 0.045635886469455 3080 -0.928720834422932 -0.541866927766378 0.012454689955687 0.045635886469455 3081 -0.928720834422932 0.012454689955687 -0.541866927766378 0.045635886469455 3082 0.012454689955687 -0.541866927766378 -0.928720834422932 0.045635886469455 3083 0.012454689955687 -0.541866927766378 -0.541866927766378 0.045635886469455 3084 0.012454689955687 -0.928720834422932 -0.541866927766378 0.045635886469455 3085 -0.926784500893605 -0.926784500893605 -0.619027916130733 0.017124153129297 3086 -0.926784500893605 -0.619027916130733 -0.926784500893605 0.017124153129297 3087 -0.926784500893605 -0.926784500893605 0.472596917917943 0.017124153129297 3088 -0.926784500893605 0.472596917917943 -0.926784500893605 0.017124153129297 3089 -0.926784500893605 -0.619027916130733 0.472596917917943 0.017124153129297 3090 -0.926784500893605 0.472596917917943 -0.619027916130733 0.017124153129297 3091 -0.619027916130733 -0.926784500893605 -0.926784500893605 0.017124153129297 3092 -0.619027916130733 -0.926784500893605 0.472596917917943 0.017124153129297 3093 -0.619027916130733 0.472596917917943 -0.926784500893605 0.017124153129297 3094 0.472596917917943 -0.926784500893605 -0.619027916130733 0.017124153129297 3095 0.472596917917943 -0.926784500893605 -0.926784500893605 0.017124153129297 3096 0.472596917917943 -0.619027916130733 -0.926784500893605 0.017124153129297 3097 Gauss quadrature constants for the reference tetrahedron order p=8 3098 3099 53 3100 -0.500000000000000 -0.500000000000000 -0.500000000000000 -1.102392306608869 3101 -0.903297922900526 -0.903297922900526 -0.903297922900526 0.014922692552682 3102 -0.903297922900526 -0.903297922900526 0.709893768701580 0.014922692552682 3103 -0.903297922900526 0.709893768701580 -0.903297922900526 0.014922692552682 3104 0.709893768701580 -0.903297922900526 -0.903297922900526 0.014922692552682 3105 -0.350841439764235 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0.277603247076406 3118 -0.736744381506260 -0.832670596765630 -0.736744381506260 0.277603247076406 3119 -0.736744381506260 -0.736744381506260 0.306159359778151 0.277603247076406 3120 -0.736744381506260 0.306159359778151 -0.736744381506260 0.277603247076406 3121 -0.736744381506260 -0.832670596765630 0.306159359778151 0.277603247076406 3122 -0.736744381506260 0.306159359778151 -0.832670596765630 0.277603247076406 3123 -0.832670596765630 -0.736744381506260 -0.736744381506260 0.277603247076406 3124 -0.832670596765630 -0.736744381506260 0.306159359778151 0.277603247076406 3125 -0.832670596765630 0.306159359778151 -0.736744381506260 0.277603247076406 3126 0.306159359778151 -0.736744381506260 -0.832670596765630 0.277603247076406 3127 0.306159359778151 -0.736744381506260 -0.736744381506260 0.277603247076406 3128 0.306159359778151 -0.832670596765630 -0.736744381506260 0.277603247076406 3129 -0.132097077177186 -0.132097077177186 -0.784460280901143 0.026820671221285 3130 -0.132097077177186 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0.003453031004456 3143 -1.002752554636276 -1.002752554636276 0.452398163998938 0.003453031004456 3144 -1.002752554636276 0.452398163998938 -1.002752554636276 0.003453031004456 3145 -1.002752554636276 -0.446893054726385 0.452398163998938 0.003453031004456 3146 -1.002752554636276 0.452398163998938 -0.446893054726385 0.003453031004456 3147 -0.446893054726385 -1.002752554636276 -1.002752554636276 0.003453031004456 3148 -0.446893054726385 -1.002752554636276 0.452398163998938 0.003453031004456 3149 -0.446893054726385 0.452398163998938 -1.002752554636276 0.003453031004456 3150 0.452398163998938 -1.002752554636276 -0.446893054726385 0.003453031004456 3151 0.452398163998938 -1.002752554636276 -1.002752554636276 0.003453031004456 3152 0.452398163998938 -0.446893054726385 -1.002752554636276 0.003453031004456 3153 Gauss quadrature constants for the reference tetrahedron order p=9 3154 3155 126 3156 -0.857142857142857 -0.857142857142857 0.571428571428571 0.362902592520648 3157 -0.857142857142857 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-0.612213575618572 5972 -0.571428571428571 -0.571428571428571 -0.857142857142857 -0.612213575618572 5973 -0.285714285714286 -0.857142857142857 -0.857142857142857 -0.612213575618572 5974 -0.857142857142857 -0.857142857142857 -0.571428571428571 -0.612213575618572 5975 -0.857142857142857 -0.571428571428571 -0.857142857142857 -0.612213575618572 5976 -0.571428571428571 -0.857142857142857 -0.857142857142857 -0.612213575618572 5977 -0.857142857142857 -0.857142857142857 -0.857142857142857 -0.612213575618572 5978 -0.833333333333333 -0.833333333333333 0.500000000000000 0.076141727180383 5979 -0.833333333333333 -0.500000000000000 0.166666666666667 0.076141727180383 5980 -0.833333333333333 -0.166666666666667 -0.166666666666667 0.076141727180383 5981 -0.833333333333333 0.166666666666667 -0.500000000000000 0.076141727180383 5982 -0.833333333333333 0.500000000000000 -0.833333333333333 0.076141727180383 5983 -0.500000000000000 -0.833333333333333 0.166666666666667 0.076141727180383 5984 -0.500000000000000 -0.500000000000000 -0.166666666666667 0.076141727180383 5985 -0.500000000000000 -0.166666666666667 -0.500000000000000 0.076141727180383 5986 -0.500000000000000 0.166666666666667 -0.833333333333333 0.076141727180383 5987 -0.166666666666667 -0.833333333333333 -0.166666666666667 0.076141727180383 5988 -0.166666666666667 -0.500000000000000 -0.500000000000000 0.076141727180383 5989 -0.166666666666667 -0.166666666666667 -0.833333333333333 0.076141727180383 5990 0.166666666666667 -0.833333333333333 -0.500000000000000 0.076141727180383 5991 0.166666666666667 -0.500000000000000 -0.833333333333333 0.076141727180383 5992 0.500000000000000 -0.833333333333333 -0.833333333333333 0.076141727180383 5993 -0.833333333333333 -0.833333333333333 0.166666666666667 0.076141727180383 5994 -0.833333333333333 -0.500000000000000 -0.166666666666667 0.076141727180383 5995 -0.833333333333333 -0.166666666666667 -0.500000000000000 0.076141727180383 5996 -0.833333333333333 0.166666666666667 -0.833333333333333 0.076141727180383 5997 -0.500000000000000 -0.833333333333333 -0.166666666666667 0.076141727180383 5998 -0.500000000000000 -0.500000000000000 -0.500000000000000 0.076141727180383 5999 -0.500000000000000 -0.166666666666667 -0.833333333333333 0.076141727180383 6000 -0.166666666666667 -0.833333333333333 -0.500000000000000 0.076141727180383 6001 -0.166666666666667 -0.500000000000000 -0.833333333333333 0.076141727180383 6002 0.166666666666667 -0.833333333333333 -0.833333333333333 0.076141727180383 6003 -0.833333333333333 -0.833333333333333 -0.166666666666667 0.076141727180383 6004 -0.833333333333333 -0.500000000000000 -0.500000000000000 0.076141727180383 6005 -0.833333333333333 -0.166666666666667 -0.833333333333333 0.076141727180383 6006 -0.500000000000000 -0.833333333333333 -0.500000000000000 0.076141727180383 6007 -0.500000000000000 -0.500000000000000 -0.833333333333333 0.076141727180383 6008 -0.166666666666667 -0.833333333333333 -0.833333333333333 0.076141727180383 6009 -0.833333333333333 -0.833333333333333 -0.500000000000000 0.076141727180383 6010 -0.833333333333333 -0.500000000000000 -0.833333333333333 0.076141727180383 6011 -0.500000000000000 -0.833333333333333 -0.833333333333333 0.076141727180383 6012 -0.833333333333333 -0.833333333333333 -0.833333333333333 0.076141727180383 6013 -0.800000000000000 -0.800000000000000 0.400000000000000 -0.004255896151021 6014 -0.800000000000000 -0.400000000000000 0.000000000000000 -0.004255896151021 6015 -0.800000000000000 0.000000000000000 -0.400000000000000 -0.004255896151021 6016 -0.800000000000000 0.400000000000000 -0.800000000000000 -0.004255896151021 6017 -0.400000000000000 -0.800000000000000 0.000000000000000 -0.004255896151021 6018 -0.400000000000000 -0.400000000000000 -0.400000000000000 -0.004255896151021 6019 -0.400000000000000 0.000000000000000 -0.800000000000000 -0.004255896151021 6020 0.000000000000000 -0.800000000000000 -0.400000000000000 -0.004255896151021 6021 0.000000000000000 -0.400000000000000 -0.800000000000000 -0.004255896151021 6022 0.400000000000000 -0.800000000000000 -0.800000000000000 -0.004255896151021 6023 -0.800000000000000 -0.800000000000000 0.000000000000000 -0.004255896151021 6024 -0.800000000000000 -0.400000000000000 -0.400000000000000 -0.004255896151021 6025 -0.800000000000000 0.000000000000000 -0.800000000000000 -0.004255896151021 6026 -0.400000000000000 -0.800000000000000 -0.400000000000000 -0.004255896151021 6027 -0.400000000000000 -0.400000000000000 -0.800000000000000 -0.004255896151021 6028 0.000000000000000 -0.800000000000000 -0.800000000000000 -0.004255896151021 6029 -0.800000000000000 -0.800000000000000 -0.400000000000000 -0.004255896151021 6030 -0.800000000000000 -0.400000000000000 -0.800000000000000 -0.004255896151021 6031 -0.400000000000000 -0.800000000000000 -0.800000000000000 -0.004255896151021 6032 -0.800000000000000 -0.800000000000000 -0.800000000000000 -0.004255896151021 6033 -0.750000000000000 -0.750000000000000 0.250000000000000 0.000083414141296 6034 -0.750000000000000 -0.250000000000000 -0.250000000000000 0.000083414141296 6035 -0.750000000000000 0.250000000000000 -0.750000000000000 0.000083414141296 6036 -0.250000000000000 -0.750000000000000 -0.250000000000000 0.000083414141296 6037 -0.250000000000000 -0.250000000000000 -0.750000000000000 0.000083414141296 6038 0.250000000000000 -0.750000000000000 -0.750000000000000 0.000083414141296 6039 -0.750000000000000 -0.750000000000000 -0.250000000000000 0.000083414141296 6040 -0.750000000000000 -0.250000000000000 -0.750000000000000 0.000083414141296 6041 -0.250000000000000 -0.750000000000000 -0.750000000000000 0.000083414141296 6042 -0.750000000000000 -0.750000000000000 -0.750000000000000 0.000083414141296 6043 -0.666666666666667 -0.666666666666667 0.000000000000000 -0.000000352698561 6044 -0.666666666666667 0.000000000000000 -0.666666666666667 -0.000000352698561 6045 0.000000000000000 -0.666666666666667 -0.666666666666667 -0.000000352698561 6046 -0.666666666666667 -0.666666666666667 -0.666666666666667 -0.000000352698561 6047 -0.500000000000000 -0.500000000000000 -0.500000000000000 0.000000000106067 6048 Gauss quadrature constants for the reference tetrahedron order p=21 6049 """ 6050 6051 6052 6053 6054 newton_cotes_table = """\ 6055 2 6056 1 1 6057 -1 1 6058 Closed Newton-Cotes quadrature constants order n=1 6059 6060 3 6061 1 1/3 6062 0 4/3 6063 -1 1/3 6064 Closed Newton-Cotes quadrature constants order n=2 6065 6066 4 6067 1 1/4 6068 1/3 3/4 6069 -1/3 3/4 6070 -1 1/4 6071 Closed Newton-Cotes quadrature constants order n=3 6072 6073 5 6074 1 7/45 6075 1/2 32/45 6076 0 12/45 6077 -1/2 32/45 6078 -1 7/45 6079 Closed Newton-Cotes quadrature constants order n=4 6080 6081 6 6082 1 19/144 6083 3/5 75/144 6084 1/5 50/144 6085 -1/5 50/144 6086 -3/5 75/144 6087 -1 19/144 6088 Closed Newton-Cotes quadrature constants order n=5 6089 6090 7 6091 1 41/420 6092 2/3 216/420 6093 1/3 27/420 6094 0 272/420 6095 -1/3 27/420 6096 -2/3 216/420 6097 -1 41/420 6098 Closed Newton-Cotes quadrature constants order n=6 6099 6100 8 6101 1 751/8640 6102 5/7 3577/8640 6103 3/7 1323/8640 6104 1/7 2989/8640 6105 -1/7 2989/8640 6106 -3/7 1323/8640 6107 -5/7 3577/8640 6108 -1 751/8640 6109 Closed Newton-Cotes quadrature constants order n=7 6110 """ 6111 6112 6113 chebyshev_table = """\ 6114 2 6115 0.5773502691896257645091488 1 6116 -0.5773502691896257645091488 1 6117 Chebyshev quadrature constants order n+1=3 6118 6119 3 6120 0.7071067811865475244008444 2/3 6121 0.0000000000000000000000000 2/3 6122 -0.7071067811865475244008444 2/3 6123 Chebyshev quadrature constants order n+1=4 6124 6125 4 6126 0.7946544722917661229555309 1/2 6127 0.1875924740850798998601393 1/2 6128 -0.1875924740850798998601393 1/2 6129 -0.7946544722917661229555309 1/2 6130 Chebyshev quadrature constants order n+1=5 6131 6132 5 6133 0.8324974870009818758925836 2/5 6134 0.3745414095535810655860444 2/5 6135 0.0000000000000000000000000 2/5 6136 -0.3745414095535810655860444 2/5 6137 -0.8324974870009818758925836 2/5 6138 Chebyshev quadrature constants order n+1=6 6139 6140 6 6141 0.8662468181078205913835981 1/3 6142 0.4225186537611115291185464 1/3 6143 0.2666354015167047203315346 1/3 6144 -0.2666354015167047203315346 1/3 6145 -0.4225186537611115291185464 1/3 6146 -0.8662468181078205913835981 1/3 6147 Chebyshev quadrature constants order n+1=7 6148 6149 7 6150 0.8838617007580490357042241 2/7 6151 0.5296567752851568113850475 2/7 6152 0.3239118105199076375196731 2/7 6153 0.0000000000000000000000000 2/7 6154 -0.3239118105199076375196731 2/7 6155 -0.5296567752851568113850475 2/7 6156 -0.8838617007580490357042241 2/7 6157 Chebyshev quadrature constants order n+1=8 6158 6159 9 6160 0.9115893077284344736649486 2/9 6161 0.6010186553802380714281279 2/9 6162 0.5287617830578799932601816 2/9 6163 0.1679061842148039430680319 2/9 6164 0.0000000000000000000000000 2/9 6165 -0.1679061842148039430680319 2/9 6166 -0.5287617830578799932601816 2/9 6167 -0.6010186553802380714281279 2/9 6168 -0.9115893077284344736649486 2/9 6169 Chebyshev quadrature constants order n+1=10 6170 """ 6171 6172 6173 lobatto_table = """\ 6174 3 6175 1.0000000000000000000000000 0.3333333333333333333333333 6176 0.0000000000000000000000000 1.3333333333333333333333333 6177 -1.0000000000000000000000000 0.3333333333333333333333333 6178 Lobatto (Radau) quadrature constants order 2n-3=3 6179 6180 4 6181 1.0000000000000000000000000 0.1666666666666666666666667 6182 0.4472135954999579392818347 0.8333333333333333333333333 6183 -0.4472135954999579392818347 0.8333333333333333333333333 6184 -1.0000000000000000000000000 0.1666666666666666666666667 6185 Lobatto (Radau) quadrature constants order 2n-3=5 6186 6187 5 6188 1.0000000000000000000000000 0.1000000000000000000000000 6189 0.6546536707079771437982925 0.5444444444444444444444444 6190 0.0000000000000000000000000 0.7111111111111111111111111 6191 -0.6546536707079771437982925 0.5444444444444444444444444 6192 -1.0000000000000000000000000 0.1000000000000000000000000 6193 Lobatto (Radau) quadrature constants order 2n-3=7 6194 6195 6 6196 1.0000000000000000000000000 0.0666666666666666666666667 6197 0.7650553239294646928510030 0.3784749562978469803166128 6198 0.2852315164806450963141510 0.5548583770354863530167205 6199 -0.2852315164806450963141510 0.5548583770354863530167205 6200 -0.7650553239294646928510030 0.3784749562978469803166128 6201 -1.0000000000000000000000000 0.0666666666666666666666667 6202 Lobatto (Radau) quadrature constants order 2n-3=9 6203 6204 7 6205 1.0000000000000000000000000 0.0476190476190476190476190 6206 0.8302238962785669298720322 0.2768260473615659480107004 6207 0.4688487934707142138037719 0.4317453812098626234178710 6208 0.0000000000000000000000000 0.4876190476190476190476190 6209 -0.4688487934707142138037719 0.4317453812098626234178710 6210 -0.8302238962785669298720322 0.2768260473615659480107004 6211 -1.0000000000000000000000000 0.0476190476190476190476190 6212 Lobatto (Radau) quadrature constants order 2n-3=11 6213 6214 8 6215 1.0000000000000000000000000 0.0357142857142857142857143 6216 0.8717401485096066153374457 0.2107042271435060393829921 6217 0.5917001814331423021445107 0.3411226924835043647642407 6218 0.2092992179024788687686573 0.4124587946587038815670530 6219 -0.2092992179024788687686573 0.4124587946587038815670530 6220 -0.5917001814331423021445107 0.3411226924835043647642407 6221 -0.8717401485096066153374457 0.2107042271435060393829921 6222 -1.0000000000000000000000000 0.0357142857142857142857143 6223 Lobatto (Radau) quadrature constants order 2n-3=13 6224 6225 9 6226 1.0000000000000000000000000 0.0277777777777777777777778 6227 0.8997579954114601573123452 0.1654953615608055250463397 6228 0.6771862795107377534458854 0.2745387125001617352807056 6229 0.3631174638261781587107521 0.3464285109730463451151315 6230 0.0000000000000000000000000 0.3715192743764172335600907 6231 -0.3631174638261781587107521 0.3464285109730463451151315 6232 -0.6771862795107377534458854 0.2745387125001617352807056 6233 -0.8997579954114601573123452 0.1654953615608055250463397 6234 -1.0000000000000000000000000 0.0277777777777777777777778 6235 Lobatto (Radau) quadrature constants order 2n-3=15 6236 6237 10 6238 1.0000000000000000000000000 0.0222222222222222222222222 6239 0.9195339081664588138289327 0.1333059908510701111262272 6240 0.7387738651055050750031062 0.2248893420631264521194578 6241 0.4779249498104444956611751 0.2920426836796837578755823 6242 0.1652789576663870246262198 0.3275397611838974566565105 6243 -0.1652789576663870246262198 0.3275397611838974566565105 6244 -0.4779249498104444956611751 0.2920426836796837578755823 6245 -0.7387738651055050750031062 0.2248893420631264521194578 6246 -0.9195339081664588138289327 0.1333059908510701111262272 6247 -1.0000000000000000000000000 0.0222222222222222222222222 6248 Lobatto (Radau) quadrature constants order 2n-3=17 6249 6250 11 6251 1.0000000000000000000000000 0.0181818181818181818181818 6252 0.9340014304080591343322741 0.1096122732669948644614034 6253 0.7844834736631444186224178 0.1871698817803052041081415 6254 0.5652353269962050064709640 0.2480481042640283140400849 6255 0.2957581355869393914319115 0.2868791247790080886792224 6256 0.0000000000000000000000000 0.3002175954556906937859319 6257 -0.2957581355869393914319115 0.2868791247790080886792224 6258 -0.5652353269962050064709640 0.2480481042640283140400849 6259 -0.7844834736631444186224178 0.1871698817803052041081415 6260 -0.9340014304080591343322741 0.1096122732669948644614034 6261 -1.0000000000000000000000000 0.0181818181818181818181818 6262 Lobatto (Radau) quadrature constants order 2n-3=19 6263 6264 12 6265 1.0000000000000000000000000 0.0151515151515151515151515 6266 0.9448992722228822234075801 0.0916845174131961306683426 6267 0.8192793216440066783486416 0.1579747055643701151646710 6268 0.6328761530318606776624049 0.2125084177610211453583021 6269 0.3995309409653489322643498 0.2512756031992012802932444 6270 0.1365529328549275548640619 0.2714052409106961770002883 6271 -0.1365529328549275548640619 0.2714052409106961770002883 6272 -0.3995309409653489322643498 0.2512756031992012802932444 6273 -0.6328761530318606776624049 0.2125084177610211453583021 6274 -0.8192793216440066783486416 0.1579747055643701151646710 6275 -0.9448992722228822234075801 0.0916845174131961306683426 6276 -1.0000000000000000000000000 0.0151515151515151515151515 6277 Lobatto (Radau) quadrature constants order 2n-3=21 6278 """ 6279