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 while 1: 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 raise RuntimeError("Found no rule for a %s with order %d in family %s" % (polygon, order, family))
54 55
56 -def load_table(polygon, family):
57 """Return list of quadrature rules matching polygon and family.""" 58 rules = parse_table_string(polygon, family) 59 rules = scale_table(rules) 60 return rules
61 62
63 -def parse_table_string(polygon, family):
64 # Fetch a table string with quadrature points and weights for different orders. 65 name = "%s_%s_table" % (family, polygon) 66 tablestring = eval(name) 67 68 nsd = domain2dim[polygon] 69 lines = tablestring.split('\n') 70 rules = [] 71 i = 0 72 while i<len(lines): 73 n = int(lines[i]) 74 i += 1 75 points = [] 76 weights = [] 77 for j in range(n): 78 values = lines[i].split() 79 points += [ map(float, values[:nsd]) ] 80 weights += [ float(values[nsd]) ] 81 i += 1 82 comment = lines[i].strip() 83 i += 2 84 order = int(comment.split('=')[-1].split()[-1]) 85 rules.append( QuadRule(polygon, nsd, order, points, weights, comment) ) 86 return rules
87 88
89 -def scale_table(rules):
90 """Scales a set of quadrature rules from [-1,+1]^nsd to [0,1]^nsd.""" 91 point_offset = 0.5 92 point_scale = 0.5 93 for rule in rules: 94 ws = point_scale**rule.nsd 95 rule.weights = [ w*ws for w in rule.weights ] 96 rule.points = [ [x*point_scale+point_offset for x in p] for p in rule.points ] 97 return rules
98 99 100 # Gauss rules 101 gauss_interval_table = """\ 102 1 103 0.0000000000000000000000000 2.0000000000000000000000000 104 Gauss quadrature points and weights order 2n-1=1 105 106 2 107 -0.5773502691896257645091488 1.0000000000000000000000000 108 0.5773502691896257645091488 1.0000000000000000000000000 109 Gauss quadrature points and weights order 2n-1=3 110 111 3 112 -0.7745966692414833770358531 0.5555555555555555555555556 113 0.0000000000000000000000000 0.8888888888888888888888889 114 0.7745966692414833770358531 0.5555555555555555555555556 115 Gauss quadrature points and weights order 2n-1=5 116 117 4 118 -0.8611363115940525752239465 0.3478548451374538573730639 119 -0.3399810435848562648026658 0.6521451548625461426269361 120 0.3399810435848562648026658 0.6521451548625461426269361 121 0.8611363115940525752239465 0.3478548451374538573730639 122 Gauss quadrature points and weights order 2n-1=7 123 124 5 125 -0.9061798459386639927976269 0.2369268850561890875142640 126 -0.5384693101056830910363144 0.4786286704993664680412915 127 0.0000000000000000000000000 0.5688888888888888888888889 128 0.5384693101056830910363144 0.4786286704993664680412915 129 0.9061798459386639927976269 0.2369268850561890875142640 130 Gauss quadrature points and weights order 2n-1=9 131 132 6 133 -0.9324695142031520278123016 0.1713244923791703450402961 134 -0.6612093864662645136613996 0.3607615730481386075698335 135 -0.2386191860831969086305017 0.4679139345726910473898703 136 0.2386191860831969086305017 0.4679139345726910473898703 137 0.6612093864662645136613996 0.3607615730481386075698335 138 0.9324695142031520278123016 0.1713244923791703450402961 139 Gauss quadrature points and weights order 2n-1=11 140 141 7 142 -0.9491079123427585245261897 0.1294849661688696932706114 143 -0.7415311855993944398638648 0.2797053914892766679014678 144 -0.4058451513773971669066064 0.3818300505051189449503698 145 0.0000000000000000000000000 0.4179591836734693877551020 146 0.4058451513773971669066064 0.3818300505051189449503698 147 0.7415311855993944398638648 0.2797053914892766679014678 148 0.9491079123427585245261897 0.1294849661688696932706114 149 Gauss quadrature points and weights order 2n-1=13 150 151 8 152 -0.9602898564975362316835609 0.1012285362903762591525314 153 -0.7966664774136267395915539 0.2223810344533744705443560 154 -0.5255324099163289858177390 0.3137066458778872873379622 155 -0.1834346424956498049394761 0.3626837833783619829651504 156 0.1834346424956498049394761 0.3626837833783619829651504 157 0.5255324099163289858177390 0.3137066458778872873379622 158 0.7966664774136267395915539 0.2223810344533744705443560 159 0.9602898564975362316835609 0.1012285362903762591525314 160 Gauss quadrature points and weights order 2n-1=15 161 162 9 163 -0.9681602395076260898355762 0.0812743883615744119718922 164 -0.8360311073266357942994297 0.1806481606948574040584720 165 -0.6133714327005903973087020 0.2606106964029354623187429 166 -0.3242534234038089290385380 0.3123470770400028400686304 167 0.0000000000000000000000000 0.3302393550012597631645251 168 0.3242534234038089290385380 0.3123470770400028400686304 169 0.6133714327005903973087020 0.2606106964029354623187429 170 0.8360311073266357942994298 0.1806481606948574040584720 171 0.9681602395076260898355762 0.0812743883615744119718922 172 Gauss quadrature points and weights order 2n-1=17 173 174 10 175 -0.9739065285171717200779640 0.0666713443086881375935688 176 -0.8650633666889845107320967 0.1494513491505805931457763 177 -0.6794095682990244062343274 0.2190863625159820439955349 178 -0.4333953941292471907992659 0.2692667193099963550912269 179 -0.1488743389816312108848260 0.2955242247147528701738930 180 0.1488743389816312108848260 0.2955242247147528701738930 181 0.4333953941292471907992659 0.2692667193099963550912269 182 0.6794095682990244062343274 0.2190863625159820439955349 183 0.8650633666889845107320967 0.1494513491505805931457763 184 0.9739065285171717200779640 0.0666713443086881375935688 185 Gauss quadrature points and weights order 2n-1=19 186 187 11 188 -0.9782286581460569928039380 0.0556685671161736664827537 189 -0.8870625997680952990751578 0.1255803694649046246346943 190 -0.7301520055740493240934162 0.1862902109277342514260976 191 -0.5190961292068118159257257 0.2331937645919904799185237 192 -0.2695431559523449723315320 0.2628045445102466621806889 193 0.0000000000000000000000000 0.2729250867779006307144835 194 0.2695431559523449723315320 0.2628045445102466621806889 195 0.5190961292068118159257257 0.2331937645919904799185237 196 0.7301520055740493240934163 0.1862902109277342514260980 197 0.8870625997680952990751578 0.1255803694649046246346940 198 0.9782286581460569928039380 0.0556685671161736664827537 199 Gauss quadrature points and weights order 2n-1=21 200 201 12 202 -0.9815606342467192506905491 0.0471753363865118271946160 203 -0.9041172563704748566784659 0.1069393259953184309602547 204 -0.7699026741943046870368938 0.1600783285433462263346525 205 -0.5873179542866174472967024 0.2031674267230659217490645 206 -0.3678314989981801937526915 0.2334925365383548087608499 207 -0.1252334085114689154724414 0.2491470458134027850005624 208 0.1252334085114689154724414 0.2491470458134027850005624 209 0.3678314989981801937526915 0.2334925365383548087608499 210 0.5873179542866174472967024 0.2031674267230659217490645 211 0.7699026741943046870368938 0.1600783285433462263346525 212 0.9041172563704748566784659 0.1069393259953184309602547 213 0.9815606342467192506905491 0.0471753363865118271946160 214 Gauss quadrature points and weights order 2n-1=23 215 216 16 217 -0.9894009349916499325961542 0.0271524594117540948517806 218 -0.9445750230732325760779884 0.0622535239386478928628438 219 -0.8656312023878317438804679 0.0951585116824927848099251 220 -0.7554044083550030338951012 0.1246289712555338720524763 221 -0.6178762444026437484466718 0.1495959888165767320815017 222 -0.4580167776572273863424194 0.1691565193950025381893121 223 -0.2816035507792589132304605 0.1826034150449235888667637 224 -0.0950125098376374401853193 0.1894506104550684962853967 225 0.0950125098376374401853193 0.1894506104550684962853967 226 0.2816035507792589132304605 0.1826034150449235888667637 227 0.4580167776572273863424194 0.1691565193950025381893121 228 0.6178762444026437484466718 0.1495959888165767320815017 229 0.7554044083550030338951012 0.1246289712555338720524763 230 0.8656312023878317438804679 0.0951585116824927848099251 231 0.9445750230732325760779884 0.0622535239386478928628438 232 0.9894009349916499325961542 0.0271524594117540948517806 233 Gauss quadrature points and weights order 2n-1=31 234 235 20 236 -0.9931285991850949247861224 0.0176140071391521183118620 237 -0.9639719272779137912676661 0.0406014298003869413310400 238 -0.9122344282513259058677524 0.0626720483341090635695065 239 -0.8391169718222188233945291 0.0832767415767047487247581 240 -0.7463319064601507926143051 0.1019301198172404350367501 241 -0.6360536807265150254528367 0.1181945319615184173123774 242 -0.5108670019508270980043641 0.1316886384491766268984945 243 -0.3737060887154195606725482 0.1420961093183820513292983 244 -0.2277858511416450780804962 0.1491729864726037467878287 245 -0.0765265211334973337546404 0.1527533871307258506980843 246 0.0765265211334973337546404 0.1527533871307258506980843 247 0.2277858511416450780804962 0.1491729864726037467878287 248 0.3737060887154195606725482 0.1420961093183820513292983 249 0.5108670019508270980043641 0.1316886384491766268984945 250 0.6360536807265150254528367 0.1181945319615184173123774 251 0.7463319064601507926143051 0.1019301198172404350367501 252 0.8391169718222188233945291 0.0832767415767047487247581 253 0.9122344282513259058677524 0.0626720483341090635695065 254 0.9639719272779137912676661 0.0406014298003869413310400 255 0.9931285991850949247861224 0.0176140071391521183118620 256 Gauss quadrature points and weights order 2n-1=39 257 258 24 259 -0.9951872199970213601799974 0.0123412297999871995468057 260 -0.9747285559713094981983919 0.0285313886289336631813078 261 -0.9382745520027327585236490 0.0442774388174198061686027 262 -0.8864155270044010342131543 0.0592985849154367807463678 263 -0.8200019859739029219539499 0.0733464814110803057340336 264 -0.7401241915785543642438281 0.0861901615319532759171852 265 -0.6480936519369755692524958 0.0976186521041138882698807 266 -0.5454214713888395356583756 0.1074442701159656347825773 267 -0.4337935076260451384870842 0.1155056680537256013533445 268 -0.3150426796961633743867933 0.1216704729278033912044632 269 -0.1911188674736163091586398 0.1258374563468282961213754 270 -0.0640568928626056260850431 0.1279381953467521569740562 271 0.0640568928626056260850431 0.1279381953467521569740562 272 0.1911188674736163091586398 0.1258374563468282961213754 273 0.3150426796961633743867933 0.1216704729278033912044632 274 0.4337935076260451384870842 0.1155056680537256013533445 275 0.5454214713888395356583756 0.1074442701159656347825773 276 0.6480936519369755692524958 0.0976186521041138882698807 277 0.7401241915785543642438281 0.0861901615319532759171852 278 0.8200019859739029219539499 0.0733464814110803057340336 279 0.8864155270044010342131543 0.0592985849154367807463678 280 0.9382745520027327585236490 0.0442774388174198061686027 281 0.9747285559130949819839199 0.0285313886289336631813078 282 0.9951872199970213601799974 0.0123412297999871995468057 283 Gauss quadrature points and weights order 2n-1=47 284 285 32 286 -0.9972638618494815635449811 0.0070186100094700966004071 287 -0.9856115115452683354001750 0.0162743947309056706051706 288 -0.9647622555875064307738119 0.0253920653092620594557526 289 -0.9349060759377396891709191 0.0342738629130214331026877 290 -0.8963211557660521239653072 0.0428358980222266806568786 291 -0.8493676137325699701336930 0.0509980592623761761961632 292 -0.7944837959679424069630973 0.0586840934785355471452836 293 -0.7321821187402896803874267 0.0658222227763618468376501 294 -0.6630442669302152009751152 0.0723457941088485062253994 295 -0.5877157572407623290407455 0.0781938957870703064717409 296 -0.5068999089322293900237475 0.0833119242269467552221991 297 -0.4213512761306353453641194 0.0876520930044038111427715 298 -0.3318686022821276497799168 0.0911738786957638847128686 299 -0.2392873622521370745446032 0.0938443990808045656391802 300 -0.1444719615827964934851864 0.0956387200792748594190820 301 -0.0483076656877383162348126 0.0965400885147278005667648 302 0.0483076656877383162348126 0.0965400885147278005667648 303 0.1444719615827964934851864 0.0956387200792748594190820 304 0.2392873622521370745446032 0.0938443990808045656391802 305 0.3318686022821276497799168 0.0911738786957638847128686 306 0.4213512761306353453641194 0.0876520930044038111427715 307 0.5068999089322293900237475 0.0833119242269467552221991 308 0.5877157572407623290407455 0.0781938957870703064717409 309 0.6630442669302152009751152 0.0723457941088485062253994 310 0.7321821187402896803874267 0.0658222227763618468376501 311 0.7944837959679424069630973 0.0586840934785355471452836 312 0.8493676137325699701336930 0.0509980592623761761961632 313 0.8963211557660521239653072 0.0428358980222266806568787 314 0.9349060759377396891709191 0.0342738629130214331026877 315 0.9647622555875064307738119 0.0253920653092620594557526 316 0.9856115115452683354001750 0.0162743947309056706051706 317 0.9972638618494815635449811 0.0070186100094700966004071 318 Gauss quadrature points and weights order 2n-1=63 319 320 40 321 -0.9982377097105592003496227 0.0045212770985331912584717 322 -0.9907262386994570064530544 0.0104982845311528136147422 323 -0.9772599499837742626633703 0.0164210583819078887128635 324 -0.9579168192137916558045410 0.0222458491941669572615043 325 -0.9328128082786765333608522 0.0279370069800234010984892 326 -0.9020988069688742967282533 0.0334601952825478473926782 327 -0.8659595032122595038207818 0.0387821679744720176399720 328 -0.8246122308333116631963202 0.0438709081856732719916747 329 -0.7783056514265193876949715 0.0486958076350722320614342 330 -0.7273182551899271032809965 0.0532278469839368243549965 331 -0.6719566846141795483793545 0.0574397690993915513666177 332 -0.6125538896679802379526125 0.0613062424929289391665380 333 -0.5494671250951282020759313 0.0648040134566010380745545 334 -0.4830758016861787129085666 0.0679120458152339038256901 335 -0.4137792043716050015248797 0.0706116473912867796954836 336 -0.3419940908257584730074925 0.0728865823958040590605107 337 -0.2681521850072536811411843 0.0747231690579682642001893 338 -0.1926975807013710997155169 0.0761103619006262423715581 339 -0.1160840706752552084834513 0.0770398181642479655883075 340 -0.0387724175060508219331934 0.0775059479784248112637240 341 0.0387724175060508219331934 0.0775059479784248112637240 342 0.1160840706752552084834513 0.0770398181642479655883075 343 0.1926975807013710997155169 0.0761103619006262423715581 344 0.2681521850072536811411843 0.0747231690579682642001893 345 0.3419940908257584730074925 0.0728865823958040590605107 346 0.4137792043716050015248797 0.0706116473912867796954836 347 0.4830758016861787129085666 0.0679120458152339038256901 348 0.5494671250951282020759313 0.0648040134566010380745545 349 0.6125538896679802379526125 0.0613062424929289391665380 350 0.6719566846141795483793545 0.0574397690993915513666177 351 0.7273182551899271032809965 0.0532278469839368243549965 352 0.7783056514265193876949715 0.0486958076350722320614342 353 0.8246122308333116631963202 0.0438709081856732719916747 354 0.8659595032122595038207818 0.0387821679744720176399720 355 0.9020988069688742967282533 0.0334601952825478473926782 356 0.9328128082786765333608522 0.0279370069800234010984892 357 0.9579168192137916558045410 0.0222458491941669572615040 358 0.9772599499837742626633703 0.0164210583819078887128635 359 0.9907262386994570064530544 0.0104982845311528136147422 360 0.9982377097105592003496227 0.0045212770985331912584717 361 Gauss quadrature points and weights order 2n-1=79 362 363 48 364 -0.9987710072524261186005415 0.0031533460523058386326773 365 -0.9935301722663507575479288 0.0073275539012762621023840 366 -0.9841245837228268577445836 0.0114772345792345394895927 367 -0.9705915925462472504614120 0.0155793157229438487281770 368 -0.9529877031604308607229607 0.0196161604573555278144607 369 -0.9313866907065543331141744 0.0235707608393243791405193 370 -0.9058791367155696728220748 0.0274265097083569482000738 371 -0.8765720202742478859056936 0.0311672278327980889020658 372 -0.8435882616243935307110898 0.0347772225647704388925486 373 -0.8070662040294426270825530 0.0382413510658307063172173 374 -0.7671590325157403392538554 0.0415450829434647492140588 375 -0.7240341309238146546744822 0.0446745608566942804194486 376 -0.6778723796326639052118513 0.0476166584924904748259066 377 -0.6288673967765136239951649 0.0503590355538544749578076 378 -0.5772247260839727038178092 0.0528901894851936670955051 379 -0.5231609747222330336782259 0.0551995036999841628682035 380 -0.4669029047509584045449289 0.0572772921004032157051502 381 -0.4086864819907167299162255 0.0591148396983956357464748 382 -0.3487558862921607381598179 0.0607044391658938800529692 383 -0.2873624873554555767358865 0.0620394231598926639041978 384 -0.2247637903946890612248654 0.0631141922862540256571260 385 -0.1612223560688917180564374 0.0639242385846481866239062 386 -0.0970046992094626989300540 0.0644661644359500822065042 387 -0.0323801709628693620333222 0.0647376968126839225030249 388 0.0323801709628693620333222 0.0647376968126839225030249 389 0.0970046992094626989300540 0.0644661644359500822065042 390 0.1612223560688917180564374 0.0639242385846481866239062 391 0.2247637903946890612248654 0.0631141922862540256571260 392 0.2873624873554555767358865 0.0620394231598926639041978 393 0.3487558862921607381598179 0.0607044391658938800529692 394 0.4086864819907167299162255 0.0591148396983956357464748 395 0.4669029047509584045449289 0.0572772921004032157051502 396 0.5231609747222330336782259 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483 economical_gauss_quadrilateral_table = """\ 484 1 485 0.000000000000000 0.000000000000000 4.000000000000000 486 Gauss quadrature points and weights on the reference quadrilateral order p=0 1 487 488 4 489 0.577350269189626 0.577350269189626 1.000000000000000 490 0.577350269189626 -0.577350269189626 1.000000000000000 491 -0.577350269189626 0.577350269189626 1.000000000000000 492 -0.577350269189626 -0.577350269189626 1.000000000000000 493 Gauss quadrature points and weights on the reference quadrilateral order p=2 3 494 495 8 496 0.683130051063973 0.000000000000000 0.816326530612245 497 -0.683130051063973 0.000000000000000 0.816326530612245 498 0.000000000000000 0.683130051063973 0.816326530612245 499 0.000000000000000 -0.683130051063973 0.816326530612245 500 0.881917103688197 0.881917103688197 0.183673469387755 501 0.881917103688197 -0.881917103688197 0.183673469387755 502 -0.881917103688197 0.881917103688197 0.183673469387755 503 -0.881917103688197 -0.881917103688197 0.183673469387755 504 Gauss quadrature points and weights on the reference quadrilateral order p=4 5 505 506 12 507 0.925820099772551 0.000000000000000 0.241975308641975 508 -0.925820099772551 0.000000000000000 0.241975308641975 509 0.000000000000000 0.925820099772551 0.241975308641975 510 0.000000000000000 -0.925820099772551 0.241975308641975 511 0.805979782918599 0.805979782918599 0.237431774690630 512 0.805979782918599 -0.805979782918599 0.237431774690630 513 -0.805979782918599 0.805979782918599 0.237431774690630 514 -0.805979782918599 -0.805979782918599 0.237431774690630 515 0.380554433208316 0.380554433208316 0.520592916667394 516 0.380554433208316 -0.380554433208316 0.520592916667394 517 -0.380554433208316 0.380554433208316 0.520592916667394 518 -0.380554433208316 -0.380554433208316 0.520592916667394 519 Gauss quadrature points and weights on the reference quadrilateral order p=6 7 520 521 20 522 1.121225763866564 0.000000000000000 0.018475842507491 523 -1.121225763866564 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2213 -0.538990774213829 0.754378926224437 0.258107746839302 0.039008180977840 2214 -0.538990774213829 0.754378926224437 -0.258107746839302 0.039008180977840 2215 -0.538990774213829 0.258107746839302 0.754378926224437 0.039008180977840 2216 -0.538990774213829 -0.258107746839302 0.754378926224437 0.039008180977840 2217 -0.538990774213829 -0.754378926224437 0.258107746839302 0.039008180977840 2218 -0.538990774213829 -0.754378926224437 -0.258107746839302 0.039008180977840 2219 -0.538990774213829 0.258107746839302 -0.754378926224437 0.039008180977840 2220 -0.538990774213829 -0.258107746839302 -0.754378926224437 0.039008180977840 2221 0.258107746839302 0.754378926224437 0.538990774213829 0.039008180977840 2222 0.258107746839302 0.754378926224437 -0.538990774213829 0.039008180977840 2223 0.258107746839302 0.538990774213829 0.754378926224437 0.039008180977840 2224 0.258107746839302 -0.538990774213829 0.754378926224437 0.039008180977840 2225 0.258107746839302 -0.754378926224437 0.538990774213829 0.039008180977840 2226 0.258107746839302 -0.754378926224437 -0.538990774213829 0.039008180977840 2227 0.258107746839302 0.538990774213829 -0.754378926224437 0.039008180977840 2228 0.258107746839302 -0.538990774213829 -0.754378926224437 0.039008180977840 2229 -0.258107746839302 0.754378926224437 0.538990774213829 0.039008180977840 2230 -0.258107746839302 0.754378926224437 -0.538990774213829 0.039008180977840 2231 -0.258107746839302 0.538990774213829 0.754378926224437 0.039008180977840 2232 -0.258107746839302 -0.538990774213829 0.754378926224437 0.039008180977840 2233 -0.258107746839302 -0.754378926224437 0.538990774213829 0.039008180977840 2234 -0.258107746839302 -0.754378926224437 -0.538990774213829 0.039008180977840 2235 -0.258107746839302 0.538990774213829 -0.754378926224437 0.039008180977840 2236 -0.258107746839302 -0.538990774213829 -0.754378926224437 0.039008180977840 2237 Gauss quadrature points and weights on the reference cube order p=18 19 2238 """ 2239 2240 2241 economical_gauss_triangle_table = """\ 2242 1 2243 -0.333333333333333 -0.333333333333333 2.000000000000000 2244 Gauss quadrature points and weights on the reference triangle order p=1 2245 2246 3 2247 -0.666666666666667 -0.666666666666667 0.666666666666667 2248 -0.666666666666667 0.333333333333333 0.666666666666667 2249 0.333333333333333 -0.666666666666667 0.666666666666667 2250 Gauss quadrature points and weights on the reference triangle order p=2 2251 2252 4 2253 -0.333333333333333 -0.333333333333333 -1.125000000000000 2254 -0.600000000000000 -0.600000000000000 1.041666666666667 2255 -0.600000000000000 0.200000000000000 1.041666666666667 2256 0.200000000000000 -0.600000000000000 1.041666666666667 2257 Gauss quadrature points and weights on the reference triangle order p=3 2258 2259 6 2260 -0.108103018168070 -0.108103018168070 0.446763179356022 2261 -0.108103018168070 -0.783793963663860 0.446763179356022 2262 -0.783793963663860 -0.108103018168070 0.446763179356022 2263 -0.816847572980458 -0.816847572980458 0.219903487310644 2264 -0.816847572980458 0.633695145960918 0.219903487310644 2265 0.633695145960918 -0.816847572980458 0.219903487310644 2266 Gauss quadrature points and weights on the reference triangle order p=4 2267 2268 7 2269 -0.333333333333333 -0.333333333333333 0.450000000000000 2270 -0.059715871789770 -0.059715871789770 0.264788305577012 2271 -0.059715871789770 -0.880568256420460 0.264788305577012 2272 -0.880568256420460 -0.059715871789770 0.264788305577012 2273 -0.797426985353088 -0.797426985353088 0.251878361089654 2274 -0.797426985353088 0.594853970706174 0.251878361089654 2275 0.594853970706174 -0.797426985353088 0.251878361089654 2276 Gauss quadrature points and weights on the reference triangle order p=5 2277 2278 12 2279 -0.501426509658180 -0.501426509658180 0.233572551452758 2280 -0.501426509658180 0.002853019316358 0.233572551452758 2281 0.002853019316358 -0.501426509658180 0.233572551452758 2282 -0.873821971016996 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-0.869739794195568 0.106694471217676 2301 -0.374269007990252 0.276888377139620 0.154227521780514 2302 0.276888377139620 -0.902619369149368 0.154227521780514 2303 -0.902619369149368 -0.374269007990252 0.154227521780514 2304 -0.374269007990252 -0.902619369149368 0.154227521780514 2305 0.276888377139620 -0.374269007990252 0.154227521780514 2306 -0.902619369149368 0.276888377139620 0.154227521780514 2307 Gauss quadrature points and weights on the reference triangle order p=7 2308 2309 16 2310 -0.333333333333333 -0.333333333333333 0.288631215355574 2311 -0.081414823414554 -0.081414823414554 0.190183268534570 2312 -0.081414823414554 -0.837170353170892 0.190183268534570 2313 -0.837170353170892 -0.081414823414554 0.190183268534570 2314 -0.658861384496480 -0.658861384496480 0.206434741069436 2315 -0.658861384496480 0.317722768992960 0.206434741069436 2316 0.317722768992960 -0.658861384496480 0.206434741069436 2317 -0.898905543365938 -0.898905543365938 0.064916995246396 2318 -0.898905543365938 0.797811086731876 0.064916995246395 2319 0.797811086731876 -0.898905543365938 0.064916995246396 2320 -0.473774340730724 0.456984785910808 0.054460628348870 2321 0.456984785910808 -0.983210445180084 0.054460628348870 2322 -0.983210445180084 -0.473774340730724 0.054460628348870 2323 -0.473774340730724 -0.983210445180084 0.054460628348870 2324 0.456984785910808 -0.473774340730724 0.054460628348870 2325 -0.983210445180084 0.456984785910808 0.054460628348870 2326 Gauss quadrature points and weights on the reference triangle order p=8 2327 2328 19 2329 -0.333333333333333 -0.333333333333333 0.194271592565598 2330 -0.020634961602524 -0.020634961602524 0.062669400454278 2331 -0.020634961602524 -0.958730076794950 0.062669400454278 2332 -0.958730076794950 -0.020634961602524 0.062669400454278 2333 -0.125820817014126 -0.125820817014126 0.155655082009548 2334 -0.125820817014126 -0.748358365971746 0.155655082009548 2335 -0.748358365971746 -0.125820817014126 0.155655082009548 2336 -0.623592928761934 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0.018843333927466 2371 0.847311867175000 -0.980918369199402 0.018843333927466 2372 -0.980918369199402 -0.866393497975600 0.018843333927466 2373 -0.866393497975600 -0.980918369199402 0.018843333927466 2374 0.847311867175000 -0.866393497975600 0.018843333927466 2375 -0.980918369199402 0.847311867175000 0.018843333927466 2376 Gauss quadrature points and weights on the reference triangle order p=10 2377 2378 27 2379 0.069222096541516 0.069222096541516 0.001854012657922 2380 0.069222096541516 -1.138444193083034 0.001854012657922 2381 -1.138444193083034 0.069222096541516 0.001854012657922 2382 -0.202061394068290 -0.202061394068290 0.154299069829626 2383 -0.202061394068290 -0.595877211863420 0.154299069829626 2384 -0.595877211863420 -0.202061394068290 0.154299069829626 2385 -0.593380199137436 -0.593380199137436 0.118645954761548 2386 -0.593380199137436 0.186760398274870 0.118645954761548 2387 0.186760398274870 -0.593380199137436 0.118645954761548 2388 -0.761298175434838 -0.761298175434838 0.072369081006836 2389 -0.761298175434838 0.522596350869674 0.072369081006836 2390 0.522596350869674 -0.761298175434838 0.072369081006836 2391 -0.935270103777448 -0.935270103777448 0.027319462005356 2392 -0.935270103777448 0.870540207554896 0.027319462005356 2393 0.870540207554896 -0.935270103777448 0.027319462005356 2394 -0.286758703477414 0.186402426856426 0.104674223924408 2395 0.186402426856426 -0.899643723379010 0.104674223924408 2396 -0.899643723379010 -0.286758703477414 0.104674223924408 2397 -0.286758703477414 -0.899643723379010 0.104674223924408 2398 0.186402426856426 -0.286758703477414 0.104674223924408 2399 -0.899643723379010 0.186402426856426 0.104674223924408 2400 -0.657022039391916 0.614978006319584 0.041415319278282 2401 0.614978006319584 -0.957955966927668 0.041415319278282 2402 -0.957955966927668 -0.657022039391916 0.041415319278282 2403 -0.657022039391916 -0.957955966927668 0.041415319278282 2404 0.614978006319584 -0.657022039391916 0.041415319278282 2405 -0.957955966927668 0.614978006319584 0.041415319278282 2406 Gauss quadrature points and weights on the reference triangle order p=11 2407 2408 33 2409 -0.023565220452390 -0.023565220452390 0.051462132880910 2410 -0.023565220452390 -0.952869559095220 0.051462132880910 2411 -0.952869559095220 -0.023565220452390 0.051462132880910 2412 -0.120551215411080 -0.120551215411080 0.087385089076076 2413 -0.120551215411080 -0.758897569177842 0.087385089076076 2414 -0.758897569177842 -0.120551215411080 0.087385089076076 2415 -0.457579229975768 -0.457579229975768 0.125716448435770 2416 -0.457579229975768 -0.084841540048464 0.125716448435770 2417 -0.084841540048464 -0.457579229975768 0.125716448435770 2418 -0.744847708916828 -0.744847708916828 0.069592225861418 2419 -0.744847708916828 0.489695417833656 0.069592225861418 2420 0.489695417833656 -0.744847708916828 0.069592225861418 2421 -0.957365299093580 -0.957365299093580 0.012332522103118 2422 -0.957365299093580 0.914730598187158 0.012332522103118 2423 0.914730598187158 -0.957365299093580 0.012332522103118 2424 -0.448573460628972 0.217886471559576 0.080743115532762 2425 0.217886471559576 -0.769313010930604 0.080743115532762 2426 -0.769313010930604 -0.448573460628972 0.080743115532762 2427 -0.448573460628972 -0.769313010930604 0.080743115532762 2428 0.217886471559576 -0.448573460628972 0.080743115532762 2429 -0.769313010930604 0.217886471559576 0.080743115532762 2430 -0.437348838020120 0.391672173575606 0.044713546404606 2431 0.391672173575606 -0.954323335555486 0.044713546404606 2432 -0.954323335555486 -0.437348838020120 0.044713546404606 2433 -0.437348838020120 -0.954323335555486 0.044713546404606 2434 0.391672173575606 -0.437348838020120 0.044713546404606 2435 -0.954323335555486 0.391672173575606 0.044713546404606 2436 -0.767496168184806 0.716028067088146 0.034632462217318 2437 0.716028067088146 -0.948531898903340 0.034632462217318 2438 -0.948531898903340 -0.767496168184806 0.034632462217318 2439 -0.767496168184806 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0.036709828212560 2910 -0.706069127893522 0.722805434309974 0.001408809355816 2911 0.722805434309974 -1.016736306416454 0.001408809355816 2912 -1.016736306416454 -0.706069127893522 0.001408809355816 2913 -0.706069127893522 -1.016736306416454 0.001408809355816 2914 0.722805434309974 -0.706069127893522 0.001408809355816 2915 -1.016736306416454 0.722805434309974 0.001408809355816 2916 -0.724546042342154 0.671173915824726 0.020225369854924 2917 0.671173915824726 -0.946627873482572 0.020225369854924 2918 -0.946627873482572 -0.724546042342154 0.020225369854924 2919 -0.724546042342154 -0.946627873482572 0.020225369854924 2920 0.671173915824726 -0.724546042342154 0.020225369854924 2921 -0.946627873482572 0.671173915824726 0.020225369854924 2922 -0.880607781701986 0.859512343113706 0.007147818771900 2923 0.859512343113706 -0.978904561411718 0.007147818771900 2924 -0.978904561411718 -0.880607781701986 0.007147818771900 2925 -0.880607781701986 -0.978904561411718 0.007147818771900 2926 0.859512343113706 -0.880607781701986 0.007147818771900 2927 -0.978904561411718 0.859512343113706 0.007147818771900 2928 Gauss quadrature points and weights on the reference triangle order p=20 2929 """ 2930 2931 2932 economical_gauss_tetrahedron_table = """\ 2933 1 2934 -0.500000000000000 -0.500000000000000 -0.500000000000000 1.333333333333333 2935 Gauss quadrature constants for the reference tetrahedron order p=1 2936 2937 4 2938 -0.723606797749979 -0.723606797749979 -0.723606797749979 0.333333333333333 2939 0.170820393249937 -0.723606797749979 -0.723606797749979 0.333333333333333 2940 -0.723606797749979 0.170820393249937 -0.723606797749979 0.333333333333333 2941 -0.723606797749979 -0.723606797749979 0.170820393249937 0.333333333333333 2942 Gauss quadrature constants for the reference tetrahedron order p=2 2943 2944 5 2945 -0.500000000000000 -0.500000000000000 -0.500000000000000 -1.066666666666667 2946 -0.666666666666667 -0.666666666666667 -0.666666666666667 0.600000000000000 2947 -0.666666666666667 -0.666666666666667 0.000000000000000 0.600000000000000 2948 -0.666666666666667 0.000000000000000 -0.666666666666667 0.600000000000000 2949 0.000000000000000 -0.666666666666667 -0.666666666666667 0.600000000000000 2950 Gauss quadrature constants for the reference tetrahedron order p=3 2951 2952 11 2953 -0.500000000000000 -0.500000000000000 -0.500000000000000 -0.105244444444444 2954 -0.857142857142857 -0.857142857142857 -0.857142857142857 0.060977777777778 2955 -0.857142857142857 -0.857142857142857 0.571428571428571 0.060977777777778 2956 -0.857142857142857 0.571428571428571 -0.857142857142857 0.060977777777778 2957 0.571428571428571 -0.857142857142857 -0.857142857142857 0.060977777777778 2958 -0.201192847666402 -0.201192847666402 -0.798807152333598 0.199111111111111 2959 -0.201192847666402 -0.798807152333598 -0.201192847666402 0.199111111111111 2960 -0.798807152333598 -0.201192847666402 -0.201192847666402 0.199111111111111 2961 -0.201192847666402 -0.798807152333598 -0.798807152333598 0.199111111111111 2962 -0.798807152333598 -0.201192847666402 -0.798807152333598 0.199111111111111 2963 -0.798807152333598 -0.798807152333598 -0.201192847666402 0.199111111111111 2964 Gauss quadrature constants for the reference tetrahedron order p=4 2965 2966 14 2967 -0.814529499378218 -0.814529499378218 -0.814529499378218 0.097990724155149 2968 0.443588498134653 -0.814529499378218 -0.814529499378218 0.097990724155149 2969 -0.814529499378218 0.443588498134653 -0.814529499378218 0.097990724155149 2970 -0.814529499378218 -0.814529499378218 0.443588498134653 0.097990724155149 2971 -0.378228161473399 -0.378228161473399 -0.378228161473399 0.150250567624021 2972 -0.865315515579804 -0.378228161473399 -0.378228161473399 0.150250567624021 2973 -0.378228161473399 -0.865315515579804 -0.378228161473399 0.150250567624021 2974 -0.378228161473399 -0.378228161473399 -0.865315515579804 0.150250567624021 2975 -0.091007408251299 -0.091007408251299 -0.908992591748701 0.056728027702775 2976 -0.091007408251299 -0.908992591748701 -0.091007408251299 0.056728027702775 2977 -0.908992591748701 -0.091007408251299 -0.091007408251299 0.056728027702775 2978 -0.091007408251299 -0.908992591748701 -0.908992591748701 0.056728027702775 2979 -0.908992591748701 -0.091007408251299 -0.908992591748701 0.056728027702775 2980 -0.908992591748701 -0.908992591748701 -0.091007408251299 0.056728027702775 2981 Gauss quadrature constants for the reference tetrahedron order p=5 2982 2983 24 2984 -0.570794257481696 -0.570794257481696 -0.570794257481696 0.053230333677557 2985 -0.287617227554912 -0.570794257481696 -0.570794257481696 0.053230333677557 2986 -0.570794257481696 -0.287617227554912 -0.570794257481696 0.053230333677557 2987 -0.570794257481696 -0.570794257481696 -0.287617227554912 0.053230333677557 2988 -0.918652082930777 -0.918652082930777 -0.918652082930777 0.013436281407094 2989 0.755956248792332 -0.918652082930777 -0.918652082930777 0.013436281407094 2990 -0.918652082930777 0.755956248792332 -0.918652082930777 0.013436281407094 2991 -0.918652082930777 -0.918652082930777 0.755956248792332 0.013436281407094 2992 -0.355324219715449 -0.355324219715449 -0.355324219715449 0.073809575391540 2993 -0.934027340853653 -0.355324219715449 -0.355324219715449 0.073809575391540 2994 -0.355324219715449 -0.934027340853653 -0.355324219715449 0.073809575391540 2995 -0.355324219715449 -0.355324219715449 -0.934027340853653 0.073809575391540 2996 -0.872677996249965 -0.872677996249965 -0.460655337083368 0.064285714285714 2997 -0.872677996249965 -0.460655337083368 -0.872677996249965 0.064285714285714 2998 -0.872677996249965 -0.872677996249965 0.206011329583298 0.064285714285714 2999 -0.872677996249965 0.206011329583298 -0.872677996249965 0.064285714285714 3000 -0.872677996249965 -0.460655337083368 0.206011329583298 0.064285714285714 3001 -0.872677996249965 0.206011329583298 -0.460655337083368 0.064285714285714 3002 -0.460655337083368 -0.872677996249965 -0.872677996249965 0.064285714285714 3003 -0.460655337083368 -0.872677996249965 0.206011329583298 0.064285714285714 3004 -0.460655337083368 0.206011329583298 -0.872677996249965 0.064285714285714 3005 0.206011329583298 -0.872677996249965 -0.460655337083368 0.064285714285714 3006 0.206011329583298 -0.872677996249965 -0.872677996249965 0.064285714285714 3007 0.206011329583298 -0.460655337083368 -0.872677996249965 0.064285714285714 3008 Gauss quadrature constants for the reference tetrahedron order p=6 3009 3010 31 3011 0.000000000000000 0.000000000000000 -1.000000000000000 0.007760141093474 3012 0.000000000000000 -1.000000000000000 0.000000000000000 0.007760141093474 3013 -1.000000000000000 0.000000000000000 0.000000000000000 0.007760141093474 3014 -1.000000000000000 -1.000000000000000 0.000000000000000 0.007760141093474 3015 -1.000000000000000 0.000000000000000 -1.000000000000000 0.007760141093474 3016 0.000000000000000 -1.000000000000000 -1.000000000000000 0.007760141093474 3017 -0.500000000000000 -0.500000000000000 -0.500000000000000 0.146113787728871 3018 -0.843573615339364 -0.843573615339364 -0.843573615339364 0.084799532195309 3019 -0.843573615339364 -0.843573615339364 0.530720846018092 0.084799532195309 3020 -0.843573615339364 0.530720846018092 -0.843573615339364 0.084799532195309 3021 0.530720846018092 -0.843573615339364 -0.843573615339364 0.084799532195309 3022 -0.756313566672190 -0.756313566672190 -0.756313566672190 -0.500141920914655 3023 -0.756313566672190 -0.756313566672190 0.268940700016569 -0.500141920914655 3024 -0.756313566672190 0.268940700016569 -0.756313566672190 -0.500141920914655 3025 0.268940700016569 -0.756313566672190 -0.756313566672190 -0.500141920914655 3026 -0.334921671107159 -0.334921671107159 -0.334921671107159 0.039131402104588 3027 -0.334921671107159 -0.334921671107159 -0.995234986678524 0.039131402104588 3028 -0.334921671107159 -0.995234986678524 -0.334921671107159 0.039131402104588 3029 -0.995234986678524 -0.334921671107159 -0.334921671107159 0.039131402104588 3030 -0.800000000000000 -0.800000000000000 -0.600000000000000 0.220458553791887 3031 -0.800000000000000 -0.600000000000000 -0.800000000000000 0.220458553791887 3032 -0.800000000000000 -0.800000000000000 0.200000000000000 0.220458553791887 3033 -0.800000000000000 0.200000000000000 -0.800000000000000 0.220458553791887 3034 -0.800000000000000 -0.600000000000000 0.200000000000000 0.220458553791887 3035 -0.800000000000000 0.200000000000000 -0.600000000000000 0.220458553791887 3036 -0.600000000000000 -0.800000000000000 -0.800000000000000 0.220458553791887 3037 -0.600000000000000 -0.800000000000000 0.200000000000000 0.220458553791887 3038 -0.600000000000000 0.200000000000000 -0.800000000000000 0.220458553791887 3039 0.200000000000000 -0.800000000000000 -0.600000000000000 0.220458553791887 3040 0.200000000000000 -0.800000000000000 -0.800000000000000 0.220458553791887 3041 0.200000000000000 -0.600000000000000 -0.800000000000000 0.220458553791887 3042 Gauss quadrature constants for the reference tetrahedron order p=7 3043 3044 43 3045 -0.500000000000000 -0.500000000000000 -0.500000000000000 -0.164001509269119 3046 -0.586340136778654 -0.586340136778654 -0.586340136778654 0.114002446582935 3047 -0.586340136778654 -0.586340136778654 -0.240979589664039 0.114002446582935 3048 -0.586340136778654 -0.240979589664039 -0.586340136778654 0.114002446582935 3049 -0.240979589664039 -0.586340136778654 -0.586340136778654 0.114002446582935 3050 -0.835792823378907 -0.835792823378907 -0.835792823378907 0.015736266505071 3051 -0.835792823378907 -0.835792823378907 0.507378470136720 0.015736266505071 3052 -0.835792823378907 0.507378470136720 -0.835792823378907 0.015736266505071 3053 0.507378470136720 -0.835792823378907 -0.835792823378907 0.015736266505071 3054 -0.988436098989604 -0.988436098989604 -0.988436098989604 0.001358672872743 3055 -0.988436098989604 -0.988436098989604 0.965308296968812 0.001358672872743 3056 -0.988436098989604 0.965308296968812 -0.988436098989604 0.001358672872743 3057 0.965308296968812 -0.988436098989604 -0.988436098989604 0.001358672872743 3058 -0.898934519962212 -0.898934519962212 -0.101065480037788 0.036637470595738 3059 -0.898934519962212 -0.101065480037788 -0.898934519962212 0.036637470595738 3060 -0.101065480037788 -0.898934519962212 -0.898934519962212 0.036637470595738 3061 -0.898934519962212 -0.101065480037788 -0.101065480037788 0.036637470595738 3062 -0.101065480037788 -0.898934519962212 -0.101065480037788 0.036637470595738 3063 -0.101065480037788 -0.101065480037788 -0.898934519962212 0.036637470595738 3064 -0.541866927766378 -0.541866927766378 -0.928720834422932 0.045635886469455 3065 -0.541866927766378 -0.928720834422932 -0.541866927766378 0.045635886469455 3066 -0.541866927766378 -0.541866927766378 0.012454689955687 0.045635886469455 3067 -0.541866927766378 0.012454689955687 -0.541866927766378 0.045635886469455 3068 -0.541866927766378 -0.928720834422932 0.012454689955687 0.045635886469455 3069 -0.541866927766378 0.012454689955687 -0.928720834422932 0.045635886469455 3070 -0.928720834422932 -0.541866927766378 -0.541866927766378 0.045635886469455 3071 -0.928720834422932 -0.541866927766378 0.012454689955687 0.045635886469455 3072 -0.928720834422932 0.012454689955687 -0.541866927766378 0.045635886469455 3073 0.012454689955687 -0.541866927766378 -0.928720834422932 0.045635886469455 3074 0.012454689955687 -0.541866927766378 -0.541866927766378 0.045635886469455 3075 0.012454689955687 -0.928720834422932 -0.541866927766378 0.045635886469455 3076 -0.926784500893605 -0.926784500893605 -0.619027916130733 0.017124153129297 3077 -0.926784500893605 -0.619027916130733 -0.926784500893605 0.017124153129297 3078 -0.926784500893605 -0.926784500893605 0.472596917917943 0.017124153129297 3079 -0.926784500893605 0.472596917917943 -0.926784500893605 0.017124153129297 3080 -0.926784500893605 -0.619027916130733 0.472596917917943 0.017124153129297 3081 -0.926784500893605 0.472596917917943 -0.619027916130733 0.017124153129297 3082 -0.619027916130733 -0.926784500893605 -0.926784500893605 0.017124153129297 3083 -0.619027916130733 -0.926784500893605 0.472596917917943 0.017124153129297 3084 -0.619027916130733 0.472596917917943 -0.926784500893605 0.017124153129297 3085 0.472596917917943 -0.926784500893605 -0.619027916130733 0.017124153129297 3086 0.472596917917943 -0.926784500893605 -0.926784500893605 0.017124153129297 3087 0.472596917917943 -0.619027916130733 -0.926784500893605 0.017124153129297 3088 Gauss quadrature constants for the reference tetrahedron order p=8 3089 3090 53 3091 -0.500000000000000 -0.500000000000000 -0.500000000000000 -1.102392306608869 3092 -0.903297922900526 -0.903297922900526 -0.903297922900526 0.014922692552682 3093 -0.903297922900526 -0.903297922900526 0.709893768701580 0.014922692552682 3094 -0.903297922900526 0.709893768701580 -0.903297922900526 0.014922692552682 3095 0.709893768701580 -0.903297922900526 -0.903297922900526 0.014922692552682 3096 -0.350841439764235 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0.277603247076406 3109 -0.736744381506260 -0.832670596765630 -0.736744381506260 0.277603247076406 3110 -0.736744381506260 -0.736744381506260 0.306159359778151 0.277603247076406 3111 -0.736744381506260 0.306159359778151 -0.736744381506260 0.277603247076406 3112 -0.736744381506260 -0.832670596765630 0.306159359778151 0.277603247076406 3113 -0.736744381506260 0.306159359778151 -0.832670596765630 0.277603247076406 3114 -0.832670596765630 -0.736744381506260 -0.736744381506260 0.277603247076406 3115 -0.832670596765630 -0.736744381506260 0.306159359778151 0.277603247076406 3116 -0.832670596765630 0.306159359778151 -0.736744381506260 0.277603247076406 3117 0.306159359778151 -0.736744381506260 -0.832670596765630 0.277603247076406 3118 0.306159359778151 -0.736744381506260 -0.736744381506260 0.277603247076406 3119 0.306159359778151 -0.832670596765630 -0.736744381506260 0.277603247076406 3120 -0.132097077177186 -0.132097077177186 -0.784460280901143 0.026820671221285 3121 -0.132097077177186 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0.003453031004456 3134 -1.002752554636276 -1.002752554636276 0.452398163998938 0.003453031004456 3135 -1.002752554636276 0.452398163998938 -1.002752554636276 0.003453031004456 3136 -1.002752554636276 -0.446893054726385 0.452398163998938 0.003453031004456 3137 -1.002752554636276 0.452398163998938 -0.446893054726385 0.003453031004456 3138 -0.446893054726385 -1.002752554636276 -1.002752554636276 0.003453031004456 3139 -0.446893054726385 -1.002752554636276 0.452398163998938 0.003453031004456 3140 -0.446893054726385 0.452398163998938 -1.002752554636276 0.003453031004456 3141 0.452398163998938 -1.002752554636276 -0.446893054726385 0.003453031004456 3142 0.452398163998938 -1.002752554636276 -1.002752554636276 0.003453031004456 3143 0.452398163998938 -0.446893054726385 -1.002752554636276 0.003453031004456 3144 Gauss quadrature constants for the reference tetrahedron order p=9 3145 3146 126 3147 -0.857142857142857 -0.857142857142857 0.571428571428571 0.362902592520648 3148 -0.857142857142857 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-0.612213575618572 5963 -0.571428571428571 -0.571428571428571 -0.857142857142857 -0.612213575618572 5964 -0.285714285714286 -0.857142857142857 -0.857142857142857 -0.612213575618572 5965 -0.857142857142857 -0.857142857142857 -0.571428571428571 -0.612213575618572 5966 -0.857142857142857 -0.571428571428571 -0.857142857142857 -0.612213575618572 5967 -0.571428571428571 -0.857142857142857 -0.857142857142857 -0.612213575618572 5968 -0.857142857142857 -0.857142857142857 -0.857142857142857 -0.612213575618572 5969 -0.833333333333333 -0.833333333333333 0.500000000000000 0.076141727180383 5970 -0.833333333333333 -0.500000000000000 0.166666666666667 0.076141727180383 5971 -0.833333333333333 -0.166666666666667 -0.166666666666667 0.076141727180383 5972 -0.833333333333333 0.166666666666667 -0.500000000000000 0.076141727180383 5973 -0.833333333333333 0.500000000000000 -0.833333333333333 0.076141727180383 5974 -0.500000000000000 -0.833333333333333 0.166666666666667 0.076141727180383 5975 -0.500000000000000 -0.500000000000000 -0.166666666666667 0.076141727180383 5976 -0.500000000000000 -0.166666666666667 -0.500000000000000 0.076141727180383 5977 -0.500000000000000 0.166666666666667 -0.833333333333333 0.076141727180383 5978 -0.166666666666667 -0.833333333333333 -0.166666666666667 0.076141727180383 5979 -0.166666666666667 -0.500000000000000 -0.500000000000000 0.076141727180383 5980 -0.166666666666667 -0.166666666666667 -0.833333333333333 0.076141727180383 5981 0.166666666666667 -0.833333333333333 -0.500000000000000 0.076141727180383 5982 0.166666666666667 -0.500000000000000 -0.833333333333333 0.076141727180383 5983 0.500000000000000 -0.833333333333333 -0.833333333333333 0.076141727180383 5984 -0.833333333333333 -0.833333333333333 0.166666666666667 0.076141727180383 5985 -0.833333333333333 -0.500000000000000 -0.166666666666667 0.076141727180383 5986 -0.833333333333333 -0.166666666666667 -0.500000000000000 0.076141727180383 5987 -0.833333333333333 0.166666666666667 -0.833333333333333 0.076141727180383 5988 -0.500000000000000 -0.833333333333333 -0.166666666666667 0.076141727180383 5989 -0.500000000000000 -0.500000000000000 -0.500000000000000 0.076141727180383 5990 -0.500000000000000 -0.166666666666667 -0.833333333333333 0.076141727180383 5991 -0.166666666666667 -0.833333333333333 -0.500000000000000 0.076141727180383 5992 -0.166666666666667 -0.500000000000000 -0.833333333333333 0.076141727180383 5993 0.166666666666667 -0.833333333333333 -0.833333333333333 0.076141727180383 5994 -0.833333333333333 -0.833333333333333 -0.166666666666667 0.076141727180383 5995 -0.833333333333333 -0.500000000000000 -0.500000000000000 0.076141727180383 5996 -0.833333333333333 -0.166666666666667 -0.833333333333333 0.076141727180383 5997 -0.500000000000000 -0.833333333333333 -0.500000000000000 0.076141727180383 5998 -0.500000000000000 -0.500000000000000 -0.833333333333333 0.076141727180383 5999 -0.166666666666667 -0.833333333333333 -0.833333333333333 0.076141727180383 6000 -0.833333333333333 -0.833333333333333 -0.500000000000000 0.076141727180383 6001 -0.833333333333333 -0.500000000000000 -0.833333333333333 0.076141727180383 6002 -0.500000000000000 -0.833333333333333 -0.833333333333333 0.076141727180383 6003 -0.833333333333333 -0.833333333333333 -0.833333333333333 0.076141727180383 6004 -0.800000000000000 -0.800000000000000 0.400000000000000 -0.004255896151021 6005 -0.800000000000000 -0.400000000000000 0.000000000000000 -0.004255896151021 6006 -0.800000000000000 0.000000000000000 -0.400000000000000 -0.004255896151021 6007 -0.800000000000000 0.400000000000000 -0.800000000000000 -0.004255896151021 6008 -0.400000000000000 -0.800000000000000 0.000000000000000 -0.004255896151021 6009 -0.400000000000000 -0.400000000000000 -0.400000000000000 -0.004255896151021 6010 -0.400000000000000 0.000000000000000 -0.800000000000000 -0.004255896151021 6011 0.000000000000000 -0.800000000000000 -0.400000000000000 -0.004255896151021 6012 0.000000000000000 -0.400000000000000 -0.800000000000000 -0.004255896151021 6013 0.400000000000000 -0.800000000000000 -0.800000000000000 -0.004255896151021 6014 -0.800000000000000 -0.800000000000000 0.000000000000000 -0.004255896151021 6015 -0.800000000000000 -0.400000000000000 -0.400000000000000 -0.004255896151021 6016 -0.800000000000000 0.000000000000000 -0.800000000000000 -0.004255896151021 6017 -0.400000000000000 -0.800000000000000 -0.400000000000000 -0.004255896151021 6018 -0.400000000000000 -0.400000000000000 -0.800000000000000 -0.004255896151021 6019 0.000000000000000 -0.800000000000000 -0.800000000000000 -0.004255896151021 6020 -0.800000000000000 -0.800000000000000 -0.400000000000000 -0.004255896151021 6021 -0.800000000000000 -0.400000000000000 -0.800000000000000 -0.004255896151021 6022 -0.400000000000000 -0.800000000000000 -0.800000000000000 -0.004255896151021 6023 -0.800000000000000 -0.800000000000000 -0.800000000000000 -0.004255896151021 6024 -0.750000000000000 -0.750000000000000 0.250000000000000 0.000083414141296 6025 -0.750000000000000 -0.250000000000000 -0.250000000000000 0.000083414141296 6026 -0.750000000000000 0.250000000000000 -0.750000000000000 0.000083414141296 6027 -0.250000000000000 -0.750000000000000 -0.250000000000000 0.000083414141296 6028 -0.250000000000000 -0.250000000000000 -0.750000000000000 0.000083414141296 6029 0.250000000000000 -0.750000000000000 -0.750000000000000 0.000083414141296 6030 -0.750000000000000 -0.750000000000000 -0.250000000000000 0.000083414141296 6031 -0.750000000000000 -0.250000000000000 -0.750000000000000 0.000083414141296 6032 -0.250000000000000 -0.750000000000000 -0.750000000000000 0.000083414141296 6033 -0.750000000000000 -0.750000000000000 -0.750000000000000 0.000083414141296 6034 -0.666666666666667 -0.666666666666667 0.000000000000000 -0.000000352698561 6035 -0.666666666666667 0.000000000000000 -0.666666666666667 -0.000000352698561 6036 0.000000000000000 -0.666666666666667 -0.666666666666667 -0.000000352698561 6037 -0.666666666666667 -0.666666666666667 -0.666666666666667 -0.000000352698561 6038 -0.500000000000000 -0.500000000000000 -0.500000000000000 0.000000000106067 6039 Gauss quadrature constants for the reference tetrahedron order p=21 6040 """ 6041 6042 6043 6044 6045 newton_cotes_table = """\ 6046 2 6047 1 1 6048 -1 1 6049 Closed Newton-Cotes quadrature constants order n=1 6050 6051 3 6052 1 1/3 6053 0 4/3 6054 -1 1/3 6055 Closed Newton-Cotes quadrature constants order n=2 6056 6057 4 6058 1 1/4 6059 1/3 3/4 6060 -1/3 3/4 6061 -1 1/4 6062 Closed Newton-Cotes quadrature constants order n=3 6063 6064 5 6065 1 7/45 6066 1/2 32/45 6067 0 12/45 6068 -1/2 32/45 6069 -1 7/45 6070 Closed Newton-Cotes quadrature constants order n=4 6071 6072 6 6073 1 19/144 6074 3/5 75/144 6075 1/5 50/144 6076 -1/5 50/144 6077 -3/5 75/144 6078 -1 19/144 6079 Closed Newton-Cotes quadrature constants order n=5 6080 6081 7 6082 1 41/420 6083 2/3 216/420 6084 1/3 27/420 6085 0 272/420 6086 -1/3 27/420 6087 -2/3 216/420 6088 -1 41/420 6089 Closed Newton-Cotes quadrature constants order n=6 6090 6091 8 6092 1 751/8640 6093 5/7 3577/8640 6094 3/7 1323/8640 6095 1/7 2989/8640 6096 -1/7 2989/8640 6097 -3/7 1323/8640 6098 -5/7 3577/8640 6099 -1 751/8640 6100 Closed Newton-Cotes quadrature constants order n=7 6101 """ 6102 6103 6104 chebyshev_table = """\ 6105 2 6106 0.5773502691896257645091488 1 6107 -0.5773502691896257645091488 1 6108 Chebyshev quadrature constants order n+1=3 6109 6110 3 6111 0.7071067811865475244008444 2/3 6112 0.0000000000000000000000000 2/3 6113 -0.7071067811865475244008444 2/3 6114 Chebyshev quadrature constants order n+1=4 6115 6116 4 6117 0.7946544722917661229555309 1/2 6118 0.1875924740850798998601393 1/2 6119 -0.1875924740850798998601393 1/2 6120 -0.7946544722917661229555309 1/2 6121 Chebyshev quadrature constants order n+1=5 6122 6123 5 6124 0.8324974870009818758925836 2/5 6125 0.3745414095535810655860444 2/5 6126 0.0000000000000000000000000 2/5 6127 -0.3745414095535810655860444 2/5 6128 -0.8324974870009818758925836 2/5 6129 Chebyshev quadrature constants order n+1=6 6130 6131 6 6132 0.8662468181078205913835981 1/3 6133 0.4225186537611115291185464 1/3 6134 0.2666354015167047203315346 1/3 6135 -0.2666354015167047203315346 1/3 6136 -0.4225186537611115291185464 1/3 6137 -0.8662468181078205913835981 1/3 6138 Chebyshev quadrature constants order n+1=7 6139 6140 7 6141 0.8838617007580490357042241 2/7 6142 0.5296567752851568113850475 2/7 6143 0.3239118105199076375196731 2/7 6144 0.0000000000000000000000000 2/7 6145 -0.3239118105199076375196731 2/7 6146 -0.5296567752851568113850475 2/7 6147 -0.8838617007580490357042241 2/7 6148 Chebyshev quadrature constants order n+1=8 6149 6150 9 6151 0.9115893077284344736649486 2/9 6152 0.6010186553802380714281279 2/9 6153 0.5287617830578799932601816 2/9 6154 0.1679061842148039430680319 2/9 6155 0.0000000000000000000000000 2/9 6156 -0.1679061842148039430680319 2/9 6157 -0.5287617830578799932601816 2/9 6158 -0.6010186553802380714281279 2/9 6159 -0.9115893077284344736649486 2/9 6160 Chebyshev quadrature constants order n+1=10 6161 """ 6162 6163 6164 lobatto_table = """\ 6165 3 6166 1.0000000000000000000000000 0.3333333333333333333333333 6167 0.0000000000000000000000000 1.3333333333333333333333333 6168 -1.0000000000000000000000000 0.3333333333333333333333333 6169 Lobatto (Radau) quadrature constants order 2n-3=3 6170 6171 4 6172 1.0000000000000000000000000 0.1666666666666666666666667 6173 0.4472135954999579392818347 0.8333333333333333333333333 6174 -0.4472135954999579392818347 0.8333333333333333333333333 6175 -1.0000000000000000000000000 0.1666666666666666666666667 6176 Lobatto (Radau) quadrature constants order 2n-3=5 6177 6178 5 6179 1.0000000000000000000000000 0.1000000000000000000000000 6180 0.6546536707079771437982925 0.5444444444444444444444444 6181 0.0000000000000000000000000 0.7111111111111111111111111 6182 -0.6546536707079771437982925 0.5444444444444444444444444 6183 -1.0000000000000000000000000 0.1000000000000000000000000 6184 Lobatto (Radau) quadrature constants order 2n-3=7 6185 6186 6 6187 1.0000000000000000000000000 0.0666666666666666666666667 6188 0.7650553239294646928510030 0.3784749562978469803166128 6189 0.2852315164806450963141510 0.5548583770354863530167205 6190 -0.2852315164806450963141510 0.5548583770354863530167205 6191 -0.7650553239294646928510030 0.3784749562978469803166128 6192 -1.0000000000000000000000000 0.0666666666666666666666667 6193 Lobatto (Radau) quadrature constants order 2n-3=9 6194 6195 7 6196 1.0000000000000000000000000 0.0476190476190476190476190 6197 0.8302238962785669298720322 0.2768260473615659480107004 6198 0.4688487934707142138037719 0.4317453812098626234178710 6199 0.0000000000000000000000000 0.4876190476190476190476190 6200 -0.4688487934707142138037719 0.4317453812098626234178710 6201 -0.8302238962785669298720322 0.2768260473615659480107004 6202 -1.0000000000000000000000000 0.0476190476190476190476190 6203 Lobatto (Radau) quadrature constants order 2n-3=11 6204 6205 8 6206 1.0000000000000000000000000 0.0357142857142857142857143 6207 0.8717401485096066153374457 0.2107042271435060393829921 6208 0.5917001814331423021445107 0.3411226924835043647642407 6209 0.2092992179024788687686573 0.4124587946587038815670530 6210 -0.2092992179024788687686573 0.4124587946587038815670530 6211 -0.5917001814331423021445107 0.3411226924835043647642407 6212 -0.8717401485096066153374457 0.2107042271435060393829921 6213 -1.0000000000000000000000000 0.0357142857142857142857143 6214 Lobatto (Radau) quadrature constants order 2n-3=13 6215 6216 9 6217 1.0000000000000000000000000 0.0277777777777777777777778 6218 0.8997579954114601573123452 0.1654953615608055250463397 6219 0.6771862795107377534458854 0.2745387125001617352807056 6220 0.3631174638261781587107521 0.3464285109730463451151315 6221 0.0000000000000000000000000 0.3715192743764172335600907 6222 -0.3631174638261781587107521 0.3464285109730463451151315 6223 -0.6771862795107377534458854 0.2745387125001617352807056 6224 -0.8997579954114601573123452 0.1654953615608055250463397 6225 -1.0000000000000000000000000 0.0277777777777777777777778 6226 Lobatto (Radau) quadrature constants order 2n-3=15 6227 6228 10 6229 1.0000000000000000000000000 0.0222222222222222222222222 6230 0.9195339081664588138289327 0.1333059908510701111262272 6231 0.7387738651055050750031062 0.2248893420631264521194578 6232 0.4779249498104444956611751 0.2920426836796837578755823 6233 0.1652789576663870246262198 0.3275397611838974566565105 6234 -0.1652789576663870246262198 0.3275397611838974566565105 6235 -0.4779249498104444956611751 0.2920426836796837578755823 6236 -0.7387738651055050750031062 0.2248893420631264521194578 6237 -0.9195339081664588138289327 0.1333059908510701111262272 6238 -1.0000000000000000000000000 0.0222222222222222222222222 6239 Lobatto (Radau) quadrature constants order 2n-3=17 6240 6241 11 6242 1.0000000000000000000000000 0.0181818181818181818181818 6243 0.9340014304080591343322741 0.1096122732669948644614034 6244 0.7844834736631444186224178 0.1871698817803052041081415 6245 0.5652353269962050064709640 0.2480481042640283140400849 6246 0.2957581355869393914319115 0.2868791247790080886792224 6247 0.0000000000000000000000000 0.3002175954556906937859319 6248 -0.2957581355869393914319115 0.2868791247790080886792224 6249 -0.5652353269962050064709640 0.2480481042640283140400849 6250 -0.7844834736631444186224178 0.1871698817803052041081415 6251 -0.9340014304080591343322741 0.1096122732669948644614034 6252 -1.0000000000000000000000000 0.0181818181818181818181818 6253 Lobatto (Radau) quadrature constants order 2n-3=19 6254 6255 12 6256 1.0000000000000000000000000 0.0151515151515151515151515 6257 0.9448992722228822234075801 0.0916845174131961306683426 6258 0.8192793216440066783486416 0.1579747055643701151646710 6259 0.6328761530318606776624049 0.2125084177610211453583021 6260 0.3995309409653489322643498 0.2512756031992012802932444 6261 0.1365529328549275548640619 0.2714052409106961770002883 6262 -0.1365529328549275548640619 0.2714052409106961770002883 6263 -0.3995309409653489322643498 0.2512756031992012802932444 6264 -0.6328761530318606776624049 0.2125084177610211453583021 6265 -0.8192793216440066783486416 0.1579747055643701151646710 6266 -0.9448992722228822234075801 0.0916845174131961306683426 6267 -1.0000000000000000000000000 0.0151515151515151515151515 6268 Lobatto (Radau) quadrature constants order 2n-3=21 6269 """ 6270