function [pts,t,table,stats_matrix]=set_leb_opt_n1n3n6_gll(deg) %-------------------------------------------------------------------------- % PURPOSE: %----------- % SET OF POINTS FOR THE UNIT SIMPLEX, HAVING VERTICES % [0,0], [1,0], [0,1], SYMMETRIES AND ASSIGNED DISTRIBUTION ON THE % SIDES (Gauss-Legendre-Lobatto). % THE LEBESGUE CONSTANTS ARE PARTICULARLY LOW. %-------------------------------------------------------------------------- % INPUT: %--------- % deg: DEGREE OF THE SET. %-------------------------------------------------------------------------- % OUTPUT: %--------- % pts: CARTESIAN COORDINATES, STORED IN A "n x 2" MATRIX. % pts_bar: BARYCENTRIC COORDINATES, STORED IN A "n x 3" MATRIX. % t: DESCRIBES THE SYMMETRY. % stats_matrix: MATRIX WITH STATISTICS WITH COLUMNS THAT CONSIST OF % DEGREE, LEBESGUE CONSTANT, ABS. VALUE OF VANDERMONDE MATRIX "V" % DETERMINANT (w.r.t DUBINER BASIS), CONDITIONING OF "V". %-------------------------------------------------------------------------- % --------------------------------------------------------------- % |DEGR| LEBESGUE | ABS.DET.VAND.| COND. VAND. | MAX. LAGR. | % --------------------------------------------------------------- % | 1 | 1.00000e+00 | 5.87878e+01 | 3.70699e+00 | 1.00000e+00 | % | 2 | 1.66667e+00 | 3.13712e+04 | 1.01810e+01 | 1.00000e+00 | % | 3 | 2.11244e+00 | 3.44701e+08 | 2.03638e+01 | 1.00000e+00 | % | 4 | 2.58801e+00 | 8.98221e+13 | 3.95876e+01 | 1.02814e+00 | % | 5 | 3.08116e+00 | 6.69351e+20 | 5.28526e+01 | 1.05314e+00 | % | 6 | 3.59605e+00 | 1.87536e+29 | 7.13009e+01 | 1.02909e+00 | % | 7 | 4.14590e+00 | 1.94719e+39 | 9.41441e+01 | 1.02500e+00 | % | 8 | 4.77382e+00 | 7.59934e+50 | 1.18408e+02 | 1.03994e+00 | % | 9 | 5.49425e+00 | 1.43106e+64 | 1.41026e+02 | 1.03863e+00 | % | 10 | 6.28492e+00 | 2.40480e+79 | 1.84334e+02 | 1.08256e+00 | % | 11 | 7.00203e+00 | 2.11926e+96 | 2.62281e+02 | 1.18378e+00 | % | 12 | 7.25934e+00 | 2.07917e+115 | 2.28046e+02 | 1.04342e+00 | % | 13 | 8.57673e+00 | 2.16493e+135 | 3.32262e+02 | 1.09934e+00 | % | 14 | 8.81565e+00 | 1.23824e+158 | 4.80175e+02 | 1.18945e+00 | % | 15 | 8.90865e+00 | 9.40546e+182 | 3.96741e+02 | 1.09615e+00 | % | 16 | 1.06553e+01 | 1.73095e+208 | 4.54660e+02 | 1.18671e+00 | % | 17 | 1.14147e+01 | 2.25105e+236 | 5.47919e+02 | 1.08768e+00 | % | 18 | 1.26889e+01 | 1.11594e+267 | 4.72717e+02 | 1.07067e+00 | % --------------------------------------------------------------- %-------------------------------------------------------------------------- %% Copyright (C) 2011 %% Francesca Rapetti, Alvise Sommariva, Marco Vianello. %% %% This program is free software; you can redistribute it and/or modify %% it under the terms of the GNU General Public License as published by %% the Free Software Foundation; either version 2 of the License, or %% (at your option) any later version. %% %% This program is distributed in the hope that it will be useful, %% but WITHOUT ANY WARRANTY; without even the implied warranty of %% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the %% GNU General Public License for more details. %% %% You should have received a copy of the GNU General Public License %% along with this program; if not, write to the Free Software %% Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA %% %% Author: %% Francesca Rapetti %% Alvise Sommariva %% Marco Vianello %% %% Date: October 23, 2011. %-------------------------------------------------------------------------- switch deg case 1 % PASSED (I). % DEGREE: 1 % POINTS: 3 % LEBESGUE CONSTANT: 1.00000e+00 % DETERMINANT: -5.87878e+01 % CONDITIONING: 3.70699e+00 % MAX LAGRANGE POLYNOMIALS: 1.00000e+00 % VANDERMONDE: ORTHONORMAL DUBINER LEGENDRE. table=[ 0 0 0 ]; t=[ ]; pts=[ 1.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 1.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 1.000000000000000000000000000000e+00 ]; case 2 % PASSED (I). % DEGREE: 2 % POINTS: 6 % LEBESGUE CONSTANT: 1.66667e+00 % DETERMINANT: 3.13712e+04 % CONDITIONING: 1.01810e+01 % MAX LAGRANGE POLYNOMIALS: 1.00000e+00 % VANDERMONDE: ORTHONORMAL DUBINER LEGENDRE. table=[ 0 0 0 ]; t=[ ]; pts=[ 1.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 1.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 1.000000000000000000000000000000e+00 5.000000000000000000000000000000e-01 5.000000000000000000000000000000e-01 0.000000000000000000000000000000e+00 5.000000000000000000000000000000e-01 0.000000000000000000000000000000e+00 5.000000000000000000000000000000e-01 0.000000000000000000000000000000e+00 5.000000000000000000000000000000e-01 5.000000000000000000000000000000e-01 ]; case 3 % PASSED (I). % DEGREE: 3 % POINTS: 10 % LEBESGUE CONSTANT: 2.11244e+00 % DETERMINANT: -3.44701e+08 % CONDITIONING: 2.03638e+01 % MAX LAGRANGE POLYNOMIALS: 1.00000e+00 % VANDERMONDE: ORTHONORMAL DUBINER LEGENDRE. table=[ 1 0 0 ]; t=[ ]; pts=[ 3.333333333333333148296162562474e-01 3.333333333333333148296162562474e-01 3.333333333333333148296162562474e-01 1.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 1.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 1.000000000000000000000000000000e+00 2.763932022500210639037732107681e-01 7.236067977499789360962267892319e-01 0.000000000000000000000000000000e+00 7.236067977499789360962267892319e-01 2.763932022500210639037732107681e-01 0.000000000000000000000000000000e+00 2.763932022500210639037732107681e-01 0.000000000000000000000000000000e+00 7.236067977499789360962267892319e-01 7.236067977499789360962267892319e-01 0.000000000000000000000000000000e+00 2.763932022500210639037732107681e-01 0.000000000000000000000000000000e+00 2.763932022500210639037732107681e-01 7.236067977499789360962267892319e-01 0.000000000000000000000000000000e+00 7.236067977499789360962267892319e-01 2.763932022500210639037732107681e-01 ]; case 4 % PASSED (I). % DEGREE: 4 % POINTS: 15 % LEBESGUE CONSTANT: 2.58801e+00 % DETERMINANT: -8.98221e+13 % CONDITIONING: 3.95876e+01 % MAX LAGRANGE POLYNOMIALS: 1.02814e+00 % VANDERMONDE: ORTHONORMAL DUBINER LEGENDRE. table=[ 0 1 0 ]; t=[ -7.113198907115936275857848158921e-01 ]; pts=[ 2.371066302371978573582111948781e-01 2.371066302371978573582111948781e-01 5.257867395256041742612751477282e-01 2.371066302371978573582111948781e-01 5.257867395256041742612751477282e-01 2.371066302371978573582111948781e-01 5.257867395256041742612751477282e-01 2.371066302371978573582111948781e-01 2.371066302371978573582111948781e-01 1.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 1.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 1.000000000000000000000000000000e+00 1.726731646460114566465904317738e-01 8.273268353539885433534095682262e-01 0.000000000000000000000000000000e+00 5.000000000000000000000000000000e-01 5.000000000000000000000000000000e-01 0.000000000000000000000000000000e+00 8.273268353539885433534095682262e-01 1.726731646460114566465904317738e-01 0.000000000000000000000000000000e+00 1.726731646460114566465904317738e-01 0.000000000000000000000000000000e+00 8.273268353539885433534095682262e-01 5.000000000000000000000000000000e-01 0.000000000000000000000000000000e+00 5.000000000000000000000000000000e-01 8.273268353539885433534095682262e-01 0.000000000000000000000000000000e+00 1.726731646460114566465904317738e-01 0.000000000000000000000000000000e+00 1.726731646460114566465904317738e-01 8.273268353539885433534095682262e-01 0.000000000000000000000000000000e+00 5.000000000000000000000000000000e-01 5.000000000000000000000000000000e-01 0.000000000000000000000000000000e+00 8.273268353539885433534095682262e-01 1.726731646460114566465904317738e-01 ]; case 5 % PASSED (I). % DEGREE: 5 % POINTS: 21 % LEBESGUE CONSTANT: 3.08116e+00 % DETERMINANT: 6.69351e+20 % CONDITIONING: 5.28526e+01 % MAX LAGRANGE POLYNOMIALS: 1.05314e+00 % VANDERMONDE: ORTHONORMAL DUBINER LEGENDRE. table=[ 0 2 0 ]; t=[ -1.237022082059360705841299932217e+00 -5.213955589468205475967010897875e-01 ]; pts=[ 4.123406940197869019470999774057e-01 4.123406940197869019470999774057e-01 1.753186119604261961058000451885e-01 1.737985196489401640285166195099e-01 1.737985196489401640285166195099e-01 6.524029607021195609206642984645e-01 4.123406940197869019470999774057e-01 1.753186119604261961058000451885e-01 4.123406940197869019470999774057e-01 1.737985196489401640285166195099e-01 6.524029607021195609206642984645e-01 1.737985196489401640285166195099e-01 1.753186119604261961058000451885e-01 4.123406940197869019470999774057e-01 4.123406940197869019470999774057e-01 6.524029607021195609206642984645e-01 1.737985196489401640285166195099e-01 1.737985196489401640285166195099e-01 1.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 1.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 1.000000000000000000000000000000e+00 1.174723380352677426508023472707e-01 8.825276619647322018380464214715e-01 0.000000000000000000000000000000e+00 3.573842417596774789778635295079e-01 6.426157582403224655109852392343e-01 0.000000000000000000000000000000e+00 6.426157582403225765332877017499e-01 3.573842417596774234667122982501e-01 0.000000000000000000000000000000e+00 8.825276619647324238826513465028e-01 1.174723380352675761173486534972e-01 0.000000000000000000000000000000e+00 1.174723380352677426508023472707e-01 0.000000000000000000000000000000e+00 8.825276619647322018380464214715e-01 3.573842417596774789778635295079e-01 0.000000000000000000000000000000e+00 6.426157582403224655109852392343e-01 6.426157582403225765332877017499e-01 0.000000000000000000000000000000e+00 3.573842417596774234667122982501e-01 8.825276619647324238826513465028e-01 0.000000000000000000000000000000e+00 1.174723380352675761173486534972e-01 0.000000000000000000000000000000e+00 1.174723380352677426508023472707e-01 8.825276619647322018380464214715e-01 0.000000000000000000000000000000e+00 3.573842417596774789778635295079e-01 6.426157582403224655109852392343e-01 0.000000000000000000000000000000e+00 6.426157582403225765332877017499e-01 3.573842417596774234667122982501e-01 0.000000000000000000000000000000e+00 8.825276619647324238826513465028e-01 1.174723380352675761173486534972e-01 ]; case 6 % PASSED (I). % DEGREE: 6 % POINTS: 28 % LEBESGUE CONSTANT: 3.59605e+00 % DETERMINANT: -1.87536e+29 % CONDITIONING: 7.13009e+01 % MAX LAGRANGE POLYNOMIALS: 1.02909e+00 % VANDERMONDE: ORTHONORMAL DUBINER LEGENDRE. table=[ 1 1 1 ]; t=[ -3.710100618799806460046397660335e-01 3.869879153567130636304227664368e-01 2.171028049671443471702048100269e-01 ]; pts=[ 3.333333333333333148296162562474e-01 3.333333333333333148296162562474e-01 3.333333333333333148296162562474e-01 1.236700206266602153348799220112e-01 1.236700206266602153348799220112e-01 7.526599587466795693302401559777e-01 1.236700206266602153348799220112e-01 7.526599587466795693302401559777e-01 1.236700206266602153348799220112e-01 7.526599587466795693302401559777e-01 1.236700206266602153348799220112e-01 1.236700206266602153348799220112e-01 3.254637175704040430446184473112e-01 5.425665225375484457259744885960e-01 1.319697598920475112294070640928e-01 3.254637175704040430446184473112e-01 1.319697598920475112294070640928e-01 5.425665225375484457259744885960e-01 5.425665225375484457259744885960e-01 3.254637175704040430446184473112e-01 1.319697598920475112294070640928e-01 5.425665225375484457259744885960e-01 1.319697598920475112294070640928e-01 3.254637175704040430446184473112e-01 1.319697598920475112294070640928e-01 3.254637175704040430446184473112e-01 5.425665225375484457259744885960e-01 1.319697598920475112294070640928e-01 5.425665225375484457259744885960e-01 3.254637175704040430446184473112e-01 1.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 1.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 1.000000000000000000000000000000e+00 8.488805186071646247114586003590e-02 9.151119481392835375288541399641e-01 0.000000000000000000000000000000e+00 2.655756032646428010934869234916e-01 7.344243967353571989065130765084e-01 0.000000000000000000000000000000e+00 5.000000000000001110223024625157e-01 4.999999999999998889776975374843e-01 0.000000000000000000000000000000e+00 7.344243967353571989065130765084e-01 2.655756032646428010934869234916e-01 0.000000000000000000000000000000e+00 9.151119481392833154842492149328e-01 8.488805186071668451575078506721e-02 0.000000000000000000000000000000e+00 8.488805186071646247114586003590e-02 0.000000000000000000000000000000e+00 9.151119481392835375288541399641e-01 2.655756032646428010934869234916e-01 0.000000000000000000000000000000e+00 7.344243967353571989065130765084e-01 5.000000000000001110223024625157e-01 0.000000000000000000000000000000e+00 4.999999999999998889776975374843e-01 7.344243967353571989065130765084e-01 0.000000000000000000000000000000e+00 2.655756032646428010934869234916e-01 9.151119481392833154842492149328e-01 0.000000000000000000000000000000e+00 8.488805186071668451575078506721e-02 0.000000000000000000000000000000e+00 8.488805186071646247114586003590e-02 9.151119481392835375288541399641e-01 0.000000000000000000000000000000e+00 2.655756032646428010934869234916e-01 7.344243967353571989065130765084e-01 0.000000000000000000000000000000e+00 5.000000000000001110223024625157e-01 4.999999999999998889776975374843e-01 0.000000000000000000000000000000e+00 7.344243967353571989065130765084e-01 2.655756032646428010934869234916e-01 0.000000000000000000000000000000e+00 9.151119481392833154842492149328e-01 8.488805186071668451575078506721e-02 ]; case 7 % PASSED (I). % DEGREE: 7 % POINTS: 36 % LEBESGUE CONSTANT: 4.14590e+00 % DETERMINANT: 1.94719e+39 % CONDITIONING: 9.41441e+01 % MAX LAGRANGE POLYNOMIALS: 1.02500e+00 % VANDERMONDE: ORTHONORMAL DUBINER LEGENDRE. table=[ 0 3 1 ]; t=[ -8.224753183886709395622460760933e-01 -2.637438419436463621359223452600e-01 2.999553236873908068638172608189e-01 3.076665664768104546666904752783e-01 3.814606641657357632979596928635e-01 ]; pts=[ 2.741584394628903131874153586978e-01 2.741584394628903131874153586978e-01 4.516831210742193736251692826045e-01 8.791461398121544479344890987704e-02 8.791461398121544479344890987704e-02 8.241707720375690549019509489881e-01 2.741584394628903131874153586978e-01 4.516831210742193736251692826045e-01 2.741584394628903131874153586978e-01 8.791461398121544479344890987704e-02 8.241707720375690549019509489881e-01 8.791461398121544479344890987704e-02 4.516831210742193736251692826045e-01 2.741584394628903131874153586978e-01 2.741584394628903131874153586978e-01 8.241707720375690549019509489881e-01 8.791461398121544479344890987704e-02 8.791461398121544479344890987704e-02 4.500074460521015229375052513205e-01 4.500074460521015229375052513205e-01 9.998510789579695412498949735891e-02 4.500074460521015229375052513205e-01 9.998510789579695412498949735891e-02 4.500074460521015229375052513205e-01 9.998510789579695412498949735891e-02 4.500074460521015229375052513205e-01 4.500074460521015229375052513205e-01 2.574575396908898028414114378393e-01 6.389182038566255661393711307028e-01 1.036242564524846310192174314579e-01 2.574575396908898028414114378393e-01 1.036242564524846310192174314579e-01 6.389182038566255661393711307028e-01 6.389182038566255661393711307028e-01 2.574575396908898028414114378393e-01 1.036242564524846310192174314579e-01 6.389182038566255661393711307028e-01 1.036242564524846310192174314579e-01 2.574575396908898028414114378393e-01 1.036242564524846310192174314579e-01 2.574575396908898028414114378393e-01 6.389182038566255661393711307028e-01 1.036242564524846310192174314579e-01 6.389182038566255661393711307028e-01 2.574575396908898028414114378393e-01 1.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 1.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 0.000000000000000000000000000000e+00 1.000000000000000000000000000000e+00 6.412992574519671418187272138312e-02 9.358700742548032858181272786169e-01 0.000000000000000000000000000000e+00 2.041499092834289652564905281906e-01 7.958500907165710902546607030672e-01 0.000000000000000000000000000000e+00 3.953503910487606298751472877484e-01 6.046496089512394256360039435094e-01 0.000000000000000000000000000000e+00 6.046496089512393146137014809938e-01 3.953503910487606853862985190062e-01 0.000000000000000000000000000000e+00 7.958500907165713122992656280985e-01 2.041499092834286877007343719015e-01 0.000000000000000000000000000000e+00 9.358700742548031747958248161012e-01 6.412992574519682520417518389877e-02 0.000000000000000000000000000000e+00 6.412992574519671418187272138312e-02 0.000000000000000000000000000000e+00 9.358700742548032858181272786169e-01 2.041499092834289652564905281906e-01 0.000000000000000000000000000000e+00 7.958500907165710902546607030672e-01 3.953503910487606298751472877484e-01 0.000000000000000000000000000000e+00 6.046496089512394256360039435094e-01 6.046496089512393146137014809938e-01 0.000000000000000000000000000000e+00 3.953503910487606853862985190062e-01 7.958500907165713122992656280985e-01 0.000000000000000000000000000000e+00 2.041499092834286877007343719015e-01 9.358700742548031747958248161012e-01 0.000000000000000000000000000000e+00 6.412992574519682520417518389877e-02 0.000000000000000000000000000000e+00 6.412992574519671418187272138312e-02 9.358700742548032858181272786169e-01 0.000000000000000000000000000000e+00 2.041499092834289652564905281906e-01 7.958500907165710902546607030672e-01 0.000000000000000000000000000000e+00 3.953503910487606298751472877484e-01 6.046496089512394256360039435094e-01 0.000000000000000000000000000000e+00 6.046496089512393146137014809938e-01 3.953503910487606853862985190062e-01 0.000000000000000000000000000000e+00 7.958500907165713122992656280985e-01 2.041499092834286877007343719015e-01 0.000000000000000000000000000000e+00 9.358700742548031747958248161012e-01 6.412992574519682520417518389877e-02 ]; case 8 % PASSED (I). % DEGREE: 8 % POINTS: 45 % LEBESGUE CONSTANT: 4.77382e+00 % DETERMINANT: 7.59934e+50 % CONDITIONING: 1.18408e+02 % MAX LAGRANGE POLYNOMIALS: 1.03994e+00 % VANDERMONDE: ORTHONORMAL DUBINER LEGENDRE. table=[ 0 3 2 ]; t=[ -6.714343268839856326835047184431e-01 -1.163743150370817991401395374851e+00 -2.024983409269499046168050426786e-01 2.554502949718992499406056140288e-01 5.775713840017694389317171044240e-01 4.958527275015719437689654114365e-01 1.780416127569382223061467129810e-01 ]; pts=[ 2.238114422946618775611682394810e-01 2.238114422946618775611682394810e-01 5.523771154106762448776635210379e-01 3.879143834569392934596976374451e-01 3.879143834569392934596976374451e-01 2.241712330861213020583022625942e-01 6.749944697564996820560168089287e-02 6.749944697564996820560168089287e-02 8.650011060487001746110991007299e-01 2.238114422946618775611682394810e-01 5.523771154106762448776635210379e-01 2.238114422946618775611682394810e-01 3.879143834569392934596976374451e-01 2.241712330861213020583022625942e-01 3.879143834569392934596976374451e-01 6.749944697564996820560168089287e-02 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