Symmetric points on the simplex.
In this homepage we propose the so called LEBGLS points on the unit simplex (i.e. the
triangle with vertices V1=[0,0], V2=[1,0], V3=[0,1]). The LEBGLS is a symmetric point set, with points on the boundary with a Gauss-Legendre-Lobatto distribution.
Each set is stored in a Matlab file, that is actually a Matlab function. Its input is a degree (that varies from set to set).
| DEGR | LEBESGUE | ABS.DET.VAND. | COND. VAND. | MAX. LAGR. |
| 1 | 1.00000 | 5.87878e+01 | 3.70699 | 1.00000 |
| 2 | 1.66667 | 3.13712e+04 | 1.01810e+01 | 1.00000 |
| 3 | 2.11244 | 3.44701e+08 | 2.03638e+01 | 1.00000 |
| 4 | 2.58801 | 8.98221e+13 | 3.95876e+01 | 1.02814 |
| 5 | 3.08116 | 6.69351e+20 | 5.28526e+01 | 1.05314 |
| 6 | 3.59605 | 1.87536e+29 | 7.13009e+01 | 1.02909 |
| 7 | 4.14590 | 1.94719e+39 | 9.41441e+01 | 1.02500 |
| 8 | 4.77382 | 7.59934e+50 | 1.18408e+02 | 1.03994 |
| 9 | 5.49425 | 1.43106e+64 | 1.41026e+02 | 1.03863 |
| 10 | 6.28492 | 2.40480e+79 | 1.84334e+02 | 1.08256 |
| 11 | 7.00203 | 2.11926e+96 | 2.62281e+02 | 1.18378 |
| 12 | 7.25934 | 2.07917e+115 | 2.28046e+02 | 1.04342 |
| 13 | 8.57673 | 2.16493e+135 | 3.32262e+02 | 1.09934 |
| 14 | 8.81565 | 1.23824e+158 | 4.80175e+02 | 1.18945 |
| 15 | 8.90865 | 9.40546e+182 | 3.96741e+02 | 1.09615 |
| 16 | 10.6553 | 1.73095e+208 | 4.54660e+02 | 1.18671 |
| 17 | 11.4147 | 2.25105e+236 | 5.47919e+02 | 1.08768 |
| 18 | 12.6889 | 1.11594e+267 | 4.72717e+02 | 1.07067 |
The output variables of the M-file set_leb_opt_n1n3n6_gll are
- the barycentric coordinates of the points are stored in the N x 3 matrix pts;
- a vector t that stores the pointsets in compressed form;
- the table table that stores the symmetries of orbits 1, 3 and 6;
- some statistics are stored in the variable in the matrix stats_matrix;
» Matlab downloads
The set is stored in Matlab files that can be downloaded by clicking on [m].
■ Other sets
Some other unisolvent sets on the simplex were proposed in the paper
M. Briani, A. Sommariva, M. Vianello, Computing Fekete and Lebesgue points: simplex, square, disk.
» Matlab downloads
The set is stored in Matlab files that can be downloaded by clicking on [m].
The latter three sets are stored as a [zip] file.
For more details, see also the following homepage