Some points on the simplex.

In this homepage we list some sets of points on the unit simplex (i.e. the triangle with vertices V1=[0,0], V2=[1,0], V3=[0,1]). 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).

The typical output of these routines are The variable stats_matrix is a matrix whose columns are

■ New sets

Our purpose is to achieve better results either in terms of Lebesgue constant either of Vandermonde determinants for unisolvent sets on the simplex. For the complete description see the paper
M. Briani, A. Sommariva, M. Vianello, Computing Fekete and Lebesgue points: simplex, square, disk.
The results are listed below. The set LGL is a set with assigned points on the sides (Gauss-Legendre-Lobatto), while the set LEB has not such constrain.
LEBESGUE CONST.
DEG LEB LGL FEK
1 1.00000e+00 1.00000e+00 1.00000e+00
2 1.48631e+00 1.66667e+00 1.66667e+00
3 1.97475e+00 2.11244e+00 2.11244e+00
4 2.41919e+00 2.58725e+00 2.72926e+00
5 2.89521e+00 3.08214e+00 3.61080e+00
6 3.39157e+00 3.59472e+00 4.17064e+00
7 3.94178e+00 4.13763e+00 5.19969e+00
8 4.54560e+00 5.21334e+00 6.89581e+00
9 5.27594e+00 5.51107e+00 6.96685e+00
10 5.62771e+00 5.93288e+00 8.24041e+00
11 6.45419e+00 6.66260e+00 8.02859e+00
12 6.90026e+00 7.12872e+00 8.56588e+00
13 7.58692e+00 7.74381e+00 9.81157e+00
14 8.30508e+00 8.30508e+00 1.08926e+01
15 9.05613e+00 9.06741e+00 1.14177e+01
16 8.58269e+00 8.58342e+00 1.27371e+01
17 9.11807e+00 9.11807e+00 1.35982e+01
18 9.88359e+00 9.88359e+00 1.44276e+01
VANDERMONDE DET.
DEG LEB LGL FEK
1 5.87878e+01 5.87878e+01 5.87878e+01
2 2.43369e+04 3.13712e+04 3.13712e+04
3 1.99873e+08 3.44701e+08 3.44701e+08
4 4.37706e+13 8.99015e+13 9.62345e+13
5 2.96436e+20 6.66154e+20 8.03325e+20
6 3.72550e+28 1.87052e+29 2.29077e+29
7 1.57972e+38 1.88500e+39 2.59212e+39
8 3.41006e+49 8.03633e+50 1.35737e+51
9 2.75650e+62 2.22811e+64 3.79476e+64
10 4.80833e+76 4.57515e+79 1.04966e+80
11 8.71343e+92 4.43645e+96 9.66903e+96
12 3.41787e+111 1.69089e+115 7.21516e+115
13 1.81466e+131 5.70766e+135 4.45658e+136
14 9.94900e+157 9.94900e+157 2.51332e+159
15 1.87857e+182 1.88891e+182 9.65617e+183
16 2.66621e+209 2.70064e+209 3.35601e+210
17 7.85856e+237 7.85856e+237 1.00679e+239
18 1.43972e+268 1.43972e+268 4.10313e+269
CONDITIONING
LEB LEB LGL FEK
1 3.70699e+00 3.70699e+00 3.70699e+00
2 1.06172e+01 1.01810e+01 1.01810e+01
3 1.94563e+01 2.03638e+01 2.03638e+01
4 3.56093e+01 3.91919e+01 3.88320e+01
5 4.66719e+01 5.23357e+01 5.18535e+01
6 6.09687e+01 6.79519e+01 6.99566e+01
7 7.87929e+01 8.89001e+01 9.26722e+01
8 1.07117e+02 1.22730e+02 1.10362e+02
9 1.39288e+02 1.36115e+02 1.55431e+02
10 1.15565e+02 1.79449e+02 1.62914e+02
11 1.45536e+02 1.75462e+02 2.07722e+02
12 1.72900e+02 2.44254e+02 2.40029e+02
13 2.06024e+02 2.90055e+02 2.90513e+02
14 3.07878e+02 3.09248e+02 3.13798e+02
15 3.73081e+02 3.63190e+02 3.51918e+02
16 4.08135e+02 4.26188e+02 4.11205e+02
17 4.64395e+02 4.33208e+02 4.09856e+02
18 5.05228e+02 4.89530e+02 4.65169e+02


Below some comparisons between the best known results. The Heinrichs set HEI is not complete (the set is available only for degrees 6, 9, 12). In the tables we compare for the available sets, respectively, the LEB set that minimizes the Lebesgue constant and maximizes the Vandermonde determinant with Taylor-Wingate-Vincent and Heinrichs set.

Set LEB that minimizes Lebesgue constant

LEBESGUE CONST.
LEB BSV TWV HEI
3 1.97475e+00 2.11244e+00
6 3.39157e+00 4.17068e+00 3.68814e+00
9 5.27594e+00 6.80260e+00 5.59459e+00
12 6.90026e+00 9.67588e+00 7.51440e+00
15 9.05613e+00 1.00178e+01
18 9.88359e+00 1.47332e+01
VANDERMONDE DET.
DEG LEB TWV HEI
3 1.99873e+08 3.44701e+08
6 3.72550e+28 2.29077e+29 1.84457e+29
9 2.75650e+62 2.39655e+64 1.42675e+64
12 3.41787e+115 6.15489e+115 2.59072e+115
15 1.87857e+182 4.60070e+183
18 1.43972e+268 1.04327e+269
CONDITIONING
DEG BSV TWV HEI
3 1.94563e+01 1.44838e+01
6 6.09687e+01 7.01598e+01 7.39846e+01
9 1.39288e+02 1.41349e+02 1.45078e+02
12 1.72900e+02 2.35424e+02 2.45598e+02
15 3.73081e+02 3.28023e+02
18 5.05228e+02 4.24963e+02


Set FEK that maximizes the Vandermonde determinant

LEBESGUE CONST.
DEG FEK TWV HEI
3 2.11244e+00 2.11244e+00
6 4.17064e+00 4.17068e+00 3.68814e+00
9 6.96685e+00 6.80260e+00 5.59459e+00
12 8.56588e+00 9.67588e+00 7.51440e+00
15 1.14177e+01 1.00178e+01
18 1.44276e+01 1.47332e+01
VANDERMONDE DET.
DEG FEK TWV HEI
3 3.44701e+08 3.44701e+08
6 2.29077e+29 2.29077e+29 1.84457e+29
9 3.79476e+64 2.39655e+64 1.42675e+64
12 7.21516e+115 6.15489e+115 2.59072e+115
15 9.65617e+183 4.60070e+183
18 4.10313e+269 1.04327e+269
CONDITIONING
DEG FEK TWV HEI
3 2.03638e+01 1.44838e+01
6 6.99566e+01 7.01598e+01 7.39846e+01
9 1.55431e+02 1.41349e+02 1.45078e+02
12 2.40029e+02 2.35424e+02 2.45598e+02
15 3.51918e+02 3.28023e+02
18 4.65169e+02 4.24963e+02

» Matlab downloads

The set is stored in Matlab files that can be downloaded by clicking on [m].

The Matlab codes that we have used are compressed in a zip file that can be downloaded by clicking on [zip].


■ Taylor-Wingate-Vincent set

This set of points is taken from the reference paper
  1. M. Taylor, B. Wingate and R. Vincent, An algorithm for computing Fekete points in the triangle, SIAM J. Numer. Anal. 38 (2000), pages 1707-1720.
The points were computed numerically by the authors and displayed in compact form via orbits only for degrees 3, 6, 9, 12, 15, 18, though they say that other degrees can be sent at request. The purpose is to maximize the absolute value of the Vandermonde determinant of the point set. Here they are available either in cartesian coordinates on the unit simplex, either in barycentric coordinates (see the .m files for more details). The tables below show the main properties of this set.

LEBESGUE CONST.
DEG TWV
3 2.11244e+00
6 4.17068e+00
9 6.80260e+00
12 9.67588e+00
15 1.00178e+01
18 1.47332e+01
VANDERMONDE DET.
DEG TWV
3 3.44701e+08
6 2.29077e+29
9 2.39655e+64
12 6.15489e+115
15 4.60070e+183
18 1.04327e+269
CONDITIONING
DEG TWV
3 1.44838e+01
6 7.01598e+01
9 1.41349e+02
12 2.35424e+02
15 3.28023e+02
18 4.24963e+02

» Matlab downloads

The set is stored in a Matlab file that can be downloaded by clicking on [m].

■ Heinrichs set

This set of points is taken from the reference paper
  1. W. Heinrichs, "Improved Lebesgue constants on the triangle, Journal of Computational Physics 207 (2005), pages 625-638.
and are almost optimal in terms of the Lebesgue function. The algorithm looks very promising but is only sketched on the paper and not available as software. Here they are available either in cartesian coordinates on the unit simplex, either in barycentric coordinates (see the .m files for more details). The tables below show the main properties of this sets HEI and its version with assigned points on the sides HEI2. In the aforementioned work, only few degrees of HEI are given and consequently we could not test thoroughly the set.

LEBESGUE CONST.
DEG HEI HEI2
6 3.68814e+00 3.87e+00
9 5.59459e+00 5.59e+00
12 7.12e+00 7.51e+00
15 8.41e+00 9.25e+00
18 10.08e+00 11.16e+00
VANDERMONDE DET.
DEG HEI
6 1.84457e+29
9 1.42675e+64
12 2.59072e+115
CONDITIONING
DEG HEI
6 7.39846e+01
9 1.45078e+02
12 2.45598e+02

» Matlab downloads

The set is stored in a Matlab file that can be downloaded by clicking on [m].

■ Warburton type sets

This set of points is taken from the reference paper
  1. T. Warburton, An explicit construction of interpolation nodes on the simplex, Journal Of Engineering Mathematics, Volume 56, Number 3, pages 247-262, (2006).
The paper includes a Matlab code for computing points on a reference equilateral triangle. Its purpose is to achieve on such a set a determinant of the Vandermonde matrix close to value obtained by Fekete points of degree deg. The main ingredients are that the points on each side are Gauss-Legendre-Lobatto of degree deg+1 (as suggested by a conjecture by Bos) and a warp and blend parameter α.

Since we are looking for point sets with low Lebesgue constant, we considered also other distributions on each side as Expanded Chebyshev, Gegenbauer or Jacobi sets. In the case of Expanded Chebyshev or Gauss-Legendre-Lobatto distribution, the optimization process worked only on the warp and blend parameter α while in the other cases also the Gegenbauer or Jacobi exponents were involved and stored as output in the variable t, whose first component is the parameter α while the other possible ones describe the Gaussian points.

The degree of each set ranges from 2 to 25. In the tables below, you can find some properties of these sets.

LEBESGUE CONSTANT
DEG ECH GGL GJL GLL
2 1.66667e+00 1.66667e+00 1.66667e+00 1.66667e+00
3 2.11244e+00 2.10843e+00 2.10843e+00 2.11244e+00
4 2.66208e+00 2.61329e+00 2.61329e+00 2.66208e+00
5 3.38154e+00 3.10826e+00 3.10828e+00 3.12228e+00
6 3.95455e+00 3.63744e+00 3.63744e+00 3.70188e+00
7 4.64350e+00 4.22296e+00 4.22330e+00 4.27495e+00
8 5.49021e+00 4.87941e+00 4.87941e+00 4.96419e+00
9 6.28683e+00 5.66404e+00 5.66414e+00 5.73726e+00
10 7.32253e+00 6.58890e+00 6.58984e+00 6.67288e+00
11 8.35666e+00 7.83046e+00 7.83047e+00 7.90430e+00
12 9.85302e+00 9.28023e+00 9.28026e+00 9.36467e+00
13 1.19219e+01 1.13534e+01 1.13534e+01 1.14613e+01
14 1.43869e+01 1.38761e+01 1.38764e+01 1.39683e+01
15 1.79858e+01 1.74234e+01 1.74232e+01 1.76433e+01
16 2.24556e+01 2.20241e+01 2.20247e+01 2.22223e+01
17 3.14902e+01 2.81686e+01 2.82197e+01 2.87561e+01
18 4.25887e+01 3.58951e+01 3.59003e+01 3.67593e+01
19 5.45072e+01 4.61123e+01 4.61123e+01 4.75758e+01
20 7.02808e+01 5.99076e+01 5.99076e+01 6.21778e+01
21 9.18698e+01 7.85535e+01 7.85539e+01 8.20457e+01
22 1.20441e+02 1.04400e+02 1.04400e+02 1.09194e+02
23 1.59655e+02 1.38510e+02 1.38485e+02 1.46403e+02
24 2.13884e+02 1.85408e+02 1.85408e+02 1.97993e+02
25 2.87110e+02 2.49910e+02 2.49910e+02 2.69042e+02
VANDERMONDE DETERMINANT
DEG ECH GGL GJL GLL
2 3.13712e+04 3.13712e+04 3.13712e+04 3.13712e+04
3 3.44701e+08 3.41339e+08 3.41339e+08 3.44701e+08
4 9.53402e+13 8.64995e+13 8.64988e+13 9.53402e+13
5 7.90833e+20 7.04916e+20 7.05425e+20 7.73131e+20
6 2.22441e+29 1.39990e+29 1.40234e+29 2.14229e+29
7 2.37217e+39 1.44421e+39 1.36815e+39 2.19882e+39
8 1.05651e+51 5.21497e+50 5.21751e+50 9.56829e+50
9 2.11594e+64 9.58622e+63 9.29522e+63 1.84381e+64
10 2.06954e+79 8.68170e+78 7.88511e+78 1.77726e+79
11 1.02866e+96 4.11928e+95 4.12293e+95 8.32292e+95
12 2.81684e+114 1.08873e+114 1.09539e+114 2.27575e+114
13 4.34066e+134 1.34255e+134 1.34139e+134 3.23409e+134
14 4.02669e+156 1.17666e+156 1.23174e+156 2.96899e+156
15 2.29307e+180 3.06668e+179 3.41469e+179 1.53545e+180
16 8.37595e+205 1.73995e+205 1.82377e+205 5.37931e+205
17 1.16298e+233 2.28838e+232 2.57669e+232 2.10351e+233
18 9.91505e+261 2.40040e+261 3.00884e+261 3.29585e+262
19 1.03550e+293 1.74753e+292 1.74812e+292 3.78572e+293
20 Inf Inf Inf Inf
21 Inf Inf Inf Inf
22 Inf Inf Inf Inf
23 Inf Inf Inf Inf
24 Inf Inf Inf Inf
25 Inf Inf Inf Inf
CONDITIONING
DEG ECH GGL GJL GLL
2 1.01810e+01 1.01810e+01 1.01810e+01 1.01810e+01
3 2.03638e+01 2.04423e+01 1.23487e+01 2.03638e+01
4 3.87556e+01 3.90831e+01 3.90832e+01 3.87556e+01
5 4.42896e+01 5.27760e+01 5.27796e+01 4.47408e+01
6 5.96204e+01 6.99833e+01 6.99829e+01 5.92238e+01
7 8.63843e+01 8.33676e+01 8.41875e+01 7.49980e+01
8 1.18629e+02 9.56059e+01 1.14295e+02 1.18602e+02
9 1.14622e+02 1.38333e+02 1.30839e+02 1.37896e+02
10 1.70291e+02 1.65693e+02 1.59700e+02 1.62722e+02
11 1.84724e+02 1.93259e+02 1.95462e+02 1.90440e+02
12 2.33242e+02 2.25960e+02 1.83316e+02 1.96597e+02
13 2.42909e+02 2.01118e+02 2.74293e+02 2.71783e+02
14 2.51446e+02 2.88934e+02 2.89030e+02 2.84779e+02
15 3.15962e+02 3.85259e+02 3.55752e+02 3.91830e+02
16 4.02997e+02 4.82362e+02 5.30730e+02 5.52640e+02
17 5.97299e+02 5.54361e+02 7.16356e+02 7.96280e+02
18 1.34706e+03 1.04473e+03 1.05162e+03 1.15026e+03
19 1.88138e+03 1.42876e+03 1.04944e+03 1.62729e+03
20 2.62447e+03 1.91181e+03 1.41683e+03 2.26159e+03
21 3.71637e+03 2.63437e+03 1.92882e+03 3.18027e+03
22 5.07671e+03 3.65191e+03 3.65192e+03 2.96448e+03
23 7.04610e+03 3.63160e+03 5.17446e+03 6.23685e+03
24 9.79771e+03 7.21598e+03 7.21596e+03 8.82524e+03
25 9.32855e+03 1.00784e+04 1.00787e+04 1.24307e+04

» Matlab downloads

The sets are stored in Matlab files that can be downloaded by clicking on [m].

■ Blyth-Pozrikidis type sets

This set of points is taken from the reference paper
  1. M.G. Blyth and C. Pozrikidis, A Lobatto interpolation grid over the triangle, IMA Journal of Applied Mathematics (2005), pages 1?17.
The set is explicit and can be easily computed. The original paper has a Gauss-Legendre-Lobatto distribution on the sides due to Bos conjecture about Fekete points, but this is restrictive in the case of the Lebesgue points. We experimented the expanded Chebyshev distribution and almost optimal Gegenbauer or Jacobi sets. In this case, the variable t of our Matlab functions stores the (almost-)optimal parameters.
LEBESGUE CONST.
DEG ECH GGL GJL GLL
2 1.66667e+00 1.66667e+00 1.48630e+00 1.66667e+00
3 2.13298e+00 2.10843e+00 1.97466e+00 2.11244e+00
4 2.61876e+00 2.61329e+00 2.44963e+00 2.66208e+00
5 3.16074e+00 3.13677e+00 2.96950e+00 3.13677e+00
6 3.92945e+00 3.87403e+00 3.65529e+00 3.87446e+00
7 4.81468e+00 4.61282e+00 4.35552e+00 4.65886e+00
8 6.19895e+00 5.82290e+00 5.44453e+00 5.92615e+00
9 7.84786e+00 7.17684e+00 6.69221e+00 7.38966e+00
10 1.05438e+01 9.45415e+00 8.70173e+00 9.82718e+00
11 1.40052e+01 1.23654e+01 1.12479e+01 1.29292e+01
12 1.93973e+01 1.69312e+01 1.53239e+01 1.77758e+01
13 2.68792e+01 2.33748e+01 2.10059e+01 2.45357e+01
14 3.81383e+01 3.30718e+01 2.96009e+01 3.46895e+01
15 5.46277e+01 4.74420e+01 4.23749e+01 4.95885e+01
16 7.93645e+01 6.90738e+01 6.14806e+01 7.18872e+01
17 1.16388e+02 1.01666e+02 9.03776e+01 1.05368e+02
18 1.72341e+02 1.51008e+02 1.34220e+02 1.56162e+02
19 2.57024e+02 2.26584e+02 2.01076e+02 2.33515e+02
20 3.85863e+02 3.42400e+02 3.03396e+02 3.51424e+02
21 5.82457e+02 5.20978e+02 4.61198e+02 5.32479e+02
22 8.83700e+02 7.95444e+02 7.04599e+02 8.09947e+02
23 1.34700e+03 1.22238e+03 1.08242e+03 1.24019e+03
24 2.06041e+03 1.88239e+03 1.66844e+03 1.90354e+03
25 3.16257e+03 2.91008e+03 2.58084e+03 2.93628e+03
VANDERMONDE DET.
DEG ECH GGL GJL GLL
2 3.13712e+04 3.13712e+04 2.43366e+04 3.13712e+04
3 3.37720e+08 3.41338e+08 2.00226e+08 3.44701e+08
4 8.90037e+13 8.64998e+13 2.84350e+13 9.53402e+13
5 6.59849e+20 7.64702e+20 1.89324e+20 7.66515e+20
6 1.52958e+29 2.01900e+29 2.51316e+28 1.99726e+29
7 1.20374e+39 1.91907e+39 1.71865e+38 1.83500e+39
8 3.43127e+50 7.01579e+50 2.65507e+49 6.35119e+50
9 3.72866e+63 1.02275e+64 2.34997e+62 8.72398e+63
10 1.60799e+78 6.73732e+78 5.45170e+76 4.95420e+78
11 2.83932e+94 1.98053e+95 8.69887e+92 1.20066e+95
12 2.10216e+112 2.71052e+113 3.73660e+110 1.27215e+113
13 6.64057e+131 1.71360e+133 1.18632e+130 5.99830e+132
14 9.05809e+152 5.04504e+154 1.09255e+151 1.27407e+154
15 5.37423e+175 6.71583e+177 7.20388e+173 1.22820e+177
16 1.39125e+200 4.06160e+202 1.35055e+198 5.39117e+201
17 1.57067e+226 1.09130e+229 1.72885e+224 1.07713e+228
18 7.70402e+253 1.28371e+257 6.33470e+251 9.75957e+255
19 1.63085e+283 7.15488e+286 1.30297e+281 3.98400e+285
20 Inf Inf Inf Inf
21 Inf Inf Inf Inf
22 Inf Inf Inf Inf
23 Inf Inf Inf Inf
24 Inf Inf Inf Inf
25 Inf Inf Inf Inf
CONDITIONING
DEG ECH GGL GJL GLL
2 1.01810e+01 1.01810e+01 1.06169e+01 1.01810e+01
3 2.05134e+01 2.04423e+01 1.94563e+01 2.03638e+01
4 3.90295e+01 3.90831e+01 3.58225e+01 3.87556e+01
5 5.29233e+01 5.34322e+01 4.53017e+01 5.34441e+01
6 7.15050e+01 6.97531e+01 5.72934e+01 7.09958e+01
7 8.55903e+01 8.12759e+01 8.02592e+01 9.34635e+01
8 1.03051e+02 1.19177e+02 1.00288e+02 9.85574e+01
9 1.51774e+02 1.36949e+02 1.29565e+02 1.45555e+02
10 1.78575e+02 1.69109e+02 1.49728e+02 1.95572e+02
11 2.45630e+02 2.26741e+02 1.57220e+02 2.02912e+02
12 3.13224e+02 2.64006e+02 1.97001e+02 2.23932e+02
13 3.04434e+02 3.04155e+02 2.70380e+02 3.62384e+02
14 5.38801e+02 4.08285e+02 3.33318e+02 4.68221e+02
15 7.06891e+02 6.39554e+02 5.11130e+02 6.34629e+02
16 9.24716e+02 7.79283e+02 7.63351e+02 7.53976e+02
17 1.44253e+03 1.21689e+03 1.20342e+03 1.20330e+03
18 2.08488e+03 2.32590e+03 1.83914e+03 1.85290e+03
19 2.93324e+03 2.95496e+03 2.86299e+03 2.83715e+03
20 4.52048e+03 4.61506e+03 4.50372e+03 4.14891e+03
21 6.67814e+03 8.54190e+03 6.97208e+03 6.84264e+03
22 1.06908e+04 1.31082e+04 1.06709e+04 1.06305e+04
23 1.58095e+04 1.65985e+04 1.31912e+04 1.65309e+04
24 2.60152e+04 2.61709e+04 2.06786e+04 2.57320e+04
25 3.91502e+04 4.71705e+04 3.24974e+04 4.07069e+04


■ Chen-Babuska sets

This set of points is taken from the reference paper
  1. Q. Chen and I. Babuska, Approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle, Computer Methods in Applied Mechanics and Engineering, Volume 128, Issues 3-4, 15 December 1995, Pages 405-417.
The points were computed numerically only for degrees from 2 to 13 and were described in compact form in terms of their orbits. Here they are available either in cartesian coordinates on the unit simplex, either in barycentric coordinates (see the .m files for more details).

In the tables below are displayed the main properties of this set of points.
LEBESGUE CONST.
DEG C-B
2 1.66667e+00
3 2.11140e+00
4 2.69195e+00
5 3.30095e+00
6 3.79096e+00
7 4.39070e+00
8 5.08912e+00
9 5.91702e+00
10 7.08454e+00
11 8.33772e+00
12 1.00790e+01
13 1.20433e+01
VANDERMONDE DET.
DEG C-B
2 3.13712e+04
3 3.44636e+08
4 9.57971e+13
5 7.90584e+20
6 2.20618e+29
7 2.31936e+39
8 1.00547e+51
9 1.93485e+64
10 1.75424e+79
11 7.88094e+95
12 1.83747e+114
13 2.29062e+134
CONDITIONING
DEG C-B
2 1.01810e+01
3 2.03441e+01
4 3.89050e+01
5 5.35011e+01
6 6.93115e+01
7 7.59088e+01
8 1.16504e+02
9 1.42991e+02
10 1.72345e+02
11 1.98905e+02
12 2.32488e+02
13 2.70781e+02

» Matlab downloads

The set is stored in a Matlab file that can be downloaded by clicking on [m].

■ Hesthaven set

This set of points is taken from the reference paper
  1. J.S. Hestaven, From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex, SIAM J. Numer. Anal., Vol. 35, No. 2, pages 655-676, April 1998.
These points are computed by the author for degrees ranging from 3 to 15, and stored in compact form in terms of their orbits. Here we compute their cartesian coordinates on a simplex and the pertinent barycentric coordinates. The results are stored in a Matlab file. In the tables below you can find the main properties of this set.
LEBESGUE CONST.
DEG HES
3 2.11471e+00
4 2.60499e+00
5 3.20959e+00
6 4.07299e+00
7 4.78680e+00
8 5.88305e+00
9 6.93624e+00
10 8.42020e+00
11 1.00916e+01
12 1.23922e+01
13 1.53180e+01
14 2.28594e+01
15 2.96805e+01
VANDERMONDE DET.
DEG HES
3 3.27066e+08
4 7.83087e+13
5 6.34808e+20
6 1.72062e+29
7 1.60512e+39
8 5.73756e+50
9 9.73357e+63
10 7.76177e+78
11 3.18086e+95
12 6.69919e+113
13 7.68316e+133
14 4.98697e+155
15 2.07518e+179
CONDITIONING
DEG HES
3 2.07655e+01
4 4.05431e+01
5 5.33526e+01
6 7.01182e+01
7 9.64710e+01
8 9.73397e+01
9 1.44958e+02
10 1.58067e+02
11 2.41356e+02
12 3.63495e+02
13 5.49954e+02
14 8.88984e+02
15 1.38471e+03

» Matlab downloads

The set is stored in a Matlab file that can be downloaded by clicking on [m].

■ Approximate Fekete and Discrete Leja sets

These points were obtained extracting Approximate Fekete Points (AFP) and Discrete Leja Points (DLP) from two Weakly Admissible Meshes (WAM) on the unit simplex. For references see:
  1. L. Bos, A. Sommariva and M. Vianello, Least-squares polynomial approximation on weakly admissible meshes: disk and triangle, J. Comput. Appl. Math. 235 (2010), pages 660-668;
  2. L. Bos, S. De Marchi, A.Sommariva and M.Vianello, Computing multivariate Fekete and Leja points by numerical linear algebra, SIAM J. Numer. Anal., to appear;
  3. L. Bos, J.P. Calvi, N. Levenberg, A. Sommariva and M. Vianello, Geometric Weakly Admissible Meshes, Discrete Least Squares Approximations and Approximate Fekete Points, Math. Comp.
We computed them for degrees ranging from 2 to 25. The tables below show the main properties of these sets (the first tables are for the first wam, the next ones are regarding the second WAM).
LEBESGUE CONST.
DEG AFP DLP
2 1.66667e+00 1.66667e+00
3 2.26358e+00 3.87761e+00
4 3.11967e+00 5.31324e+00
5 4.14542e+00 6.80272e+00
6 5.34774e+00 1.46844e+01
7 1.29709e+01 1.10897e+01
8 1.00947e+01 2.37668e+01
9 1.13274e+01 1.77868e+01
10 1.16343e+01 2.49430e+01
11 1.46370e+01 2.52255e+01
12 1.48108e+01 3.13713e+01
13 1.84545e+01 3.04025e+01
14 2.38765e+01 3.38272e+01
15 2.95172e+01 3.30685e+01
16 2.66694e+01 9.77858e+01
17 3.64149e+01 5.68717e+01
18 3.63378e+01 8.31842e+01
19 2.76153e+01 5.73026e+01
20 3.98396e+01 8.66610e+01
21 3.51484e+01 8.93065e+01
22 5.45720e+01 8.11744e+01
23 6.66861e+01 1.16117e+02
24 5.47263e+01 8.26428e+01
25 5.87654e+01 1.15177e+02


LEBESGUE CONST.
DEG AFP DLP
2 1.66667e+00 1.66667e+00
3 2.20392e+00 5.36501e+00
4 2.92786e+00 5.13196e+00
5 3.75993e+00 7.59071e+00
6 7.26269e+00 8.72518e+00
7 7.38824e+00 1.59270e+01
8 1.32970e+01 1.52265e+01
9 1.17000e+01 2.36535e+01
10 1.63163e+01 2.01590e+01
11 1.00108e+01 2.27660e+01
12 2.14654e+01 2.98377e+01
13 2.65904e+01 3.13749e+01
14 2.00226e+01 3.56759e+01
15 2.08303e+01 5.17546e+01
16 2.93308e+01 4.20068e+01
17 2.72535e+01 4.17110e+01
18 4.63427e+01 1.37383e+02
19 3.80479e+01 7.10551e+01
20 4.44063e+01 9.58671e+01
21 3.11092e+01 7.32212e+01
22 4.36009e+01 1.71721e+02
23 4.87372e+01 9.43460e+01
24 5.25890e+01 9.60096e+01
25 6.39276e+01 1.34176e+02
VANDERMONDE DET.
DEG AFP DLP
2 3.13712e+04 3.13712e+04
3 3.00881e+08 1.03596e+08
4 6.49867e+13 2.88865e+13
5 4.63337e+20 1.09259e+20
6 1.00463e+29 1.39084e+28
7 6.54778e+38 6.95495e+37
8 3.08220e+50 7.24101e+48
9 5.17014e+63 8.29497e+61
10 9.61635e+78 7.55506e+76
11 3.39858e+95 1.05838e+93
12 1.90954e+114 9.76984e+110
13 7.05130e+134 3.02699e+129
14 1.15297e+157 2.39139e+153
15 1.72071e+181 6.15971e+176
16 3.58895e+207 3.75267e+201
17 4.19607e+235 2.69850e+229
18 1.18074e+265 6.16277e+259
19 1.22631e+297 1.06384e+291
20 Inf Inf
21 Inf Inf
22 Inf Inf
23 Inf Inf
24 Inf Inf
25 Inf Inf


VANDERMONDE DET.
DEG AFP DLP
2 3.13712e+04 3.13712e+04
3 3.24116e+08 9.57079e+07
4 7.84258e+13 2.51088e+13
5 5.90750e+20 4.93133e+19
6 1.00796e+29 2.64441e+28
7 1.24310e+39 8.34300e+37
8 5.31595e+50 7.47029e+48
9 1.54887e+64 5.62148e+61
10 1.92182e+79 2.02828e+76
11 1.60553e+96 1.40273e+94
12 7.54964e+114 1.33869e+111
13 2.59848e+135 3.24109e+132
14 1.12505e+158 1.41490e+154
15 1.66350e+182 6.72810e+177
16 7.71761e+208 3.19631e+203
17 4.28066e+236 1.64071e+231
18 1.90516e+267 1.44769e+260
19 3.00295e+299 3.65539e+292
20 Inf Inf
21 Inf Inf
22 Inf Inf
23 Inf Inf
24 Inf Inf
25 Inf Inf
CONDITIONING
DEG AFP DLP
2 1.01810e+01 1.01810e+01
3 2.16896e+01 2.66430e+01
4 3.92273e+01 5.90810e+01
5 5.53913e+01 8.05595e+01
6 7.23234e+01 8.46814e+01
7 1.06979e+02 1.83457e+02
8 1.21893e+02 2.08649e+02
9 1.48251e+02 1.68471e+02
10 1.92608e+02 2.69562e+02
11 1.88537e+02 4.97223e+02
12 2.76966e+02 4.49599e+02
13 3.40641e+02 5.43775e+02
14 4.84052e+02 6.79332e+02
15 4.70563e+02 6.07371e+02
16 4.81914e+02 2.16964e+03
17 8.08041e+02 1.11027e+03
18 7.39718e+02 7.85703e+02
19 7.94453e+02 2.47833e+03
20 9.85078e+02 3.30141e+03
21 1.05270e+03 1.12564e+03
22 1.21491e+03 2.24164e+03
23 1.45467e+03 2.38626e+03
24 1.13203e+03 3.52126e+03
25 2.22248e+03 3.56795e+03


CONDITIONING
DEG AFP DLP
2 1.01810e+01 1.01810e+01
3 2.10919e+01 2.86304e+01
4 3.98776e+01 5.00485e+01
5 5.22721e+01 6.67243e+01
6 8.91108e+01 1.04876e+02
7 8.73718e+01 1.33591e+02
8 1.21875e+02 1.66198e+02
9 1.53385e+02 2.74312e+02
10 2.01772e+02 2.70747e+02
11 2.09667e+02 3.43299e+02
12 3.58483e+02 4.16564e+02
13 2.62970e+02 3.33601e+02
14 4.12196e+02 4.11040e+02
15 4.71148e+02 7.10774e+02
16 5.92727e+02 8.63825e+02
17 5.73209e+02 7.17337e+02
18 7.51134e+02 2.43820e+03
19 9.72120e+02 1.21566e+03
20 1.08031e+03 1.36392e+03
21 8.43606e+02 1.36677e+03
22 1.28007e+03 3.77716e+03
23 1.75566e+03 2.72116e+03
24 1.65590e+03 3.23390e+03
25 1.75561e+03 3.47619e+03

» Matlab downloads

The set is stored in a Matlab file that can be downloaded by clicking on [m].