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 cartesian coordinates of the points are stored in the N x 2 matrix pts;
• the barycentric coordinates of the points are stored in the N x 3 matrix pts_bar;
• for certain sets, some optimization coefficients are stored in the vector t; see below for the meaning of this variable;
• some statistics are stored in the variable in the matrix stats_matrix;
The variable stats_matrix is a matrix whose columns are
• the degree of the set;
• a good approximation of the Lebesgue constant of the set;
• the Vandermonde determinant with respect to orthonormal Dubiner basis (w.r.t. Legendre weight on the unit simplex) evaluated on the set;
• the conditioning of the Vandermonde matrix with respect to orthonormal Dubiner basis (w.r.t. Legendre weight on the unit simplex) evaluated on the set.

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

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

The set is stored in a Matlab file that can be downloaded by clicking on [m].
• Taylor-Wingate-Vincent set, [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

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

The sets are stored in Matlab files that can be downloaded by clicking on [m].
• Set based on expanded Chebyshev points on sides [m];
• Set based on Gauss-Legendre-Lobatto points on sides [m];
• Set based on near optimal Gegenbauer points on sides [m];
• Set based on near optimal Gauss-Jacobi-Lobatto points on sides [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
• Set based on expanded Chebyshev points on sides [m];
• Set based on Gauss-Legendre-Lobatto points on sides [m];
• Set based on near optimal Gegenbauer points on sides [m];
• Set based on near optimal Gauss-Jacobi-Lobatto points on sides [m].

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

The set is stored in a Matlab file that can be downloaded by clicking on [m].
• Chen-Babuska set, [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

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