Some points on the square.

In this homepage we list some sets of points on the square (having vertices V1=[-1,-1], V2=[-1,1], V3=[1,1], V4=[1,-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 square. 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 LEB is a set with low Lebesgue constant, FEK is a set with high Vandermonde determinant (w.r.t. the tensorial Chebyshev basis), PDL is a set of Padua-Jacobi points with low Lebesgue constant, PDF with high Vandermonde determinant (w.r.t. the tensorial Chebyshev basis) and PDC is the classic Padua points set (based on Chebyshev-Lobatto points).
1 2.00000e+00 2.00007e+00 2.00000e+00 2.00006e+00 2.00000e+00
2 2.39485e+00 2.93426e+00 2.65381e+00 2.88249e+00 3.00000e+00
3 2.73320e+00 3.72371e+00 3.70371e+00 4.03151e+00 3.77614e+00
4 3.23913e+00 3.99693e+00 3.73895e+00 4.56669e+00 4.40973e+00
5 3.59073e+00 4.72823e+00 4.13883e+00 5.37282e+00 4.94781e+00
6 3.99479e+00 6.07333e+00 4.58195e+00 5.70494e+00 5.41750e+00
7 4.33866e+00 5.48077e+00 4.93966e+00 6.36755e+00 5.83566e+00
8 4.89553e+00 5.95343e+00 5.30089e+00 6.58034e+00 6.21346e+00
9 5.18625e+00 6.20421e+00 5.59999e+00 7.15173e+00 6.55871e+00
10 5.32219e+00 6.63758e+00 5.92505e+00 7.28742e+00 6.87710e+00
11 6.18386e+00 6.86824e+00 6.18699e+00 7.79854e+00 7.17289e+00
12 6.44339e+00 7.37532e+00 6.44564e+00 7.88368e+00 7.44938e+00
13 6.69596e+00 7.54755e+00 6.70169e+00 8.35034e+00 7.70917e+00
14 6.91222e+00 7.94861e+00 6.93431e+00 8.40244e+00 7.95435e+00
15 7.10613e+00 8.05725e+00 7.16080e+00 8.83302e+00 8.18663e+00
16 7.34757e+00 8.27049e+00 7.36923e+00 8.86223e+00 8.40744e+00
17 7.58169e+00 8.33798e+00 7.58169e+00 9.26325e+00 8.61795e+00
18 7.76413e+00 8.59410e+00 7.76413e+00 9.27598e+00 8.81917e+00
19 7.89256e+00 8.83642e+00 7.99687e+00 9.65320e+00 9.01197e+00
20 8.14862e+00 8.99242e+00 8.13987e+00 9.65344e+00 9.19709e+00
1 4.00000e+00 4.00000e+00 4.00000e+00 4.00000e+00 4.00000e+00
2 4.79020e+01 6.54507e+01 5.49172e+01 5.58567e+01 5.40000e+01
3 3.58493e+03 6.27131e+03 3.83600e+03 3.98318e+03 3.88800e+03
4 1.23699e+06 3.48883e+06 1.62724e+06 2.05370e+06 1.97642e+06
5 5.12471e+09 1.72862e+10 5.84191e+09 9.04643e+09 8.72405e+09
6 2.02930e+14 9.23161e+14 2.51732e+14 4.13028e+14 3.95986e+14
7 9.46637e+19 2.90540e+20 1.15681e+20 2.22205e+20 2.13439e+20
8 5.22685e+26 2.19789e+27 7.51674e+26 1.61662e+27 1.54891e+27
9 7.14536e+34 2.37937e+35 7.94454e+34 1.76239e+35 1.69170e+35
10 1.69903e+44 4.48320e+44 1.50769e+44 3.20565e+44 3.07382e+44
11 5.04246e+54 1.38813e+55 5.07902e+54 1.05952e+55 1.01771e+55
12 3.25802e+66 9.45981e+66 3.33306e+66 6.95055e+66 6.67398e+66
13 4.67513e+79 1.21419e+80 4.82393e+79 9.73516e+79 9.36253e+79
14 1.52197e+94 4.30204e+94 1.58875e+94 3.13782e+94 3.01779e+94
15 1.25675e+110 2.95460e+110 1.33144e+110 2.48061e+110 2.38910e+110
16 2.63861e+127 6.17533e+127 2.70554e+127 5.13356e+127 4.94524e+127
17 1.52319e+146 3.12736e+146 1.52319e+146 2.94290e+146 2.83856e+146
18 2.70592e+166 5.83991e+166 2.70592e+166 4.95044e+166 4.77640e+166
19 1.40584e+188 2.51552e+188 1.54235e+188 2.57101e+188 2.48347e+188
20 2.27964e+211 4.24219e+211 2.30006e+211 4.34090e+211 4.19459e+211
1 3.00000e+00 3.00000e+00 3.00000e+00 3.00000e+00 3.00000e+00
2 7.01531e+00 9.00000e+00 7.97630e+00 7.77237e+00 8.00000e+00
3 1.24849e+01 1.01962e+01 1.61901e+01 1.41022e+01 1.60948e+01
4 2.38346e+01 1.64466e+01 2.35014e+01 2.05075e+01 2.34377e+01
5 3.36092e+01 3.22364e+01 2.74553e+01 2.79869e+01 2.71830e+01
6 4.46782e+01 3.98261e+01 4.97824e+01 4.79785e+01 3.59517e+01
7 5.75369e+01 4.80111e+01 5.82183e+01 5.84603e+01 5.80954e+01
8 7.08739e+01 7.24969e+01 8.05971e+01 5.92046e+01 5.75877e+01
9 8.32196e+01 9.20774e+01 8.86064e+01 9.10086e+01 7.03850e+01
10 1.10530e+02 8.62110e+01 1.09033e+02 1.06727e+02 8.43201e+01
11 1.27539e+02 1.25201e+02 1.26173e+02 1.25199e+02 9.96510e+01
12 1.44984e+02 1.38720e+02 1.55558e+02 1.47162e+02 1.16147e+02
13 1.71609e+02 1.70173e+02 1.80078e+02 1.70828e+02 1.34016e+02
14 1.96207e+02 2.05317e+02 2.04246e+02 1.97776e+02 1.53068e+02
15 2.16993e+02 2.26357e+02 2.20163e+02 2.12889e+02 1.73477e+02
16 2.49677e+02 2.49005e+02 2.41269e+02 2.48333e+02 1.95083e+02
17 2.78630e+02 2.78939e+02 2.78875e+02 2.78864e+02 2.18034e+02
18 2.41843e+02 2.53548e+02 3.09293e+02 3.03614e+02 2.42191e+02
19 3.31655e+02 2.67685e+02 3.45987e+02 3.39596e+02 2.67685e+02
20 3.74733e+02 2.94393e+02 3.98350e+02 3.79903e+02 2.94393e+02

» Matlab downloads

The sets are 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].