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 cartesian coordinates of the points are stored in the N x 2 matrix pts;
• 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 tensorial Chebyshev basis evaluated on the set;
• the conditioning of the Vandermonde matrix with respect to tensorial Chebyshev basis 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 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).

LEBESGUE CONST. DEG LEB FEK PDL PDF PDC 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

VANDERMONDE DET. DEG LEB FEK PDL PDF PDC 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

CONDITIONING DEG LEB FEK PDL PDF PDC 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].

• LEB set, [m];
• FEK set, [m];
• PDJ set (Lebesgue optimal), [m];
• PDJ set (Fekete optimal), [m];
• PDC set (classic Padua points set), [m];

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

• Matlab codes, [zip].