Some points on the unit disk.

In this homepage we list some sets of points on the unit disk. 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;
• 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 Koornwinder II basis evaluated on the set;
• the conditioning of the Vandermonde matrix with respect to Koornwinder II 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 unit disk. 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 has particularly low Lebesgue constant, while the set FEK has particularly high absolute values of Vandermonde determinants (w.r.t. Koornwinder II basis).

LEBESGUE CONST. DEG LEB FEK 1 1.66666e+00 1.66666e+00 2 1.98882e+00 1.98889e+00 3 2.46712e+00 2.63051e+00 4 2.97321e+00 3.19006e+00 5 3.50396e+00 3.85218e+00 6 4.30163e+00 4.38492e+00 7 5.08102e+00 5.58331e+00 8 5.42679e+00 6.44660e+00 9 6.72990e+00 7.02124e+00 10 8.48685e+00 7.84479e+00 11 1.03165e+01 8.91868e+00 12 1.14822e+01 1.48496e+01 13 1.19771e+01 1.70223e+01 14 1.23882e+01 1.85145e+01 15 1.41646e+01 2.14413e+01 16 1.74375e+01 2.89097e+01

VANDERMONDE DET. DEG LEB FEK 1 4.94884e-01 4.94884e-01 2 2.12581e+00 2.12596e+00 3 2.97745e+01 3.43700e+01 4 2.07842e+03 3.48899e+03 5 7.03580e+05 2.03386e+06 6 1.10256e+10 1.12671e+10 7 4.15306e+13 4.20169e+14 8 2.87535e+19 1.81089e+20 9 1.74423e+25 1.02462e+27 10 2.28398e+34 6.71855e+34 11 2.96657e+38 4.88383e+43 12 1.06066e+48 6.50595e+53 13 4.08248e+58 7.10538e+64 14 2.34232e+76 3.03916e+77 15 2.38002e+89 1.57432e+91 16 3.51709e+97 6.73852e+105

CONDITIONING DEG LEB FEK 1 2.98787e+00 2.98787e+00 2 5.13352e+00 5.33265e+00 3 1.08730e+01 1.06179e+01 4 1.32763e+01 1.38758e+01 5 1.55726e+01 1.64950e+01 6 2.84505e+01 2.64576e+01 7 2.55160e+01 3.28434e+01 8 3.58843e+01 4.32800e+01 9 5.34490e+01 4.79974e+01 10 6.37297e+01 6.08951e+01 11 7.85848e+01 6.88040e+01 12 1.06035e+02 7.40490e+01 13 9.55036e+01 1.09216e+02 14 1.18076e+02 1.11901e+02 15 1.52596e+02 1.25213e+02 16 1.43086e+02 2.31940e+02

» Matlab downloads

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

• LEB set on the unit disk [m];
• FEK set on the unit disk [m].

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

• Matlab codes [zip].