Sets



Last update: February 06, 2021.

In this homepage we have stored several pointsets suitable for interpolation or cubature on intervals, simplex (triangle), square, disk and polygons.

We point out that concerning cubature rules in Phyton, a comprehensive list can be found at https://awesomeopensource.com/project/nschloe/quadpy.

INTERVAL:

» Interpolation

  1. General set with low Lebesgue constant: [.m]
  2. Extended Chebyshev: [.m]
  3. Gauss-Legendre-Lobatto (Fekete points in the interval): [.m]

SIMPLEX (TRIANGLE):

» Interpolation/Least squares sets

  • General set with low Lebesgue constant: [.m] (last update: Jan 08, 2017, old version: [.m]).
  • General set with low Lebesgue constant and Gauss-Legendre-Lobatto distribution on the side: [.m]
  • General set with high absolute value of the Vandermonde matrix, i.e. (quasi-) Fekete points: [.m]
  • Symmetric set with low Lebesgue constant: [.m]
  • Weakly Admissible Mesh: [.m]
  • Approximate Fekete points (with degree from 1 to 70): [.m]

» Cubature

  • Best cubature sets on the triangle (up to degree 50): [.m]

» Comparisons

  • For a comparison on several interpolation sets see also: [html].
  • For a comparison on several cubature sets see also: [html].

» Other contributions

    You may find useful the M-file for the evaluation of the Vandermonde matrix w.r.t.
  • Dubiner Legendre basis [.m] as orthonormal basis;
  • Proriol-Dubiner [.m] as orthogonal basis with its derivatives (in a form suitable for cubature, i.e. it is the transpose of the Vandermonde matrix for interpolation purposes).

SQUARE:

» Interpolation

  • General set with low Lebesgue constant: [.m] (last update: Jan 08, 2017, old version: [.m]).
  • General set with high absolute value of the Vandermonde matrix, i.e. (quasi-) Fekete points: [.m]
  • Padua-Jacobi points with low Lebesgue constant: [.m]
  • Padua-Jacobi points with high absolute value of the Vandermonde matrix: [.m]
  • Padua points: [.m]

» Cubature

  • Almost minimal rules on the square (Legendre weight): [.m]

DISK:

» Interpolation/Least squares sets

  • Good interpolation sets on the unit disk [.html]
  • Weakly Admissible Mesh: [.m]
» Cubature

  • Cubature rules (Legendre weight): [.m]

SPHERE:

» Interpolation

» Cubature