triangle_fekete_rule, a MATLAB code which can return information defining any of seven Fekete rules for high order interpolation and quadrature over the interior of a triangle in 2D.
Fekete points can be defined for any region OMEGA. To define the Fekete points for a given region, let Poly(N) be some finite dimensional vector space of polynomials, such as all polynomials of degree less than L, or all polynomials whose monomial terms have total degree less than some value L.
Let P(1:M) be any basis for Poly(N). For this basis, the Fekete points are defined as those points Z(1:M) which maximize the determinant of the corresponding Vandermonde matrix:
V = [ P1(Z1) P1(Z2) ... P1(ZM) ]
[ P2(Z1) P2(Z2) ... P2(ZM) ]
...
[ PM(ZM) P2(ZM) ... PM(ZM) ]
The seven rules have the following orders and precisions:
| Rule | Order | Precision |
|---|---|---|
| 1 | 10 | 3 |
| 2 | 28 | 6 |
| 3 | 55 | 9 |
| 4 | 91 | 12 |
| 5 | 91 | 12 |
| 6 | 136 | 15 |
| 7 | 190 | 18 |
On the triangle, it is known that some Fekete points will lie on the boundary, and that on each side of the triangle, these points will correspond to a set of Gauss-Lobatto points.
The computer code and data files described and made available on this web page are distributed under the MIT license
triangle_fekete_rule is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.
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