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.
annulus_rule, a MATLAB code which computes a quadrature rule for estimating integrals of a function over the interior of a circular annulus in 2d.
cube_felippa_rule, a MATLAB code which returns the points and weights of a felippa quadrature rule over the interior of a cube in 3d.
line_fekete, a MATLAB code which approximates the location of fekete points in an interval [a,b]. a family of sets of fekete points, indexed by size n, represents an excellent choice for defining a polynomial interpolant.
pyramid_felippa_rule, a MATLAB code which returns felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3d.
simplex_gm_rule, a MATLAB code which defines grundmann-moeller quadrature rules over the interior of a simplex in m dimensions.
square_felippa_rule, a MATLAB code which returns the points and weights of a felippa quadrature rule over the interior of a square in 2d.
stroud, a MATLAB code which contains quadrature rules for a variety of unusual areas, surfaces and volumes in 2d, 3d and n-dimensions.
tetrahedron_felippa_rule, a MATLAB code which returns felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3d.
triangle_dunavant_rule, a MATLAB code which sets up a dunavant quadrature rule over the interior of a triangle in 2d.
triangle_felippa_rule, a MATLAB code which returns felippa's quadratures rules for approximating integrals over the interior of a triangle in 2d.
triangle_integrals, a MATLAB code which implements test functions for integration over the interior of a unit triangle in 2d.
triangle_interpolate, a MATLAB code which shows how vertex data can be interpolated at any point in the interior of a triangle.
triangle_lyness_rule, a MATLAB code which returns lyness-jespersen quadrature rules over the interior of a triangle in 2d.
triangle_monte_carlo, a MATLAB code which uses the monte carlo method to estimate integrals over the interior of a triangle in 2d.
triangle_ncc_rule, a MATLAB code which defines newton-cotes closed quadrature rules over the interior of a triangle in 2d.
triangle_nco_rule, a MATLAB code which defines newton-cotes open quadrature rules over the interior of a triangle in 2d.
triangle_symq_rule, a MATLAB code which returns efficient symmetric quadrature rules, with exactness up to total degree 50, over the interior of an arbitrary triangle in 2d, by hong xiao and zydrunas gimbutas.
triangle_wandzura_rule, a MATLAB code which sets up a quadrature rule of exactness 5, 10, 15, 20, 25 or 30 over the interior of a triangle in 2d.