TRIANGLE_FEKETE_RULE is a C++ library which returns Fekete rules for 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 GNU LGPL license.
TRIANGLE_FEKETE_RULE is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.
CUBE_FELIPPA_RULE, a C++ library which returns the points and weights of a Felippa quadrature rule over the interior of a cube in 3D.
LINE_FEKETE, a C++ library 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 C++ library which returns Felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3D.
SIMPLEX_GM_RULE, a C++ library which defines Grundmann-Moeller quadrature rules over the interior of the simplex in M dimensions.
SQUARE_FELIPPA_RULE, a C++ library which returns the points and weights of a Felippa quadrature rule over the interior of a square in 2D.
STROUD, a C++ library which contains quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and M-dimensions.
TETRAHEDRON_FELIPPA_RULE, a C++ library which returns Felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.
TRIANGLE_DUNAVANT_RULE, a C++ library which sets up a Dunavant quadrature rule over the interior of a triangle in 2D.
TRIANGLE_EXACTNESS, a C++ program which investigates the polynomial exactness of a quadrature rule over the interior of a triangle in 2D.
TRIANGLE_FELIPPA_RULE, a C++ library which returns Felippa's quadratures rules for approximating integrals over the interior of a triangle in 2D.
TRIANGLE_INTEGRALS, a C++ library which returns the exact value of the integral of any monomial over the interior of the unit triangle in 2D.
TRIANGLE_LYNESS_RULE, a C++ library which returns Lyness-Jespersen quadrature rules over the interior of a triangle in 2D.
TRIANGLE_MONTE_CARLO, a C++ program which uses the Monte Carlo method to estimate integrals over a triangle.
TRIANGLE_NCC_RULE, a C++ library which defines Newton-Cotes Closed (NCC) quadrature rules over the interior of a triangle in 2D.
TRIANGLE_NCO_RULE, a C++ library which defines Newton-Cotes Open (NCO) quadrature rules over the interior of a triangle in 2D.
TRIANGLE_SYMQ_RULE, a C++ library 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 C++ library which sets up a quadrature rule of exactness 5, 10, 15, 20, 25 or 30 over the interior of a triangle in 2D.
One of the tests in the sample calling program creates EPS files of the abscissas in the unit triangle. These have been converted to PNG files for display here.
You can go up one level to the C++ source codes.
