triangle_fekete_rule


triangle_fekete_rule, a C code which returns any of seven 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:
RuleOrderPrecision
1 10 3
2 28 6
3 55 9
4 9112
5 9112
613615
719018

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.

Licensing:

The computer code and data files described and made available on this web page are distributed under the MIT license

Languages:

triangle_fekete_rule is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

ANNULUS_RULE, a C 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 C code which returns the points and weights of a Felippa quadrature rule over the interior of a cube in 3D.

LINE_FEKETE_RULE, a C 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 C code which returns Felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3D.

SIMPLEX_GM_RULE, a C code which defines Grundmann-Moeller quadrature rules over the interior of a simplex in M dimensions.

SQUARE_FELIPPA_RULE, a C code which returns the points and weights of a Felippa quadrature rule over the interior of a square in 2D.

STROUD, a C code which contains quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and M-dimensions.

TETRAHEDRON_FELIPPA_RULE, a C code which returns Felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.

TRIANGLE_EXACTNESS, a C code which investigates the polynomial exactness of a quadrature rule over the interior of a triangle in 2D.

triangle_fekete_rule_test

TRIANGLE_FELIPPA_RULE, a C code which returns Felippa's quadratures rules for approximating integrals over the interior of a triangle in 2D.

TRIANGLE_INTEGRALS, a C code which returns the exact value of the integral of any monomial over the interior of the unit triangle in 2D.

TRIANGLE_INTERPOLATE, a C code which shows how vertex data can be interpolated at any point in the interior of a triangle.

TRIANGLE_MONTE_CARLO, a C code which uses the Monte Carlo method to estimate integrals over the interior of a triangle in 2D.

TRIANGLE_NCC_RULE, a C code which defines Newton-Cotes Closed (NCC) quadrature rules over the interior of a triangle in 2D.

TRIANGLE_NCO_RULE, a C code which defines Newton-Cotes Open (NCO) quadrature rules over the interior of a triangle in 2D.

TRIANGLE_SYMQ_RULE, a C 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.

Reference:

  1. SF Bockman,
    Generalizing the Formula for Areas of Polygons to Moments,
    American Mathematical Society Monthly,
    Volume 96, Number 2, February 1989, pages 131-132.
  2. Hermann Engels,
    Numerical Quadrature and Cubature,
    Academic Press, 1980,
    ISBN: 012238850X,
    LC: QA299.3E5.
  3. Arthur Stroud,
    Approximate Calculation of Multiple Integrals,
    Prentice Hall, 1971,
    ISBN: 0130438936,
    LC: QA311.S85.
  4. Mark Taylor, Beth Wingate, Rachel Vincent,
    An Algorithm for Computing Fekete Points in the Triangle,
    SIAM Journal on Numerical Analysis,
    Volume 38, Number 5, 2000, pages 1707-1720.
  5. Stephen Wandzura, Hong Xiao,
    Symmetric Quadrature Rules on a Triangle,
    Computers and Mathematics with Applications,
    Volume 45, 2003, pages 1829-1840.

Source Code:


Last revised on 21 August 2019.