TRIANGLE_FEKETE_RULE
High Order Interpolation and Quadrature in Triangles

TRIANGLE_FEKETE_RULE is a FORTRAN90 library 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) ]
      

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 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

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL 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:

CUBE_FELIPPA_RULE, a FORTRAN90 library which returns the points and weights of a Felippa quadrature rule over the interior of a cube in 3D.

LINE_FEKETE, a FORTRAN90 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 FORTRAN90 library which returns Felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3D.

SIMPLEX_GM_RULE, a FORTRAN90 library which defines Grundmann-Moeller quadrature rules over the interior of a simplex in M dimensions.

SQUARE_FELIPPA_RULE, a FORTRAN90 library which returns the points and weights of a Felippa quadrature rule over the interior of a square in 2D.

STROUD, a FORTRAN90 library which contains quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and M-dimensions.

TETRAHEDRON_FELIPPA_RULE, a FORTRAN90 library which returns Felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.

TRIANGLE_DUNAVANT_RULE, a FORTRAN90 library which sets up a Dunavant quadrature rule over the interior of a triangle in 2D.

TRIANGLE_EXACTNESS, a FORTRAN90 program which investigates the polynomial exactness of a quadrature rule over the interior of a triangle in 2D.

TRIANGLE_FELIPPA_RULE, a FORTRAN90 library which returns Felippa's quadratures rules for approximating integrals over the interior of a triangle in 2D.

TRIANGLE_INTERPOLATE, a FORTRAN90 library which shows how vertex data can be interpolated at any point in the interior of a triangle.

TRIANGLE_LYNESS_RULE, a FORTRAN90 library which returns Lyness-Jespersen quadrature rules over the interior of a triangle in 2D.

TRIANGLE_MONTE_CARLO, a FORTRAN90 program which uses the Monte Carlo method to estimate integrals over the interior of a triangle in 2D.

TRIANGLE_NCC_RULE, a FORTRAN90 library which defines Newton-Cotes Closed (NCC) quadrature rules over the interior of a triangle in 2D.

TRIANGLE_NCO_RULE, a FORTRAN90 library which defines Newton-Cotes Open (NCO) quadrature rules over the interior of a triangle in 2D.

TRIANGLE_SVG, a FORTRAN90 library which uses Scalable Vector Graphics (SVG) to plot a triangle and any number of points, to illustrate quadrature rules and sampling techniques.

TRIANGLE_SYMQ_RULE, a FORTRAN90 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 FORTRAN90 library which sets up a quadrature rule of exactness 5, 10, 15, 20, 25 or 30 over the interior of a triangle in 2D.

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:

• triangle_fekete_rule.f90, the source code.

Examples and Tests:

• triangle_fekete_rule_test.f90, a sample calling program.
• triangle_fekete_rule_test.txt, the output file.

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.

• fekete_rule_1.png, a plot of rule 1.
• fekete_rule_2.png, a plot of rule 2.
• fekete_rule_3.png, a plot of rule 3.
• fekete_rule_4.png, a plot of rule 4.
• fekete_rule_5.png, a plot of rule 5.
• fekete_rule_6.png, a plot of rule 6.
• fekete_rule_7.png, a plot of rule 7.

List of Routines:

• FEKETE_DEGREE returns the degree of a given Fekete rule for the triangle.
• FEKETE_RULE returns the points and weights of a Fekete rule.
• FEKETE_RULE_NUM returns the number of Fekete rules available.
• FEKETE_ORDER_NUM returns the order of a given Fekete rule for the triangle.
• FEKETE_SUBORDER returns the suborders for a Fekete rule.
• FEKETE_SUBORDER_NUM returns the number of suborders for a Fekete rule.
• FEKETE_SUBRULE returns a compressed Fakete rule.
• FILE_NAME_INC increments a partially numeric filename.
• GET_UNIT returns a free FORTRAN unit number.
• I4_MODP returns the nonnegative remainder of integer division.
• I4_WRAP forces an integer to lie between given limits by wrapping.
• REFERENCE_TO_PHYSICAL_T3 maps T3 reference points to physical points.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TRIANGLE_AREA computes the area of a triangle.
• TRIANGLE_POINTS_PLOT plots a triangle and some points.

You can go up one level to the FORTRAN90 source codes.