TRIANGLE_FEKETE_RULE
High Order Interpolation and Quadrature in Triangles

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.

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:

ANNULUS_RULE, a C++ library 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++ 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_ANALYZE, a C++ program which reads a triangle defined in a file, and uses the triangle_properties() library to compute angles, area, centroid, circumcircle, edge lengths, incircle, orientation, orthocenter, and quality.

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

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.

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.cpp, the source code.
• triangle_fekete_rule.hpp, the include code.

Examples and Tests:

• triangle_fekete_rule_test.cpp, a sample calling program.
• triangle_fekete_rule_test.sh, commands to compile and run the sample 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_ORDER_NUM returns the order 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_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 Fekete rule.
• FEKETE_SUBRULE_1 returns a compressed Fekete rule 1.
• FEKETE_SUBRULE_2 returns a compressed Fekete rule 2.
• FEKETE_SUBRULE_3 returns a compressed Fekete rule 3.
• FEKETE_SUBRULE_4 returns a compressed Fekete rule 4.
• FEKETE_SUBRULE_5 returns a compressed Fekete rule 5.
• FEKETE_SUBRULE_6 returns a compressed Fekete rule 6.
• FEKETE_SUBRULE_7 returns a compressed Fekete rule 7.
• FILE_NAME_INC increments a partially numeric file name.
• I4_MAX returns the maximum of two integers.
• I4_MIN returns the smaller of two integers.
• I4_MODP returns the nonnegative remainder of integer division.
• I4_WRAP forces an integer to lie between given limits by wrapping.
• R8_HUGE returns a "huge" R8.
• R8_NINT returns the nearest integer to an R8.
• REFERENCE_TO_PHYSICAL_T3 maps T3 reference points to physical points.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TIMESTRING returns the current YMDHMS date as a string.
• 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 C++ source codes.