FEKETE
High Order Interpolation and Quadrature in Triangles

FEKETE is a FORTRAN90 library which can return information defining any of seven Fekete rules for high order interpolation and quadrature in a triangle.

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:

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

Related Data and Programs:

DUNAVANT, a FORTRAN90 library which defines Dunavant rules for quadrature on a triangle.

FELIPPA, a FORTRAN90 library which defines quadrature rules for lines, triangles, quadrilaterals, pyramids, wedges, tetrahedrons and hexahedrons.

GM_RULE, a FORTRAN90 library which defines a Grundmann-Moeller rule for quadrature over a triangle, tetrahedron, or general M-dimensional simplex.

LYNESS_RULE a FORTRAN90 library which returns Lyness-Jespersen quadrature rules for the triangle.

NCC_TRIANGLE, a FORTRAN90 library which defines Newton-Cotes closed quadrature rules on a triangle.

NCO_TRIANGLE, a FORTRAN90 library which defines Newton-Cotes open quadrature rules on a triangle.

QUADRATURE_RULES_TRI, a dataset directory which contains triples of files which defines various quadrature rules on triangles.

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

TEST_TRI_INT, a FORTRAN90 library which tests algorithms for quadrature over a triangle.

TOMS612, a FORTRAN77 library which estimates the integral of a function over a triangle.

TRIANGLE_EXACTNESS, a FORTRAN90 program which investigates the polynomial exactness of a quadrature rule for the triangle.

TRIANGLE_MONTE_CARLO, a FORTRAN90 program which uses the Monte Carlo method to estimate integrals over a triangle.

WANDZURA, a FORTRAN90 library which definines Wandzura rules for quadrature on a triangle.

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:

• fekete.f90, the source code.
• fekete.sh, commands to compile the source code.

Examples and Tests:

• fekete_prb.f90, a sample calling program.
• fekete_prb.sh, commands to compile and run the sample program.
• fekete_prb_output.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.