FEKETE
High Order Interpolation and Quadrature in Triangles

FEKETE is a C++ library, using double precision arithmetic, 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.

Related Data and Programs:

CLENSHAW_CURTIS is a C++ library which can set up a Clenshaw Curtis quadrature grid in multiple dimensions.

DUNAVANT is a C++ library of routines for defining Dunavant rules for quadrature on a triangle.

FEKETE is also available in a FORTRAN90 version and a MATLAB version.

FEM_SAMPLE is a FORTRAN90 library of routines for evaluating a finite element function defined on an order 3 or order 6 triangulation.

FEMPACK is a C++ library of routines for finite element calculations.

GM_RULES is a C++ library of routines for defining a Grundmann-Moeller rule for quadrature over a triangle, tetrahedron, or general M-dimensional simplex.

KEAST is a C++ library which defines a number of quadrature rules for a tetrahedron.

NCC_TRIANGLE is a C++ library defining Newton-Cotes closed quadrature rules on a triangle.

NCO_TRIANGLE is a C++ library defining Newton-Cotes open quadrature rules on a triangle.

NINT_EXACTNESS_TRI is an executable C++ program which investigates the polynomial exactness of a quadrature rule for the triangle.

NINTLIB is a FORTRAN90 library containing a variety of routines for numerical estimation of integrals in multiple dimensions.

QUADRULE is a C++ library for defining quadrature rules on a variety of intervals with different weight functions.

STROUD is a C++ library containing quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.

TEST_TRI_INT is a FORTRAN90 library of functions that can be used to test algorithms for quadrature over a triangle.

TOMS612 is a FORTRAN77 library of routines which can estimate the integral of a function over a triangle.

TOMS706 is a FORTRAN77 library which estimates the integral of a function over a triangulated region.

WANDZURA is a C++ library of routines for defining 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.C, the source code.
• fekete.H, the include code.
• fekete.csh, commands to compile the source code.

Examples and Tests:

• fekete_prb.C, a sample calling program.
• fekete_prb.csh, commands to compile and run the sample program.
• fekete_prb.out, the output from a run of the sample program.

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.