FEKETE
High Order Interpolation and Quadrature in Triangles

FEKETE is a MATLAB 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 MATLAB library which can set up a Clenshaw Curtis quadrature grid in multiple dimensions.

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

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

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

FEMPACK is a MATLAB library of routines for finite element calculations.

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

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

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

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

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

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

QUADRULE_FAST is a MATLAB library of routines defining efficient versions of a few 1D quadrature rules.

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

TEST_TRI_INT is a MATLAB 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 MATLAB library of routines for defining Dunavant 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.

Tar File:

A GZIP'ed TAR file of the contents of this directory is available. This is only done as a convenience for users who want ALL the files, and don't want to download them individually. This is not a convenience for me, so don't be surprised if the tar file is somewhat out of date.

• fekete_m_src.tar.csh the commands used to create the GZIP'ed TAR file.
• fekete_m_src.tar.gz the GZIP'ed TAR file.

Source Code:

• fekete_degree.m returns the degree of a given Fekete rule for the triangle.
• fekete_order_num.m returns the order of a given Fekete rule for the triangle.
• fekete_rule.m returns the points and weights of a Fekete rule.
• fekete_rule_num.m returns the number of Fekete rules available.
• fekete_suborder.m returns the suborders for a Fekete rule.
• fekete_suborder_num.m returns the number of suborders for a Fekete rule.
• fekete_subrule.m returns a compressed Fakete rule.
• file_name_inc.m increments a partially numeric filename.
• i4_modp.m returns the nonnegative remainder of integer division.
• i4_wrap.m forces an integer to lie between given limits by wrapping.
• reference_to_physical_t3.m maps T3 reference points to physical points.
• timestamp.m prints the current YMDHMS date as a time stamp.
• timestring.m writes the current YMDHMS date into a string.
• triangle_area.m computes the area of a triangle.
• triangle_points_plot.m plots a triangle and some points.

Examples and Tests:

• fekete_test.m, runs all the tests.
• fekete_test01.m, tests FEKETE_RULE_NUM, FEKETE_DEGREE, and FEKETE_ORDER_NUM.
• fekete_test02.m, tests FEKETE_RULE by summing the weights.
• fekete_test03.m, tests FEKETE_RULE by summing the barycentric coordinates.
• fekete_test04.m, tests FEKETE_ORDER by integrating monomials in the unit triangle.
• fekete_test05.m, tests FEKETE_RULE by plotting the points.
• fekete_test06.m, tests REFERENCE_TO_PHYSICAL_T3 by transforming a Fekete rule from the unit triangle to another triangle.
• fekete_test.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.

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