FEKETE
High Order Interpolation and Quadrature in Triangles

FEKETE is a MATLAB 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) ]
      

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:

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

Related Data and Programs:

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

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

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

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

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

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

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

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

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

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

WANDZURA, a MATLAB library which defines 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.

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.
• 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_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.

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