TRIANGLE_EXACTNESS
Exactness of Quadrature on Triangles

TRIANGLE_EXACTNESS is a MATLAB program which investigates the polynomial exactness of a quadrature rule for a triangle.

The polynomial exactness of a quadrature rule is defined as the highest total degree D such that the quadrature rule is guaranteed to integrate exactly all polynomials of total degree DEGREE_MAX or less, ignoring roundoff. The total degree of a polynomial is the maximum of the degrees of all its monomial terms. For a triangle, the degree of a monomial term is the sum of the exponents of x and y. Thus, for instance, the DEGREE of

x²y⁵

To be thorough, the program starts at DEGREE = 0, and then proceeds to DEGREE = 1, 2, and so on up to a maximum degree DEGREE_MAX specified by the user. At each value of DEGREE, the program generates every possible monomial term, applies the quadrature rule to it, and determines the quadrature error. The program uses a scaling factor on each monomial so that the exact integral should always be 1; therefore, each reported error can be compared on a fixed scale.

The program is very flexible and interactive. The quadrature rule is defined by three files, to be read at input, and the maximum degree is specified by the user as well.

Note that the three files that define the quadrature rule are assumed to have related names, of the form

• prefix_x.txt
• prefix_w.txt
• prefix_r.txt

For information on the form of these files, see the QUADRATURE_RULES_TRI directory listed below.

The exactness results are written to an output file with the corresponding name:

• prefix_exact.txt

Usage:

triangle_exactness ( 'prefix', degree_max )

• prefix is the common prefix for the files containing the abscissa, weight and region information of the quadrature rule;
• degree_max is the maximum total monomial degree to check. This should be a relatively small nonnegative number, particularly if the spatial dimension is high. A value of 5 or 10 might be reasonable, but a value of 50 or 100 is probably never a good input!

If the arguments are not supplied on the command line, the program will prompt for them.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

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

Related Data and Programs:

DUNAVANT, a MATLAB library which sets up a Dunavant quadrature rule for a triangle.

FEKETE, a MATLAB library which sets up a Fekete quadrature rule for a triangle.

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.

NINT_EXACTNESS, a MATLAB program which tests the exactness of integration rules over the interval, rectangle, or multidimensional rectangle.

NINT_EXACTNESS_MIXED, a MATLAB program which measures the polynomial exactness of a multidimensional quadrature rule based on a mixture of 1D quadrature rule factors.

PYRAMID_EXACTNESS, a MATLAB program which investigates the polynomial exactness of a quadrature rule for the pyramid.

QUADRATURE_RULES_TRI, a dataset directory which contains sets of files that define quadrature rules over the triangle.

SPHERE_EXACTNESS, a MATLAB program which tests the polynomial exactness of a quadrature rule for the unit sphere;

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 defines integrand functions for testing quadrature routines for the triangle.

TETRAHEDRON_EXACTNESS, a MATLAB program which investigates the polynomial exactness of a quadrature rule for the tetrahedron.

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

WANDZURA, a MATLAB library which sets up a Wandzura quadrature rule for a triangle.

Reference:

1. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.

Source Code:

• triangle_exactness.m, is the source code.

Examples and Tests:

The "W", "X" and "R" files associated with the point sets tested here may be found in the QUADRATURE_RULES_TRI directory.

• centroid_exact.txt.
• gauss4x4_exact.txt (the results here have significant error. The abscissas and weights were not available to the full precision.)
• gauss8x8_exact.txt (the results here have significant error. The abscissas and weights were not available to the full precision.)
• seven_point_exact.txt.
• strang1_exact.txt.
• strang2_exact.txt.
• strang3_exact.txt.
• strang4_exact.txt.
• strang5_exact.txt.
• strang6_exact.txt.
• strang7_exact.txt.
• strang8_exact.txt.
• strang9_exact.txt.
• strang10_exact.txt.
• toms584_19_exact.txt.
• toms612_19_exact.txt
• toms612_28_exact.txt.
• toms706_37_exact.txt.
• vertex_exact.txt.

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