NINT_EXACTNESS_TRI
Exactness of Quadrature on Triangles

NINT_EXACTNESS_TRI is a C++ program which investigates the polynomial exactness of a multidimensional 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:

nint_exactness_tri 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. 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.

Related Data and Programs:

DUNAVANT is a C++ library which sets up a Dunavant quadrature rule for a triangle.

FEKETE is a C++ library which sets up a Fekete quadrature rule for a triangle.

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

INTEGRAL_TEST is a FORTRAN90 program which uses test integrals to measure the effectiveness of certain sets of quadrature rules.

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

NINT_EXACTNESS_MIXED is a C++ program which measures the polynomial exactness of a multidimensional quadrature rule based on a mixture of 1D quadrature rule factors.

NINT_EXACTNESS_PYRAMID is a C++ program which investigates the polynomial exactness of a quadrature rule for the pyramid.

NINT_EXACTNESS_TET is a C++ program which tests the polynomial exactness of integration rules for the tetrahedron.

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

NINTLIB is a C++ library which numerically estimates integrals in multiple dimensions.

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

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

STROUD is a FORTRAN90 library which defines quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and multiple dimensions.

TEST_TRI_INT is a C++ library which defines integrand functions for testing quadrature routines for the triangle.

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

WANDZURA is a C++ 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:

• nint_exactness_tri.C, the source code.
• nint_exactness_tri.csh, commands to compile 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 (the results here don't seem as good as they should be.)
• toms612_28_exact.txt.
• toms706_37_exact.txt.
• vertex_exact.txt.

List of Routines:

• MAIN is the main program for NINT_EXACTNESS_TRI.
• CH_CAP capitalizes a single character.
• CH_EQI is true if two characters are equal, disregarding case.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• COMP_NEXT computes the compositions of the integer N into K parts.
• DTABLE_DATA_READ reads the data from a DTABLE file.
• DTABLE_HEADER_READ reads the header from a DTABLE file.
• FILE_COLUMN_COUNT counts the columns in the first line of a file.
• FILE_ROW_COUNT counts the number of row records in a file.
• MONOMIAL_INT_TRI_UNIT integrates a monomial over the unit triangle.
• MONOMIAL_QUADRATURE_TRI applies quadrature to a monomial in a triangle.
• MONOMIAL_VALUE evaluates a monomial.
• R8VEC_DOT computes the dot product of a pair of R8VEC's.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• S_TO_I4 reads an I4 from a string.
• S_TO_R8 reads an R8 from a string.
• S_TO_R8VEC reads an R8VEC from a string.
• S_WORD_COUNT counts the number of "words" in a string.
• 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_ORDER3_PHYSICAL_TO_REFERENCE maps physical points to reference points.

You can go up one level to the C++ source codes.