TETRAHEDRON_EXACTNESS
Exactness of Quadrature on Tetrahedrons

TETRAHEDRON_EXACTNESS is a FORTRAN90 program which investigates the polynomial exactness of a quadrature rule over the interior of a tetrahedron in 3D.

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 tetrahedron, the degree of a monomial term is the sum of the exponents of x, y and z. Thus, for instance, the DEGREE of

x²yz⁵

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

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

• prefix_exact.txt

Usage:

tetrahedron_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. 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:

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

Related Data and Programs:

CUBE_EXACTNESS, a FORTRAN90 library which investigates the polynomial exactness of quadrature rules over the interior of a cube in 3D.

HYPERCUBE_EXACTNESS, a FORTRAN90 program which measures the monomial exactness of an M-dimensional quadrature rule over the interior of the unit hypercube in M dimensions.

PYRAMID_EXACTNESS, a FORTRAN90 program which investigates the polynomial exactness of a quadrature rule over the interior of the unit pyramid in 3D.

SPHERE_EXACTNESS, a FORTRAN90 program which tests the polynomial exactness of a quadrature rule over the surface of the unit sphere in 3D.

SQUARE_EXACTNESS, a FORTRAN90 library which investigates the polynomial exactness of quadrature rules for f(x,y) over the interior of a rectangle in 2D.

TETRAHEDRON_ARBQ_RULE, a FORTRAN90 library which returns quadrature rules, with exactness up to total degree 15, over the interior of a tetrahedron in 3D, by Hong Xiao and Zydrunas Gimbutas.

TETRAHEDRON_FELIPPA_RULE, a FORTRAN90 library which returns Felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.

TETRAHEDRON_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3D.

TETRAHEDRON_KEAST_RULE, a FORTRAN90 library which defines ten quadrature rules, with exactness degrees 0 through 8, over the interior of a tetrahedron in 3D.

TETRAHEDRON_MONTE_CARLO, a FORTRAN90 program which uses the Monte Carlo method to estimate integrals over a tetrahedron.

TETRAHEDRON_NCC_RULE, a FORTRAN90 library which defines Newton-Cotes closed quadrature rules on a tetrahedron.

TETRAHEDRON_NCO_RULE, a FORTRAN90 library which defines Newton-Cotes open quadrature rules on a tetrahedron.

TRIANGLE_EXACTNESS, a FORTRAN90 program which investigates the polynomial exactness of a quadrature rule for the triangle.

WEDGE_EXACTNESS, a FORTRAN90 program which investigates the monomial exactness of a quadrature rule over the interior of the unit wedge in 3D.

Reference:

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

Source Code:

• tetrahedron_exactness.f90, 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_TET directory.

• keast0_exact.txt.
• keast1_exact.txt.
• keast2_exact.txt.
• keast3_exact.txt.
• keast4_exact.txt.
• keast5_exact.txt.
• keast6_exact.txt.
• keast7_exact.txt.
• keast8_exact.txt.
• keast9_exact.txt.
• ncc0_exact.txt.
• ncc1_exact.txt.
• ncc2_exact.txt.
• ncc3_exact.txt.
• ncc4_exact.txt.
• ncc5_exact.txt.
• ncc6_exact.txt.
• nco0_exact.txt.
• nco1_exact.txt.
• nco2_exact.txt.
• nco3_exact.txt.
• nco4_exact.txt.
• nco5_exact.txt.
• nco6_exact.txt.

List of Routines:

• MAIN is the main program for TETRAHEDRON_EXACTNESS.
• CH_CAP capitalizes a single character.
• CH_EQI is a case insensitive comparison of two characters for equality.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• COMP_NEXT computes the compositions of the integer N into K parts.
• FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
• FILE_ROW_COUNT counts the number of row records in a file.
• GET_UNIT returns a free FORTRAN unit number.
• MONOMIAL_INT_TET_UNIT integrates a monomial over the unit tetrahedron.
• MONOMIAL_QUADRATURE_TET applies quadrature to a monomial in a tetrahedron.
• MONOMIAL_VALUE evaluates a monomial.
• R8MAT_DATA_READ reads data from an R8MAT file.
• R8MAT_DET_4D computes the determinant of a 4 by 4 matrix.
• R8MAT_HEADER_READ reads the header from an R8MAT file.
• 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.
• TETRAHEDRON_ORDER4_PHYSICAL_TO_REFERENCE maps physical points to reference points.
• TETRAHEDRON_VOLUME computes the volume of a tetrahedron.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.