NINT_EXACTNESS_TET
Exactness of Quadrature on Tetrahedrons

NINT_EXACTNESS_TET is a MATLAB program which investigates the polynomial exactness of a quadrature rule for a tetrahedron.

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

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

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

• prefix_exact.txt

Usage:

nint_exactness_tet ( 'prefix', degree_max )

prefix

the common prefix for the files containing the abscissa, weight and region information of the quadrature rule;

degree_max

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.

Related Data and Programs:

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.

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

KEAST is a MATLAB library of routines which set up a Keast quadrature rule for a tetrahedron.

NCC_TETRAHEDRON is a MATLAB library of routines which set up a Newton Cotes Closed quadrature rule for a tetrahedron.

NCO_TETRAHEDRON is a MATLAB library of routines which set up a Newton Cotes Open quadrature rule for a tetrahedron.

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

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

NINT_EXACTNESS_TRI is a MATLAB program which tests the exactness of integration rules over a triangle.

NINTLIB is a MATLAB library of routines which numerically estimate integrals in multiple dimensions.

QUADRATURE_RULES_TET is a dataset directory which contains sets of files that define quadrature rules over a tetrahedron.

QUADRULE is a MATLAB library of routines which define quadrature rules on a variety of intervals with different weight functions.

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

TEST_INT_TET is a MATLAB library of routines which define integrand functions for testing quadrature routines for a tetrahedron.

Reference:

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

Source Code:

• nint_exactness_tet.m, is the main program.
• ch_cap.m, capitalizes a single character.
• ch_eqi.m, is TRUE if two characters are equal, ignoring case.
• ch_to_digit.m, converts a character to a digit.
• comp_next.m, computes the compositions of the integer N into K parts.
• dtable_data_read.m, reads the data from a TABLE file.
• dtable_header_read.m, reads the header from a TABLE file.
• file_column_count.m, counts the columns in a file
• file_row_count.m, counts the rows in a file
• monomial_int_tet_unit, returns the integral of a monomial over the unit tetrahedron.
• monomial_quadrature_tet.m, applies a quadrature rule to a monomial in a tetrahedron.
• monomial_value.m, evaluates a monomial.
• r8mat_det_4d.m, computes the determinant of a 4 by 4 matrix.
• s_len_trim.m, returns the length of a string to the last nonblank.
• s_to_i4.m, converts a string to an I4.
• s_to_r8.m, converts a string to an R8.
• s_to_r8vec.m, converts a string to an R8VEC.
• s_word_count.m, returns the number of words in a string.
• tetrahedron_volume.m, returns the volume of a tetrahedron.
• tetrahedron_order4_physical_to_reference.m, maps physical points to reference points.
• timestamp.m, prints the current YMDHMS date as a time stamp.
• timestring.m, returns the current YMDHMS date as a string.
• tuple_next.m, computes the next element of a tuple space.

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.

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