WEDGE_EXACTNESS
Precision Test for Wedge Quadrature Rules

WEDGE_EXACTNESS is a FORTRAN90 program which measures the precision of a quadrature rule over the interior of the unit wedge in 3D.

The interior of the unit wedge in 3D is defined by the constraints:

        0 <= X
        0 <= Y
             X + Y <= 1
       -1 <= Z <= +1
      

Usage:

wedge_exactness filename degree_max

• filename is the common prefix of the filenames containing the abscissas and the weights of the quadrature rule.
• degree_max is the maximum degree of the monomials to be checked.

Licensing:

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

Languages:

WEDGE_EXACTNESS is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

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

HYPERCUBE_EXACTNESS, a C++ 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 C++ program which investigates the polynomial exactness of a quadrature rule over the interior of the unit pyramid in 3D.

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

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

TETRAHEDRON_EXACTNESS, a C++ program which investigates the polynomial exactness of a quadrature rule over the interior of a tetrahedron in 3D.

TRIANGLE_EXACTNESS, a C++ program which investigates the polynomial exactness of a quadrature rule over the interior of a triangle in 2D.

WEDGE_FELIPPA_RULE, a C++ library which returns quadratures rules for approximating integrals over the interior of the unit wedge in 3D.

WEDGE_GRID, a C++ library which computes a grid of points over the interior of the unit wedge in 3D.

WEDGE_INTEGRALS, a C++ library which returns the exact value of the integral of any monomial over the interior of the unit wedge in 3D.

WEDGE_MONTE_CARLO, a C++ library which uses the Monte Carlo method to estimate integrals over the interior of the unit wedge in 3D.

Reference:

1. Carlos Felippa,
   A compendium of FEM integration formulas for symbolic work,
   Engineering Computation,
   Volume 21, Number 8, 2004, pages 867-890.
2. Arthur Stroud,
   Approximate Calculation of Multiple Integrals,
   Prentice Hall, 1971,
   ISBN: 0130438936,
   LC: QA311.S85.

Source Code:

• wedge_exactness.cpp, the source code.
• wedge_exactness.sh, commands to compile the source code.

Examples and Tests:

WEDGE_FELIPPA_3x2 is a quadrature rule based on a 3 point triangle rule and a 2 point Legendre rule for the line.

• wedge_felippa_3x2_w.txt, the weight file.
• wedge_felippa_3x2_x.txt, the abscissa file.
• wedge_felippa_3x2_output.txt, the exactness file produced by the program.

List of Routines:

• MAIN is the main program for WEDGE_EXACTNESS.
• CH_CAP capitalizes a single character.
• CH_EQI is true if two characters are equal, disregarding case.
• CH_TO_DIGIT returns the value of a base 10 digit.
• COMP_NEXT computes the compositions of the integer N into K parts.
• 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.
• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the minimum of two I4's.
• MONOMIAL_VALUE evaluates a monomial.
• R8MAT_DATA_READ reads the data from an R8MAT file.
• R8MAT_HEADER_READ reads the header from an R8MAT file.
• R8VEC_DOT_PRODUCT 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.
• WEDGE01_INTEGRAL returns the integral of a monomial in the unit wedge in 3D.
• WEDGE01_VOLUME returns the volume of the unit wedge in 3D.

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