PYRAMID_EXACTNESS
Precision Test for Pyramid Quadrature Rules

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

The integration region is:

       - ( 1 - Z ) <= X <= 1 - Z
       - ( 1 - Z ) <= Y <= 1 - Z
                 0 <= Z <= 1.
      

Usage:

pyramid_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:

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

PYRAMID_GRID, a FORTRAN90 library which computes a grid of points over the interior of the unit pyramid in 3D;

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

PYRAMID_MONTE_CARLO, a FORTRAN90 library which applies a Monte Carlo method to estimate integrals of a function over the interior of the unit pyramid in 3D;

PYRAMID_RULE, a FORTRAN90 program which can compute a quadrature rule over the interior of the unit pyramid in 3D.

QUADRATURE_RULES_PYRAMID, a dataset directory which contains quadrature rules 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_EXACTNESS, a FORTRAN90 program which investigates the polynomial exactness of a quadrature rule over the interior of the tetrahedron in 3D.

TRIANGLE_EXACTNESS, a FORTRAN90 program which investigates the polynomial exactness of a quadrature rule over the interior of a triangle in 2D.

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. 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:

• pyramid_exactness.f90, the source code.

Examples and Tests:

PYRAMID_L3X3_J3 is a pyramid rule formed by a conical product of a 3x3 Legendre rule and an order 3 Jacobi rule.

• pyramid_l3x3_j3_w.txt, the weight file.
• pyramid_l3x3_j3_x.txt, the abscissa file.
• pyramid_l3x3_j3_exactness.txt, the exactness file produced by the program.

List of Routines:

• MAIN is the main program for PYRAMID_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.
• DTABLE_DATA_READ reads data from a DTABLE file.
• DTABLE_HEADER_READ reads the header from a DTABLE file.
• 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_VALUE evaluates a monomial.
• PYRA_UNIT_MONOMIAL: monomial integral in a unit pyramid.
• PYRA_UNIT_VOLUME: volume of a unit pyramid with square base.
• R8_CHOOSE computes the binomial coefficient C(N,K) as an R8.
• R8_MOP returns the I-th power of -1 as an R8.
• 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.

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