PYRAMID_EXACTNESS
Precision Test for Pyramid Quadrature Rules

PYRAMID_EXACTNESS is a MATLAB program which measures the precision of a quadrature rule defined on a pyramid.

The integration region is:

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

When Z is zero, the integration region is a square lying in the (X,Y) plane, centered at (0,0,0) with "radius" 1. As Z increases to 1, the radius of the square diminishes, and when Z reaches 1, the square has contracted to the single point (0,0,1).

Usage:

pyramid_exactness ( 'filename', degree_max )

where

'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 FORTRAN90 version and a MATLAB version.

Related Data and Programs:

FELIPPA, a MATLAB library which defines quadrature rules for lines, triangles, quadrilaterals, pyramids, wedges, tetrahedrons and hexahedrons.

PYRAMID_RULE, a MATLAB program which can compute a quadrature rule for the unit pyramid.

QUADRATURE_RULES_PYRAMID, a dataset directory which contains quadrature rules for a pyramid with a square base.

SPHERE_EXACTNESS, a MATLAB program which tests the polynomial exactness of a quadrature rule for the unit sphere;

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

Reference:

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

Source Code:

pyramid_exactness.m, 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.

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

Last revised on 28 July 2009.

PYRAMID_EXACTNESS Precision Test for Pyramid Quadrature Rules