# pyramid_exactness

pyramid_exactness, a C code which measures the precision of a quadrature rule over the interior of a pyramid in 3D.

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.

### 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 C code which investigates the polynomial exactness of quadrature rules over the interior of a cube in 3D.

PYRAMID_FELIPPA_RULE, a C code which returns Felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3D.

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

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

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

PYRAMID_RULE, a C code 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 C code which tests the polynomial exactness of a quadrature rule over the surface of the unit sphere in 3D.

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

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

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

WEDGE_EXACTNESS, a C code 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:

Last revised on 29 July 2019.