# pyramid_exactness

pyramid_exactness, a Python code which measures the precision of a quadrature rule defined 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).

### Languages:

pyramid_exactness is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave versionand a Python version.

### Related Data and Programs:

cube_exactness, a Python code which investigates the polynomial exactness of quadrature rules over the interior of a cube in 3d.

hypercube_exactness, a Python code 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 Python code which returns a Felippa quadrature rule for approximating integrals over the interior of a pyramid in 3d.

pyramid_grid, a Python code which computes a grid of points over the interior of the unit pyramid in 3d;

pyramid_integrals, a Python code which returns the exact value of the integral of any monomial over the interior of the unit pyramid in 3d.

pyramid_jaskowiec_rule, a Python code which returns quadrature rules, with exactness up to total degree 20, over the interior of a pyramid in 3D, by Jan Jaskowiec, Natarajan Sukumar.

pyramid_monte_carlo, a Python 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 Python code which computes a conical product quadrature rule over the interior of the unit pyramid in 3d.

pyramid_witherden_rule, a Python code which returns a Witherden quadrature rule, with exactness up to total degree 10, over the interior of a pyramid in 3D.

sphere_exactness, a Python code which tests the polynomial exactness of a quadrature rule over the surface of the unit sphere in 3d.

square_exactness, a Python code which investigates the polynomial exactness of quadrature rules for f(x,y) over the interior of a square (rectangle/quadrilateral) in 2d.

tetrahedron_exactness a Python code which investigates the polynomial exactness of a quadrature rule over the interior of a tetrahedron in 3d.

triangle_exactness, a Python code which investigates the monomial exactness quadrature rule over the interior of a triangle in 2d.

wedge_exactness, a Python 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:

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

Last revised on 16 May 2023.