# pyramid_rule

pyramid_rule, a MATLAB code which generates a quadrature rule over the interior of the unit pyramid in 3D.

The quadrature rules generated by PYRAMID_RULE are all examples of conical product rules, and involve a kind of direct product of the form:

Legendre rule in X * Legendre rule in Y * Jacobi rule in Z
where the Jacobi rule includes a factor of (1-Z)^2.

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_rule ( legendre_order, jacobi_order, 'filename' )
where
• legendre_order is the order of the 1D Legendre quadrature rule to be used. The X and Y dimensions will use a product of this rule.
• jacobi_order is the order of the 1D Jacobi quadrature rule to be used. The Z dimension will use this rule which will include a factor of (1-X)^2 which accounts for the narrowing of the pyramid.
• 'filename' is the common prefix for the files containing the region, weight and abscissa information of the quadrature rule;

### Languages:

pyramid_rule is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

annulus_rule, a MATLAB code which computes a quadrature rule for estimating integrals of a function over the interior of a circular annulus in 2d.

circle_rule, a MATLAB code which computes a quadrature rule over the circumference of the unit circle in 2d.

cube_felippa_rule, a MATLAB code which returns the points and weights of a felippa quadrature rule over the interior of a cube in 3d.

disk_rule, a MATLAB code which computes quadrature rules over the interior of a disk in 2d.

geometry, a MATLAB code which performs geometric calculations in 2, 3 and n dimensional space.

jacobi_rule, a MATLAB code which can compute and print a gauss-jacobi quadrature rule.

legendre_rule, a MATLAB code which can compute and print a gauss-legendre quadrature rule.

pyramid_exactness, a MATLAB code which investigates the polynomial exactness of a quadrature rule over the interior of the unit pyramid in 3d.

pyramid_felippa_rule, a MATLAB code which returns felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3d.

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

pyramid_integrals, a MATLAB 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 MATLAB code which applies a monte carlo method to estimate integrals of a function 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.

square_felippa_rule, a MATLAB code which returns the points and weights of a felippa quadrature rule over the interior of a square in 2d.

stroud, a MATLAB code which defines quadrature rules for a variety of unusual areas, surfaces and volumes in 2d, 3d and n-dimensions.

tetrahedron_felippa_rule, a MATLAB code which returns felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3d.

triangle_fekete_rule, a MATLAB code which defines fekete rules for quadrature or interpolation over the interior of a triangle in 2d.

triangle_felippa_rule, a MATLAB code which returns felippa's quadratures rules for approximating integrals over the interior of a triangle in 2d.

wedge_felippa_rule, a MATLAB code which returns quadratures rules for approximating 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.

04 March 2019