pyramid_rule, a FORTRAN90 code which generates a quadrature rule over the interior of the unit pyramid in 3D.
The quadrature rules generated 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 Zwhere 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).
pyramid_rule legendre_order jacobi_order filenamewhere
The computer code and data files described and made available on this web page are distributed under the MIT license
pyramid_rule is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.
CIRCLE_RULE, a FORTRAN90 code which computes quadrature rules over the circumference of the unit circle in 2D.
CUBE_FELIPPA_RULE, a FORTRAN90 code which returns the points and weights of a Felippa quadrature rule over the interior of a cube in 3D.
DISK_RULE, a FORTRAN90 code which computes quadrature rules over the unit disk in 2D.
GEOMETRY, a FORTRAN90 code which performs geometric calculations in 2, 3 and N dimensional space.
JACOBI_RULE, a FORTRAN90 code which can compute and print a Gauss-Jacobi quadrature rule.
LEGENDRE_RULE, a FORTRAN90 code which can compute and print a Gauss-Legendre quadrature rule.
LEGENDRE_RULE_FAST, a FORTRAN90 code which uses a fast (order N) algorithm to compute a Gauss-Legendre quadrature rule of given order.
PYRAMID_EXACTNESS, a FORTRAN90 code which investigates the polynomial exactness of a quadrature rule over the interior of the unit pyramid in 3D.
PYRAMID_FELIPPA_RULE, a FORTRAN90 code which returns Felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3D.
PYRAMID_GRID, a FORTRAN90 code which computes a grid of points over the interior of the unit pyramid in 3D;
PYRAMID_INTEGRALS, a FORTRAN90 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 FORTRAN90 code which applies a Monte Carlo method to estimate integrals of a function over the interior of the unit pyramid in 3D;
SQUARE_FELIPPA_RULE, a FORTRAN90 code which returns the points and weights of a Felippa quadrature rule over the interior of a square in 2D.
STROUD, a FORTRAN90 code which defines quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.
TETRAHEDRON_FELIPPA_RULE, a FORTRAN90 code which returns Felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.
TRIANGLE_FELIPPA_RULE, a FORTRAN90 code which returns Felippa's quadratures rules for approximating integrals over the interior of a triangle in 2D.
WEDGE_FELIPPA_RULE, a FORTRAN90 code which returns quadratures rules for approximating integrals over the interior of the unit wedge in 3D.