PYRAMID_INTEGRALS
Integrals Inside the Unit Pyramid in 3D

PYRAMID_INTEGRALS is a FORTRAN77 library which returns the exact value of the integral of any monomial x^i y^j z^k over the interior of the unit pyramid in 3D.

The unit pyramid has a square base of area 4, and a height of 1. Specifically, the integration region is:

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

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

PYRAMID_INTEGRALS is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

BALL_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of the unit ball in 3D.

CIRCLE_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the circumference of the unit circle in 2D.

CUBE_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of the unit cube in 3D.

DISK_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of the unit disk in 2D.

HYPERBALL_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of the unit hyperball in M dimensions.

HYPERCUBE_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of the unit hypercube in M dimensions.

HYPERSPHERE_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the surface of the unit hypersphere in M dimensions.

LINE_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the unit line segment in 1D.

POLYGON_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of a polygon in 2D.

PYRAMID_EXACTNESS, a FORTRAN77 program which investigates the monomial exactness of a quadrature rule over the interior of a pyramid in 3D.

PYRAMID_FELIPPA_RULE, a FORTRAN77 library which returns Felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3D.

PYRAMID_GRID, a FORTRAN77 library which computes a grid of points over the interior of the unit pyramid in 3D;

PYRAMID_MONTE_CARLO, a FORTRAN77 library which applies a Monte Carlo method to estimate integrals of a function over the interior of the unit pyramid in 3D;

PYRAMID_RULE, a FORTRAN77 program which computes a quadrature rule over the interior of the unit pyramid in 3D;

SIMPLEX_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of the unit simplex in M dimensions.

SPHERE_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the surface of the unit sphere in 3D.

SQUARE_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of the unit square in 2D.

TETRAHEDRON_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3D.

TRIANGLE_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of the unit triangle in 2D.

WEDGE_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of the unit wedge in 3D.

Source Code:

• pyramid_integrals.f, the source code.
• pyramid_integrals.sh, BASH commands to compile the source code.

Examples and Tests:

• pyramid_integrals_prb.f, a sample calling program.
• pyramid_integrals_prb.sh, BASH commands to compile and run the sample program.
• pyramid_integrals_prb_output.txt, the output file.

List of Routines:

• MONOMIAL_VALUE evaluates a monomial.
• PYRAMID01_INTEGRAL: monomial integral in a unit pyramid.
• PYRAMID01_SAMPLE: sample the unit pyramid.
• PYRAMID01_VOLUME: volume of a unit pyramid with square base.
• R8_CHOOSE computes the binomial coefficient C(N,K) as an R8.
• R8_MOP returns the I-th power of -1 as an R8.
• R8MAT_TRANSPOSE_PRINT prints an R8MAT, transposed.
• R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.
• R8MAT_UNIFORM_01 fills an R8MAT with unit pseudorandom numbers.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN77 source codes.