HYPERBALL_VOLUME_MONTE_CARLO
Monte Carlo Estimate of Hyperball Volume

HYPERBALL_VOLUME_MONTE_CARLO is a FORTRAN77 program which investigates the behavior of a Monte Carlo procedure when it is applied to compute the integral of a discontinuous function. In particular, our integration region is the M-dimensional unit hypercube and our function f(x) is 1 if the point x is inside the unit hyperball of radius 1, and 0 otherwise.

The program uses the Monte Carlo method to estimate the volume. Estimates are made starting with 2^0 (=1) points and doubling repeatedly up to 2^25 points.

Because the integrand is discontinuous, any quadrature rule based on the idea of interpolation will probably be unable to do a good job. A family of quadrature rules, which rely on increasing the order of interpolation to improve accuracy, will probably get increasingly bad answers.

By contrast, a basic Monte Carlo rule, which assumes nothing about the function, integrates this function just as well as it integrates most any other square-integrable function. (That's both the strength and weakness of the blunt instrument we call Monte Carlo integration.)

Usage:

hyperball_volume_monte_carlo dim_num seed

• dim_num is the spatial dimension.
• seed is an optional seed for the random number generator. If it is not specified on the command line, a default value is used.

Licensing:

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

Languages:

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

Related Data and Programs:

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

CIRCLE_MONTE_CARLO, a FORTRAN77 library which applies a Monte Carlo method to estimate the integral of a function on the circumference of the unit circle in 2D;

CUBE_MONTE_CARLO, a FORTRAN77 library which applies a Monte Carlo method to estimate the integral of a function over the interior of the unit cube in 3D;

DISK_MONTE_CARLO, a FORTRAN77 library which applies a Monte Carlo method to estimate the integral of a function over the interior of the unit disk in 2D;

ELLIPSE_MONTE_CARLO a FORTRAN77 library which uses the Monte Carlo method to estimate the value of integrals over the interior of an ellipse in 2D.

ELLIPSOID_MONTE_CARLO a FORTRAN77 library which uses the Monte Carlo method to estimate the value of integrals over the interior of an ellipsoid in M dimensions.

HYPERBALL_INTEGRALS, a FORTRAN77 library which defines test functions for integration over the interior of the unit hyperball in M dimensions.

HYPERBALL_MONTE_CARLO, a FORTRAN77 library which applies a Monte Carlo method to estimate the integral of a function over the interior of the unit hyperball in M dimensions;

HYPERSPHERE_MONTE_CARLO, a FORTRAN77 library which applies a Monte Carlo method to estimate the integral of a function on the surface of the unit sphere in M dimensions;

LINE_MONTE_CARLO, a FORTRAN77 library which uses the Monte Carlo method to estimate integrals over the length of the unit line in 1D.

POLYGON_MONTE_CARLO, a FORTRAN77 library which applies a Monte Carlo method to estimate the integral of a function over the interior of a polygon in 2D.

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;

SIMPLEX_MONTE_CARLO, a FORTRAN77 library which uses the Monte Carlo method to estimate integrals over the interior of the unit simplex in M dimensions.

SPHERE_MONTE_CARLO, a FORTRAN77 library which applies a Monte Carlo method to estimate the integral of a function over the surface of the unit sphere in 3D;

SPHERE_TRIANGLE_MONTE_CARLO, a FORTRAN77 library which applies a Monte Carlo method to estimate the integral of a function over a spherical triangle on the surface of the unit sphere in 3D;

SQUARE_MONTE_CARLO, a FORTRAN77 library which applies a Monte Carlo method to estimate the integral of a function over the interior of the unit square in 2D.

TETRAHEDRON_MONTE_CARLO, a FORTRAN77 library which uses the Monte Carlo method to estimate integrals over a tetrahedron.

TRIANGLE_MONTE_CARLO, a FORTRAN77 library which uses the Monte Carlo method to estimate integrals over the interior of a triangle in 2D.

WEDGE_MONTE_CARLO, a FORTRAN77 library which uses the Monte Carlo method to estimate integrals over the interior of the unit wedge in 3D.

Reference:

1. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.
2. Gerald Folland,
   How to Integrate a Polynomial Over a Sphere,
   American Mathematical Monthly,
   Volume 108, Number 5, May 2001, pages 446-448.

Source Code:

• hyperball_volume_monte_carlo.f, the source code.
• ball_volume_monte_carlo.sh, commands to compile the source code.

Examples and Tests:

• hyperball_volume_monte_carlo_output.txt, the output file.

List of Routines:

• MAIN is the main program for HYPERBALL_VOLUME_MONTE_CARLO.
• HYPERBALL01_INDICATOR evaluates the unit hyperball indicator function.
• HYPERBALL01_VOLUME computes the volume of the unit hyperball.
• R8MAT_UNIFORM_01 returns a unit pseudorandom R8MAT.
• S_TO_I4 reads an I4 from a string.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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