HYPERBALL_VOLUME_MONTE_CARLO is a FORTRAN90 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.)
hyperball_volume_monte_carlo dim_num seedwhere
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
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.
BALL_MONTE_CARLO, a FORTRAN90 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 FORTRAN90 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 FORTRAN90 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 FORTRAN90 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 FORTRAN90 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 FORTRAN90 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 FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit hyperball in M dimensions.
HYPERBALL_MONTE_CARLO, a FORTRAN90 library which applies a Monte Carlo method to estimate the integral of a function over the interior of the unit hyperball in M dimensions;
HYPERBALL_VOLUME_QUAD, a FORTRAN90 program which applies a quadrature rule to estimate the volume of the unit hyperball in M dimensions.
HYPERCUBE_MONTE_CARLO, a FORTRAN90 library which applies a Monte Carlo method to estimate the integral of a function over the interior of the unit hypercube in M dimensions;
HYPERSPHERE_MONTE_CARLO, a FORTRAN90 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 FORTRAN90 library which applies a Monte Carlo method to estimate the integral of a function over the length of the unit line in 1D;
POLYGON_MONTE_CARLO, a FORTRAN90 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 FORTRAN90 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 FORTRAN90 library which uses the Monte Carlo method to estimate integrals over the interior of the unit simplex in M dimensions.
SPHERE_MONTE_CARLO, a FORTRAN90 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 FORTRAN90 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 FORTRAN90 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 FORTRAN90 library which uses the Monte Carlo method to estimate integrals over the interior of the unit tetrahedron in 3D.
TRIANGLE_MONTE_CARLO, a FORTRAN90 library which uses the Monte Carlo method to estimate integrals over the interior of the unit triangle in 2D.
WEDGE_MONTE_CARLO, a FORTRAN90 library which uses the Monte Carlo method to estimate integrals over the interior of the unit wedge in 3D.
You can go up one level to the FORTRAN90 source codes.