HYPERBALL_VOLUME_MONTE_CARLO is a C 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 FORTRAN90 version and a MATLAB version.
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hyperball_volume_monte_carlo_test
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