The plain Monte Carlo algorithm samples points randomly from the integration region to estimate the integral and its error. Using this algorithm the estimate of the integral E(f; N) for N randomly distributed points x_i is given by,
E(f; N) = = V <f> = (V / N) \sum_i^N f(x_i)
where V is the volume of the integration region. The error on this estimate \sigma(E;N) is calculated from the estimated variance of the mean,
\sigma^2 (E; N) = (V^2 / N^2) \sum_i^N (f(x_i) - <f>)^2.
For large N this variance decreases asymptotically as \Var(f)/N, where \Var(f) is the true variance of the function over the integration region. The error estimate itself should decrease as \sigma(f)/\sqrt{N}. The familiar law of errors decreasing as 1/\sqrt{N} applies—to reduce the error by a factor of 10 requires a 100-fold increase in the number of sample points.
The functions described in this section are declared in the header file gsl_monte_plain.h.
This function allocates and initializes a workspace for Monte Carlo integration in dim dimensions.
This function initializes a previously allocated integration state. This allows an existing workspace to be reused for different integrations.
This routines uses the plain Monte Carlo algorithm to integrate the function f over the dim-dimensional hypercubic region defined by the lower and upper limits in the arrays xl and xu, each of size dim. The integration uses a fixed number of function calls calls, and obtains random sampling points using the random number generator r. A previously allocated workspace s must be supplied. The result of the integration is returned in result, with an estimated absolute error abserr.