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Importance sampling

The idea of importance sampling technique is based on diminution of variance by a proper choice of the density function g(x). The general solution of this problem could be in choosing

$$g(x) = \vert f(x)\vert \; / \int \! \vert f(x)\vert \,dx.$$

However this solution is useless because it returns us to the problem of evaluation of $f(x)$ integral and requires a generation of sampling points for complicated density function.

To bypass these problems VEGAS seeks this function in the factored form

$$g(x_1, x_2, \ldots ,x_n) = g_1(x_1) \, g_2(x_2) \ldots g_n(x_n).$$

The optimal functions $g_i(x)$ could be easily evaluated in terms of $f(x)$ [Lepage-1978,Press-1992]. VEGAS is an adaptive program. For the first iteration it puts $g_i(x)=1.$ The information about $f(x)$ which VEGAS gets during the iteration is used to refine the density function. Generally VEGAS performs several iterations improving the density function after each of them.

The following parameters manage VEGAS work:

1. Itmx is a number of iterations;
2. Ncall is a number of integrand calls for one iteration.