Numerical evaluation of high-dimensional integrals

the first column gives the number of points used in the calculation. The results in the last column, although not as good as Simpson, are considerably better than those obtained by the plain trapezoid rule. Reason for the improvement is obvious: the integrand of (2) has the property (shared with the Jacobian of transformation (ii)) that it vanishes, together with all its derivatives, at the endpoints of the interval of integration. As we shall frequently encounter this property, we shall briefly refer to it as property P. For integrands with property P the trapezoid rule is just as good as Simpson. For if f(x) has property P then we can replace f f(x) dx by either of the integrals 1 ~ ~~~~~~~ 1?1/ N (3) I f(x) dx, f*(x) dx