- But wait! Consider an integral in
**k**dimensions -- for other methods, we must make a grid in each direction with points for each coordinate. - Thus, error for Trapezoidal is
which has a slower
**n**dependence as**k**increases. - However, Monte Carlo method still keeps its
behavior and for , converges faster than Trapezoidal method.
- The other methods appear better than Monte
Carlo until larger
**k**. - However, in practice, this is not so
because there the implicit condition before the
error formulae even apply.
- For instance,
gives large () values of
**n**while gives ridiculous values for**n**() with**k=5**. - This example shows that the problem of multidimensional integrals is not a trivial generalization of that in one dimension.

Geoffrey Fox, Northeast Parallel Architectures Center at Syracuse University, gcf@npac.syr.edu