Algorithms and Tests

The assumed input for the algorithms that are described in this section will consist of the list ${\bf a}$, ${\bf b}$, $\mbox{\boldmath$\Sigma$}$, $\nu$, $\mbox{\boldmath$\delta$}$, $\epsilon$ and $N_{max}$. The vector $\mbox{\boldmath$\delta$}$ is assumed to be ${\bf0}$ for MVT problems. The parameters $\epsilon$, an absolute error tolerance parameter, and $N_{max}$, a limit on the number of simulations or integrand values allowed, are introduced because of the nature of the MVT algorithms, where, for many practical problems, we cannot expect to feasibly compute high accuracy MVT values. We expect a reliable algorithm to produce output consisting of an approximation $\hat{\bf T}$ and an approximate error $\hat{\epsilon}$ satisfying $\vert{\bf T}- \hat{\bf T}\vert < \hat{\epsilon}$ most of the time, and with $\hat{\epsilon}\leq \epsilon$ in those cases where $N_{max}$ is large enough to allow algorithm termination.

Figure 2: Average Monte Carlo MVT Algorithm Times, $\epsilon = 10^{-2}$