Error Control

When numerical integration is used to evaluate $h(t)$, what is actually computed is $\hat{h}(t)=h(t)+\epsilon_I$, where $\epsilon_I$ is the numerical integration error. This error can, in principle, be made arbitrarily small, but at the expense of more work (computer time). Let $\hat{t} = t_\alpha+\epsilon_t$ be an approximation to $t_\alpha$ with error $\epsilon_t$. What is actually computed at each step in a combined numerical optimization-integration algorithm is

$$\hat{h}(\hat{t})=h(\hat{t})+\epsilon_I \approx \epsilon_t h'(t_\alpha)+\epsilon_I,$$ (1)

for sufficiently small $\vert\epsilon_t\vert$. In order to balance the numerical integration and optimization errors, an estimate for $\vert h'(t_\alpha)\vert$ is needed. This quantity does not need to be very accurate, so a simple difference quotient approximation, in the form

$$h'(t_\alpha) \approx \hat{h'}=\frac{\hat{h}(t_b)-\hat{h}(t_a)}{t_b-t_a},$$

can be determined using quantities from the Pegasus method iterations. If the Newton method is being used, then an accurate value of $h'(t_\alpha)$ is already available.

Given $\hat{h}'$ and $\tau$ (the desired error tolerance for $t_\alpha$), the numerical integration error tolerance must be set at a level at least as small as $\hat{h}'\tau$. Otherwise the numerical integration errors may dominate the total error in $\hat{h}(\hat{t})$ and it will not be possible to reliably determine when the optimization method has converged. The strategy used for the example test results in the next section was to set the error tolerance for the numerical integration at $\tau \vert\hat{h}'\vert$.