which also coincides with $\Im(\delta) = 0$ so the box size is easily set as equal to $L = r \sqrt{2 \delta_A}$.

Optimisations {#sec:optimisations}
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S91 describes a number of optimisations that can be used in the FGT with source-dependent scales:

1.  S91 introduces a single set of Taylor coefficients, positioned at the center of the $uv$ plane, to capture Gaussians with the largest scales. Visibilities larger than an upper scale size $\delta_j > b$ can be added to this single set, rather than the sets associated with the boxes. S91 recommends the upper scale size be at least 10% of the size of the evaluation region, in order to allow for practical precision in evaluating the large number of Taylor coefficients. In practice, most array configurations would have relatively few baselines with such a large $w$ that their convolution functions would span $>$ 10% of the $uv$ plane, so this optimisation has limited impact.

2.  S91 introduces a lower scale $a$ below which the Gaussian sum is directly evaluated. If there are only a few Gaussians with scales smaller than $a$, we can increase the box size, which reduces the memory and arithmetic capacity required, while incurring minimal penalty 