Optimal domain splitting in Chebyshev collocation

Chebfun uses a simple rule, essentially a binary search, to automatically split an interval when it detects that a piecewise Chebyshev polynomial representation will be more efficient than a global one. Given the complex singularity structure of the function being approximated, one can find an optimal splitting location explicitly. It turns out that Chebfun really does get the optimal location in most cases, albeit not in the most efficient manner. In cases where the function is expensive to evaluate, such as the solution to a differential equation, it can be preferable to use Chebyshev-Padé approximation to locate the complex singularities and split accordingly.