3 . Exception Handling in Derivative Computation with Nonarchimedean Calculus

While conventional computational differentiation based on the forward or reverse modes allows highly accurate computation of derivatives, there are situations where these modes fail to produce the values of derivatives, although the underlying function is differentiable. Typical examples of this phenomenon are connected to the occurrence of branch points in coding as in IF-ELSE structures as well as the occurrence of some non-differentiable parts that do not affect the differentiability of the end result. We show that based on ideas of nonarchimedean calculus on Levi-Civita fields, these problems can be avoided. It is possible to rigorously decide whether a function is differentiable or not at any given point, and if it is, to determine its derivatives to any order, even if the coding exhibits branch points or non-differentiable pieces. We give details of an implementation of the method and examples for its use for typical pathological problems.