Computing the Bouligand derivative of a class of piecewise-differentiable flows

Event–selected $C^r$ vector fields yield piecewise-differentiable flows, which possess a continuous and piecewise-linear Bouligand (or B-)derivative; here we provide an algorithm for computing this B-derivative. The number of “pieces” of the piecewise-linear B-derivative is factorial ($d!$) in the dimension ($d$) of the space, precluding a polynomial-time algorithm. We show how an exponential number ($2^d$) of points can be used to represent the B-derivative as a piecewise-linear homeomorphism in such a way that evaluating the derivative reduces to linear algebra computations involving a matrix constructed from $d$ of these points.