  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.
