  We devise the fast linear response algorithm for differentiating fractal
invariant measures of chaos with respect to perturbations of governing
equations. We first derive the first computable expansion formula for the
unstable divergence, then a `fast' formula by renormalized second-order tangent
equations. The algorithm's cost is solving only one orbit and $u$, the unstable
dimension, many first-order and second-order tangent equations; the main error
is the sampling error of the orbit. We demonstrate it on an numerical example
which is difficult for previous methods.
