A Rigorous Computational Approach to Linear Response

We present a general framework in which the formula describing the linear response of the physical measure of a perturbed system can be obtained. In this general setting we obtain an algorithm to rigorously compute the linear response. We apply our results to expanding circle maps. In particular, we present an example where we compute, up to a pre-specified error in the $L^{\infty}$-norm, the derivative of the physical measure of an expanding circle map under stochastic perturbations.