We devise a new algorithm, called the fast linear response algorithm, for accurately differentiating SRB measures with respect to some parameters of the dynamical system, where SRB measures are fractal limiting stationary measures of chaotic systems. The algorithm is illustrated in an example which is difficult for previous methods.

The core of our algorithm is the first numerical treatment of the unstable divergence, a central object in the linear response theory for fractal attractors. We first derive an computable expansion formula of the unstable divergence, then we give a new characterization of the expansion by the renormalized second-order tangent equations, whose second derivative is taken in a modified shadowing direction, computed by the non-intrusive shadowing algorithm.

The algorithm works for chaos on Riemannian manifolds with any unstable dimension, $u$. The algorithm is efficient and robust: its main cost is solving $u$ many first-order and second-order tangent equations, and it does not compute oblique projections. The convergence to the true derivative is proved for uniform hyperbolic systems.  

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