  SRB measures are limiting stationary distributions describing the statistical
behavior of chaotic dynamical systems. Directional derivatives of SRB measure
densities conditioned on unstable manifolds are critical in the sensitivity
analysis of hyperbolic chaos. These derivatives, known as the SRB density
gradients, are by-products of the regularization of Lebesgue integrals
appearing in the original linear response expression. In this paper, we propose
a novel trajectory-driven algorithm for computing the SRB density gradient
defined for systems with high-dimensional unstable manifolds. We apply the
concept of measure preservation together with the chain rule on smooth
manifolds. Due to the recursive one-step nature of our derivations, the
proposed procedure is memory-efficient and can be naturally integrated with
existing Monte Carlo schemes widely used in computational chaotic dynamics. We
numerically show the exponential convergence of our scheme, analyze the
computational cost, and present its use in the context of Monte Carlo
integration.
