  Ruelle gave a formula for linear response of transitive Anosov
diffeomorphisms. Recently, practically computable realizations of Ruelle's
formula have emerged that potentially enable sensitivity analysis of certain
high-dimensional chaotic numerical simulations encountered in the applied
sciences. In this paper, we provide full mathematical justification for the
convergence of one such efficient computation, the space-split sensitivity, or
S3, algorithm. In S3, Ruelle's formula is computed as a sum of two terms
obtained by decomposing the perturbation vector field into a coboundary and a
remainder that is parallel to the unstable direction. Such a decomposition
results in a splitting of Ruelle's formula that is amenable to efficient
computation. We prove the existence of the S3 decomposition and the convergence
of the computations of both resulting components of Ruelle's formula.
