  We present the Finite Difference Non-Intrusive Least Squares Shadowing
(FD-NILSS) algorithm for computing sensitivities of long-time averaged
quantities in chaotic dynamical systems. FD-NILSS does not require tangent
solvers, and can be implemented with little modification to existing numerical
simulation software. We also give a formula for solving the least-squares
problem in FD-NILSS, which can be applied in NILSS as well. Finally, we apply
FD-NILSS for sensitivity analysis of a chaotic flow over a 3-D cylinder at
Reynolds number 525, where FD-NILSS computes accurate sensitivities and the
computational cost is in the same order as the numerical simulation.
