Finite-difference Jacobian

An approximation to a Jacobian matrix obtained using finite differences instead of analytic partial derivatives. For a vector-valued function $F$, the forward-difference entry is $J_{ij}\approx (F_i(x+h{e}_j)-F_i(x))/h$, where $e_j$ is a coordinate basis vector. The step $h$ balances truncation and rounding errors; the resulting matrix can be used in a Newton root-finding iteration.