Let $F : R^k \to R^k$ be a smooth map whose Jacobian 
$J(F): R^k \to R$ vanishes on a discrete set $S$, so that if 
$O$ is the complement of $S$, then $f: O \to R^k$, the 
restriction of $F$, is a local diffeomorphism, and in particular 
there are only finitely many points $x$ inside the unit ball of 
$R^k$ such that $f(x) = 0$. What I would like to know is if 
there any a good algorithm for finding such points $x$. 
Of course, if $k=1$ this is easy, using the Intermediate Value 
Theorem; just partition the interval $[-1,1]$ into a reasonably large
number $N$ of subintervals of length $2/N$ and test on which 
subintervals $f$ changes sign, and then use bisection to locate 
a zero in those subintervals. If $k>1$ I suspect that some sort 
of application of Marching Cubes may work, but the usual search 
methods have not turned up anything. 



Algorithm for finding inverse images of a local diffeomorphism
With a Lipschitz bound on the derivative, Newton's method gives an algorithm to efficiently approximate $F^{-1}$. [Hubbard's calculus textbook](http://www.math.cornell.edu/~hubbard/vectorcalculus.html) has a write-up using this perspective, viewing the result as a consequence of Kantorovich's Theorem. There's a fairly extensive analysis of the absolute error. 


