Solving an overdetermined system of nonlinear equations I'm wondering what the "best" way to approach solving a system of the following form would be:
$A_1X + Be^{CY} = A_2$
$A_3X + Be^{CY} = A_4$
$A_5X + Be^{CY} = A_6$
etc.
EDIT: Coefficients $A_i, B, C$ are all real numbers - X and Y are unknowns.
With this system it looks like I could take the log of both sides and then use least squares to fit a line to the log of the data.  What would be the best way to go about it in the more general case if it were not possible to linearize the equation in that way? Apply Newton's method?
 A: If your system were linear (for instance, because of the change of variables suggested by @anon), then the right way to go would be to solve the system in the least-squares sense. In general, an over-determined system has no solution, so you want to get "as close as possible", i.e., minimize the squared $\ell_2$-norm of the residual. To do this, you can use a multitude of methods (see for example the excellent book by Lawson and Hanson: http://www.ec-securehost.com/SIAM/CL15.html). If your coefficient matrix is anywhere near large and sparse I'd recommend Mike Saunders' LSQR or MINRES: http://www.stanford.edu/group/SOL/software).
Now you can also attack your nonlinear system directly using a nonlinear least-squares method such as that of Levenberg and Marquardt (I believe Matlab has an implementation of it, as does MINPACK: http://www.netlib.org/minpack), Gauss-Newton or NL2SOL (http://dl.acm.org/citation.cfm?id=355966). The basic idea is similar to Newton's method but at each iteration, a linear least-squares problem is solved.
