Levenberg Marquardt Algorithm for "non-square" systems - Any bounds on (the no. of unknowns, minus the no. of eqns.) ?

Hi,

This question concerns a nuclear physics problem, for which I need to have self consistent solutions to 7 coupled nonlinear equations, bearing 9 unknowns. This is obviously a "non-square" system, which I believed wasn't solvable, until I read MATLAB documentation, which claims that the Levenberg Marquardt algorithm can even work for a non square system. (This is my first exposure to this technique)

Physically, the problem is well posed - I have good initial guesses and expect that the solutions won't depart significantly from them. I also realize that the last statement doesn't guarantee that, viewed as a purely mathematical problem, it will have a solution; I realize that the maths problem is quite ill-posed.

I run fsolve with the above algorithm, and it DOES converge to an answer. I put this back into the equations, and the function value comes out to be of the order of 10^(-4), which is reasonably accurate for my purpose (the terms in the original equations were ~ 10^8, so I have an accuracy of ~10^(-12), which is good enough surely).

So, even though I (apparently) have a solution at hand, a natural question here is the believability of this answer, or rather of the claim that this algorithm can handle non square systems too.

Therefore, I would request any one familiar with this algorithm or the mathematical problem, to advice me with the following two issues -

1) Are there any constraints on (U-E) for this algorithm to work, where U = no. of unknowns, and E = no. of (nonlinear) equations? (I mean, it certainly cannot work reliably if I demand 10 unknowns from one coupled equation, connecting all of them).

2) Even though my initial guesses are quite reliable (from a physics viewpoint), what else determines the reliability of the answers that MATLAB is claiming ?

I tried digging out the answer from the references provided in the MATLAB documentation, but they are the original maths papers and naturally too hard to follow for someone who is from a different discipline.

Thanks

PS - Sorry for the long question, but I wanted to make my problem absolutely clear.