Okay, you asked for it. I'm going to assume this means you want to help. Hopefully you can save me the shame of additional newbie-ish posts to the help-gsl mailing list.

I don't fully understand the GSL calls that allow the govern to test whether the iteration (all the solvers are iterative) should stop.

One is based on "absolute error" and "relative error", while the others use gradient values.

Relevant docs:

http://www.gnu.org/software/gsl/manual/html_node/Search-Stopping-Parameters-for-Minimization-Algorithms.html

My question concerns the first method, gsl_multifit_test_delta(). When I pass in values for epsabs and epsrel, what exactly does the function compare them against?

If after j iterations my system is:

F_1(a_j, b_j, c_j) = error_1_j
F_2(a_j, b_j, c_j) = error_2_j
...
F_i(a_j, b_j, c_j) = error_i_j
...
F_m(a_j, b_j, c_j) = error_m_j

Then after the next iteration it will be:

F_1(a_k, b_k, c_k) = error_1_k
F_2(a_k, b_k, c_k) = error_2_k
...
F_i(a_k, b_k, c_k) = error_i_k
...
F_m(a_k, b_k, c_k) = error_m_k

where k = j + 1, each parameter (a,b,c) has been perturbed slightly, and the residual error values have (hopefully) decreased.

What I'm interested in minimizing is the root mean squared error, or the square root of the mean squared error over the set of residuals. Is that what the docs are talking about when they use the term "absolute error"?