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Hi! Ideally, these algorithms should just work without any knowledge of the underlying math. The idea is that being a physicist is already good enough:)
However, exponential fitting is extremely difficult. No, it is EXTREMELY difficult. In particular, its success sometimes depends on algorithm implementation in a non-obvious manner. Some design decision that greatly improve robustness in many important cases turn out to degrade performance in this case. The be specific, I am a bit concerned by the fact that ALGLIB way of regularizing steps may slow down convergence on convolved problems exactly like yours. I explain it in order for you to understand that there is a possibility that you did everything right, and that is ALGLIB that needs to get better.
Now, several questions first:
* did you fail to get past the first iteration when using analytic derivatives - or numerical gradient?
* how many variables do you have?
* sometimes exponential problems get easier when you solve them with tight box constraints first (to prevent accidental escape from the initial point), and then relax constraints and re-solve. Did you try something like that?
And recommendations:
* for your case (accurate computations over inaccurate experimental data) a small differentiation step is recommended. Something proportional to sqrt(machine_epsilon), like 1E-7 or 1E-6. But not less, you do not want rounding errors to amplify.
* epsx can be chosen higher than the numerical differentiation step, like 1e-3 or 1e-4 for the beginning.
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