Computed decay time is incorrect

[pkienzle]

See scattering/activation#17

[pkienzle]

Derivative used in e4f8ea1 appears to be incorrect:

f = lambda t: sum(Ia*exp(-La*(t-To)) for Ia, La in data) - target
df = lambda t: sum(La*Ia*(To-1)*exp(-La*(t-To)) for Ia, La in data)

It is okay when To = 0, which is probably why it seems to work. This function needs a fuller review.

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[correction] It is okay when t=1 since the derivative with respect to t should be:

df = lambda t: sum(-La*Ia*exp(-La*(t-To)) for Ia, La in data)

This is only used in the root-finding code. It appears to be finding the correct value despite the incorrect derivative otherwise it would be throwing a runtime error. Using the correct code should make it converge faster.

[pkienzle]

Even with the corrected df/dt the resulting time does not have total activation matching the target value. Need to verify that formulas for decay time and activation are consistent.

[pkienzle]

After further investigation, the step sizes in the fit are too tiny because the exponential. I will instead adjust the intensities to the time at the guess and fit relative to that. Then the fit ends in a few short Newton steps.