[Sane-analysis] A1 fit error bands

[O. A. Rondon]

Since the A1 and A2 fit forms are quite simple, it's possible to
calculate the fit errors analytically, using the covariance matrices for
each case to get fit error bands at constant W or Q^2, etc.

For accuracy and ease of procedure, I've used Maple V. The results for
the 1/W and 1/W^c A1 fits are posted as Maple output, including graphs.
I plotted error bands vs Q^2 and vs W, including only diagonal terms
(first plots in each case,) and including parameter covariances (second
plots)
https://userweb.jlab.org/~rondon/analysis/asym/world/fit_A1-invW.pdf
https://userweb.jlab.org/~rondon/analysis/asym/world/fit_A1-powW2.pdf

The simple 1/W form is not too sensitive to not including covariances,
but the 1/W^c is sensitive, mainly because of the correlation between
the constant term and the power of W, and the resulting much larger
error for the constant term for this form compared to plain 1/W (a0 =
-.1602+/-0.0075 for 1/W vs a0 = -.3064+/-0.0805 for 1/W^c, or 4.7% vs
26.3% relative errors - there was a typo on a0's error  for 1/W, 0.0082
instead of 0.0075, in my report on the fits, fixed already).

In either case, the correct, full matrix, error bands are quite narrow,
assuming that the data's errors are normally distributed, which is
largely the case for our statistics dominated (SANE and SLAC, A1 and A2)
uncertainties.

This method can be implemented in code (C, Fortran, ...) to calculate
numerically the errors of the moments, as we did for RSS, with 17
parameters fits.

Cheers,

Oscar