[Ifeffit] Athena: problems with LCF

[Matt Newville]

Hi Nina,

As others have said, normalization is critical for getting reasonable
answers from LCF.    One of the drawbacks of a linear analysis is that
it always gives an answer, even when the assumptions going into the
question are poor.

> Even though the results of the LCF of the mathematical mixtures were better compared to
>  the real mixtures, LCF was also not able to reliable deconvolute these spectra into the individual
> reference spectra.
> Does anybody have an explanation for that?

I'm not sure what your tests entailed, but I would have expected that
your 'A_D_1to1' or 'A_D_1to1_calculated'  spectra would have been a
1-to-1 mixture of the A and D spectra.

A 1-to-1 mixture would a) have several isobestic point (for all E
where A(E)=D(E), all linear combinations of A and D have the same
value), and b) be half-way between A(E) and D(E) where the difference
is largest, say near E=3568 eV.   From the attached plot (simply
plotting the data from your project), it's pretty obvious that neither
of these is true.   This might be related to poor normalization, a
simple calculation mistake, or maybe I misunderstand your intent.

> It would be nice if somebody could give me information about the mathematical fitting algorithm
> implemented in Athena.

Non-linear least-squares, using the Levenberg-Marquardt algorithm.  See

   http://en.wikipedia.org/wiki/Levenberg-Marquardt_algorithm

and do feel free to read the docs:

   http://cars9.uchicago.edu/~ravel/software/doc/Athena/html/analysis/lcf.html
   http://cars9.uchicago.edu/~ifeffit/refman/node63.html

Admittedly, Levenberg-Marquardt may not be the most obvious choice for
linear analysis, but is usually quite robust and fast for finding
optimal solutions for linear problems, and is needed for other parts
of XAFS analysis.

--Matt
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