Howard,
You raise an important point below, but one which does not invalidate
my approach. Specifically, is you scale the parameters (for a non-
singular problem) you can indeed obtain any eigenvalue spectrum
you choose. HOWEVER the new eigenvalue spectrum reflects the
precision of the *scaled* parameters, and so the relative precision
of each *unscaled* parameter is unaffected.
Many of the techniques used in numerical solutions of linear equations
are validated by their effect on the eigenvalue spectrum -- and in
fact this is how one shows that solving normal equations is not
necessarily the best way to do least squares problems. Forming the
matrix product A(transpose).A x = A(tranpose)r, where A is the m by n
design matrix and r is the residual vector, to get the n by n normal
equations from the m by n design equations squares the eigenvalues of
the system and thus squares the condition number (ratio of smallest to
largest eigenvalue). This does not affect the numbers found for the
solution if infinite precision arithmetic is used, but has a large
effect if the condition number is close to the precision of computation.
But I digress. The bottom line is that if you apply a linear
transformation to the parameters I don't think you gain anything.
The revised eigenvalue spectrum will accurately show the precision
of the scaled parameters, but when you convert your coordinates and
e.s.d.'s back to Angstroms you will find yourself in the same position
as when you started -- because you will be applying the inverse
transformation.
I am not quite sure what you mean by refining on arbitrarily scaled
data -- one of the things the eigenvalues are sensitive to is the
weighting scheme, and scaling the observations in a non-uniform manner
will definitely affect the refinement. However, I don't think that
can be what you meant.
Best regards,
Lynn Ten Eyck
Howard Flack wrote:
> Lynn,
>> In some of my spare moments (so few), I think about the presentation
> you made at the computing school in Western Washington with the
> analysis of eigenvalues and vectors of the normal-equations matrix for
> macromolecular refinements. One thing that bothers me is that the
> eigenvalue spectrum can be changed by making a linear transformation
> of the variables from x(i) to s(i)x(i) where s(i) is an arbitrary
> scale factor. [You can also instead of 'refining on |F(h)|' refine on
> k(h).|F(h)| where k(h) is some arbitrary scale factor for each
> reflection. As the weights transform inversely to |F|^2^, this has no
> effect on the eigenvalues.] I would not expect the real nature of the
> refinement to be altered by such a silly trick but one can nevertheless
> easily arrange for all the eigenvalues to have the same value. This
> seems to suggest that the eigenvalues are not the invariants that one
> needs to examine or may be I am talking utter nonsense?
> Best wishes,
> Howard
> --
> Howard Flack http://www.unige.ch/crystal/ahdf/Howard.Flack.html> Laboratoire de Cristallographie Phone:(+41 22) 702 62 49
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