XSPEC uses a variant of Marquardt's algorithm described in §11.5
of ``Data Reduction and Error Analysis for the Physical Sciences"
by Bevington. (The reader is advised that this description is designed
to be read in conjunction with Bevington.) The algorithm turns on
finding a matrix 
and a vector 
such that the
equation :
gives sensible values of the change in parameters, 
, for
the fitting function. Bevington §11.4 gives the derivation
of 
and 
and shows that is parallel to the
gradient of 
.
Now the C statistic has a gradient with respect to the parameters of the fitting function of :
So, following Bevington, expand 
about 
:
substitute into C and minimize with respect to the changes in the parameters :
so to first order in the parameter changes :
or :
where :
These and then are substituted for those used in the
case and the algorithm works as required.
There is one further difference in XSPEC between the and likelihood
methods, which is caused by the fact that XSPEC uses an analytic formula for
setting the model normalisation. In the case, this means multiplying
the current model by :
where 
is the error on 
. In the likelihood case the corresponding
factor is :
