The theoretical underpinnings of all refinement methods used in the
latter stages of high resolution refinement are the same. The analysis
begins by making a Taylor series expansion of the function being minimized
about the current guess for the values of the parameters of the model
( 
). The Taylor series expansion is
where 
is the gradient of the function,
is the second derivative or normal matrix,
and 
is the shift vector which takes to 
.
The higher order terms are always assumed equal to zero.
To find the value of where 
is minimal we take the
derivative of
Equation (1) with respect to and solve for
when is 
. The result is
defines the minimum in all cases where the higher order
terms are, in fact, zero, and when 
is positive definite,
which is always the case in this application.
