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In order to demonstrate the use of the derivative method for computing
the RGCV criterion, we will derive the update for 
for the
steepest-descent algorithm. We have presented this method in a previous paper,
but we did not present the vigorous justification at that time (Reeves and Perry
1992).
The basic form of the steepest-descent method is
Note that 
, the relaxation parameter, can be a fixed scalar, or
it can be computed as a scalar function of the current estimate 
.
In any case, it is considered to be a constant with respect to 
.
The first step is to take the derivative of 
with respect
to 
. The result is
Then, multiplying both sides by 
and taking the summation over 
you have
Finally, making the substitution for the definition of 
, we get
which is the steepest-descent algorithm operating on 
, but using the
same 
that was used for 
.