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As another practical example of the use of the template file approach,
consider writing a general program for solving a system of equations of
the form:
using Newton's method [32]. A bold typeface is again used to
denote a vector.
The system of iterates can be written as:
where 
denotes the
Jacobian matrix and 
is a given set of initial values used to start
the iteration process. 
denotes the inverse of 
and the
superscript 
denotes evaluation at iterate 
. The
iterative process is continued until some stopping criteria are satisfied at
iteration
itmax. When the system to be solved is scalar valued, 
is
simply the reciprocal of 
. For practical implementation,
(
) is rewritten at each iteration as:
where 
satisfies
The linear system () is typically
solved using LU decomposition of the Jacobian matrix at each iteration. The
value of 
thus obtained is substituted into ()
to obtain 
.