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
.