In practice, modifications of the classic Newton iteration are essential
for guaranteeing global convergence, with quadratic convergence
rate near the solution.
First, when is not positive-definite, the search direction may not
exist or may not be a descent direction. Strategies to produce a related
positive-definite matrix , or alternative search directions,
become necessary. Second, far away from , the quadratic
approximation of (43) may be poor, and the Newton direction must
be adjusted. A line search, for example, can * dampen* (scale)
the Newton direction when it exists, ensuring sufficient decrease
and guaranteeing uniform progress towards a solution. These modifications
lead to the following * modified* Newton framework using a line
search: