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2.5.1 Newton Methods Overview     continued...

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: