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The computation and storage of Jacobian matrix as well as solving n linear equations in each iteration make the Newton's method iterations
expensive. Despite its plainness and general consistency, Newton's method has some major
well-known drawbacks (Dennis, 1983). Numerous approaches exist to overcome the
widely known drawbacks, such as Quasi-Newton (QN) method, Inexact Newton (IN)
method and Fixed Newton (FN) method. QN method is the famous method that
replaces Jacobian or its inverse with an approximation, which can be updated at each
iteration.
The anticipation behind QN method is to reduce the evaluation cost of
the Jacobian matrix, in which if the function evaluations are very expensive, the cost
of a solution by QN method could be much lesser than with IN method
(Drangoslav and Natasa, 1996). Nevertheless, it requires to store the full elements of the
Jacobian in each iteration. The use of IN method eliminates some of the
well-known shortcomings.
The FN method is significantly slow
and consumes more CPU time, as well as high number of iterations as the
system's dimension increases, in comparison to other variants of Newton's
method (Waziri et al., 2010a and 2010b). Due to insufficient information of
the Jacobian in each iteration, many efforts have recently been made by a
number of authors (Drangoslav and Natasa, 1996; Natasa and Zorna, 2001;
and Waziri et al., 2010a) to overcome the shortcomings of various
Newton's methods (Waziri et al., 2010b). The most critical idea common to all
these efforts is forming and storing a full-matrix approximation to the
Jacobian (directly or indirectly), which can be a very expensive task for
large-scale problems (Waziri et al., 2010b). Moreover, the FN method is the
easiest and simplest variant of Newton's method to overcome the
well-known shortcomings. |
