times using the pairs
and go to 2 if the residual power is not small enough.
is not formed explicitly; instead we compute
with an iterative formula
Nocedal (1980). Liu and Nocedal (1989) propose scaling the initial
symmetric positive definite
at each iteration as follows:
This scaling greatly improves the performances of the method.
Liu and Nocedal (1989) show that the storage limit for large-scale problems
has little effects.
A common choice for
. In practice, the initial guess
for the Hessian is the identity matrix
then it might be scaled as proposed in equation (12). The
nonlinear solver as detailed in the previous algorithm converges to
a local minimizer