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## The Linear Equation Solver

keywords:

CG-METHOD-USED
DIAGONAL-PRECONDITIONED-CG-SOLVER-USED or
MUNKSGAARDS-PRECONDITIONED-CG-SOLVER-USED or
EXPANDING-BAND-PRECONDITIONED-CG-SOLVER-USED or
FULL-MATRIX-PRECONDITIONED-CG-SOLVER-USED or
GILL-MURRAY-PONCELEON-SAUNDERS-PRECONDITIONED-CG-SOLVER-USED or
MODIFIED-MA27-PRECONDITIONED-CG-SOLVER-USED or
SCHNABEL-ESKOW-PRECONDITIONED-CG-SOLVER-USED or
USERS-PRECONDITIONED-CG-SOLVER-USED or
BANDSOLVER-PRECONDITIONED-CG-SOLVER-USED (integer) or
MULTIFRONTAL-SOLVER-USED or
DIRECT-MODIFIED-MULTIFRONTAL-SOLVER-USED

In the second stage of the process to find an (approximate) optimum value of the model,  the variables which lie on their bounds  at the Cauchy  point are fixed  and the optimum value of the model  with respect to the remaining variables sought. This optimization is equivalent to the solution of one or more systems of linear equations. The coefficient matrix of each system is the Hessian matrix of the merit function, taken with respect to the variables that are not fixed at the Cauchy point.   The user may choose between a number of appropriate linear equation solvers. The best choice will depend on the structure  and conditioning  of the coefficient matrix.

If the keyword CG-METHOD-USED is included,  the conjugate gradient method without preconditioning is used.  Conjugate gradient methods with preconditioning are also available. These range from the simple use of diagonal scalings, to preconditioners suitable for matrices with a band structure,   through incomplete factorization preconditioners, to full factorization preconditioners.

The diagonal scaling preconditioner  is invoked with the DIAGONAL-PRECONDITIONED-CG-SOLVER-USED keyword. The band matrix preconditioner  is called with the keyword BANDSOLVER-PRECONDITIONED-CG-SOLVER-USED; in this case, the semi-bandwidth   must be specified as an non-negative integer following the keyword.

Two incomplete factorization preconditioners are available,    The Munksgaard preconditioner may be specified with the keywords MUNKSGAARDS-PRECONDITIONED-CG-SOLVER-USED. Alternatively, a so-called expanding band preconditioner, invoked with the keywords EXPANDING-BAND-PRECONDITIONED-CG-SOLVER-USED, is also available.

Finally, full factorization preconditioners may be used.  The modification method due to Gill, Murray, Ponceléon and Saunders may be called by one of the keywords FULL-MATRIX-PRECONDITIONED-CG-SOLVER-USED or GILL-MURRAY-PONCELEON-SAUNDERS-PRECONDITIONED-CG-SOLVER-US ED. The alternative modification proposed by Schnabel and Eskow is invoked with either the MODIFIED-MA27-PRECONDITIONED-CG-SOLVER-USED or SCHNABEL-ESKOW-PRECONDITIONED-CG-SOLVER-USED keywords.

The systems may also be solved using direct methods.  Options available are a multifrontal method, in which negative curvature is  exploited if encountered, called with the keyword MULTIFRONTAL-SOLVER-USED, and a modified Cholesky   method, which is included when the keyword DIRECT-MODIFIED-MULTIFRONTAL-SOLVER-USED is present. A description of all these techniques is given in the methods section. The default solver

BANDSOLVER-PRECONDITIONED-CG-SOLVER-USED 5
is used.