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The Generalized Minimum
Residual (GMRES) method
is designed to solve nonsymmetric linear
systems (see Saad and Schultz [185]). The most popular
form of GMRES is based on the modified Gram-Schmidt procedure, and
uses restarts to control storage requirements.
If no restarts are used, GMRES
(like any orthogonalizing Krylov-subspace method) will
converge in no more than n steps (assuming exact arithmetic). Of
course this is of no practical value when n is large; moreover, the
storage and computational requirements in the absence of restarts are
prohibitive. Indeed, the crucial element for successful application
of GMRES(m) revolves around the decision of when to restart; that
is, the choice of m. Unfortunately, there exist examples for which
the method stagnates and convergence takes place only
at the nth step. For such systems, any choice of m less than n
fails to converge.
Saad and Schultz [185] have proven several useful results.
In particular, they show that if the coefficient matrix is real
and nearly positive definite, then a ``reasonable'' value for m
may be selected. Implications of the choice of m are discussed
below.