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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 *n*th 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.