TABLE OF CONTENTS The generalized minimal residual method (GMRES) is a projection method based on taking 
= 
and 
, in which 
is the mth Krylov subspace, with ? 1 = r 0/ ? r 0 ? 2. As seen in Chapter 5, such a technique minimizes the residual norm over all vectors in x 0 + 
. The implementation of an algorithm based on this approach is similar to that of the FOM algorithm. We first describe the basic idea and then discuss practical details and a few variations.
There are two ways to derive the algorithm. The first way exploits the optimality property and the relation (6.7). Any vector x in x 0 + 
can be written as
where y is an m-vector. Defining
the relation (6.7) results in
Since the column vectors of V m+1 are orthonormal, then
The GMRES approximation is the unique vector of x 0 + 
that minimizes (6.26). By (6.25) and (6.28), this approximation can be obtained quite simply as x m = x 0 + V my m, where y m minimizes the function J(y) = ? e 1 ? H my ? 2; i.e.,
The minimizer y m is inexpensive to compute since it requires the solution of an ( m + 1) m least-squares problem,...
