TABLE OF CONTENTS Given an initial guess x 0 to the original linear system Ax = b, we now consider an orthogonal projection method, as defined in the previous chapter, which takes 
, with
in which r 0 = b ? Ax 0. This method seeks an approximate solution x m from the affine subspace x 0 + 
of dimension m by imposing the Galerkin condition
If v 1 = r 0/ ? r 0 ? 2 in Arnoldi's method and we set ? = ? r 0 ? 2, then
by (6.8) and
As a result, the approximate solution using the above m-dimensional subspaces is given by
A method based on this approach and called the full orthogonalization method (FOM) is described next. MGS is used in the Arnoldi procedure.
Compute r 0 = b ? Ax 0, ? := ? r 0 ? 2, and ? 1 := r 0/ ?
Define the m m matrix H m = { h ij} i, j=1, , m; Set H m = 0
For j = 1, 2, , m, Do
Compute w j := Av j
For i = 1, , j, Do
h ij = ( w j, ? i)
w j
