W“

Y ¢F

y

¢ ( ¢ e( ¢ b ¢ ¥ ( ¢ ¥F ( ¢ ¥b

Replacing by , respectively, in (6.44), a relation similar to (6.45)

¡ ¡

F

would result except that is replaced by which, by (6.42) and (6.43), satis¬es ¢ ¥F

¡ ¡

¢F ¢ ¢¥

the relation

£

¡ ¡

¢¥

¢

¬

¢

¢ ¥F

¢F

The result follows immediately.

If the FOM and GMRES iterates are denoted by the superscripts and , respectively,

' ¦

then the relation (6.41) implies that

£ ¡

© ² (© ¬ © ² © (

(¢ (¢

¢ ¢ ¢

W“ W“

or,

a± ¯ „ ®

i

© £ £¬ ( © £ ¡

©

¢ ¢w ¢

(¢

¢ ¢

W“

This leads to the following relation for the residual vectors obtained by the two methods,

a± ¯ „ ®

i

£ £ ¡

¨ ¢ w

£¬

¢

(¢ ¨ (¢ ¨ ¢ ¢

W“

¨ ¡¡µ£ „ ¢

§|5¥ yq¢| ¢ £ ¥§

„ 5| j C¦£¥

5§

¡ C

"–© ¡"

§

1B

which indicates that, in general, the two residual vectors will evolve hand in hand. In par-

¢ ¬

ticular, if , then GMRES will not progress at step , a phenomenon known as

Y !

stagnation. However, in this situation, according to the de¬nitions (6.31) of the rotations,

¡

¢ ¡

¬ ©

which implies that is singular and, therefore, is not de¬ned. In fact,

P W “ ¢ B£

G¢ Y ¢ ¢

the reverse of this is also true, a result due to Brown [43], which is stated without proof in

the following proposition.

h¤ ¡ v A£

£¦ ¡CDA

C

' ' ©' )$ )

§

If at any given step , the GMRES iterates make no progress, !

¡ ¡

¡

(© ¬ (© ©

i.e., if then is singular and is not de¬ned. Conversely, if is

¢ ¢

¢ ¢ ¢

W“ ! § ¬ (©

singular at step , i.e., if FOM breaks down at step , and is nonsingular, then ! ¢

© .

(¢

W“

Note also that the use of the above lemma is not restricted to the GMRES-FOM pair.

Some of the iterative methods de¬ned in this chapter and the next involve a least-squares

problem of the form (6.24). In such cases, the iterates of the least-squares method and those

of the orthogonal residual (Galerkin) method will be related by the same equation.

t¨ £R

©§ ²

R

Another important observation from (6.40) is that if is the residual norm

obtained at step , then E

¬

(¢ 0 ¢ 0¢

(¢

W“

The superscripts and are used again to distinguish between GMRES and FOM quan-

¦ '

tities. A consequence of this is that,

r± ¯ „ ®

i

¬ £

¢W ¢

0¢ ¦ 0S ¢

(¢

¡

©

Now consider the FOM iterates, assuming that is de¬ned, i.e., that is nonsingular. ¢ ¢

An equation similar to (6.48) for FOM can be derived. Using the same notation as in the

proof of the lemma, and recalling that T

¡ ¢ 0¡ X