in which , and is computed as before, as the solution to the least-

h h

squares problem in line 11. These are the only changes that lead from the right precondi-

tioned algorithm to the ¬‚exible variant, described below.

ªm¢ HUm¢

˜

˜

¤0 " · wE¤©n¦§¥£¢

¡ ¨¦ ¤¢ ¤0

££ ¡

¢

u7 C &% ¨‚¢ C C

£

«§ §

1. Compute , , and

f

±„¬BDB% m¥

% ¬¬

2. For Do:

£ 'd ¡ P G

3. Compute f

P

4. Compute

¬ ¬ ¬ ª

¥ %„DBD%

5. For , Do:

£ % ` G S ¥

6. aS

£ S ¦¨ C

7. S

C¥

8. EndDo

¤ & £

§¦ ¤¥h u«P § % D¬ Bz% ¢P C f ¥ h

¥ §

9. Compute and

C f S ( S ¥ f h

h ¬ W ¬B f

10. De¬ne ,

¨ ¨ f f ¨ ¨©f C

11. EndDo §¦ ¨ § & % % h

w ‘ h «§ h f

12. Compute , and .

h h

h 5¥'

¤

13. If satis¬ed Stop, else set and GoTo 1.

As can be seen, the main difference with the right preconditioned version, Algorithm

£ 'd

¢

¡

9.5, is that the preconditioned vectors must be saved and the solution updated P f

¡

¡ s±% DBD% G

¬¬¬

using these vectors. It is clear that when for , then this method ¥

is equivalent mathematically to Algorithm 9.5. It is important to observe that can be P

de¬ned in line 3 without reference to any preconditioner. That is, any given new vector

can be chosen. This added ¬‚exibility may cause the algorithm some problems. Indeed,

P

may be so poorly selected that a breakdown could occur, as in the worst-case scenario

P

when is zero. P

One difference between FGMRES and the usual GMRES algorithm is that the action

¡

£ h

¡

of on a vector of the Krylov subspace is no longer in the span of . Instead,

'd

f

f

it is easy to show that

¦§ f h ¡ h qW v' £

¦¥ ‘

h

h ¦ § h ¡ h ¡ a d ¡ x`

in replacement of the simpler relation which holds for the f

f h§

±

±¯

standard preconditioned GMRES; see (6.5). As before, denotes the matrix

¦§ ¦ £

obtained from by deleting its last row and is the vector which is normalized

h

f

£

in line 9 of Algorithm 9.6 to obtain . Then, the following alternative formulation of

fE

h¥

(9.21) is valid, even when : h

C

f § h¡ ¬ ˜h f GG v' £

¦ ‘

h ¦£ w h

h

An optimality property similar to the one which de¬nes GMRES can be proved.

w

&

P

Consider the residual vector for an arbitrary vector in the af¬ne space h

w

7£ ¢

®

. This optimality property is based on the relations

(h

w

‚

¨ o`(©‚3mP

¨ C h a

GG v' £

¦µ ‘

h ©U

¨

¶

” 8

¨¥ "¡¢0'¡ ¡0

$ ¥¡ © ¤¡

©

¦ § § f h ¡ ¨ f £ ¡ h R! v' £

¦6 ‘

¬ ¤ h ¦ ¨ f ¢ f h

If denotes the function

a`h

w

«§¤

¢ ”D#”a

¨ §

`h h %