(X W“

¢

W“ ¢ ¢

¡ ¡

W“

which proves the expression (6.84).

This lemma opens up many different ways to obtain algorithms that are mathemati-

cally equivalent to the full GMRES. The simplest option computes the next basis vector

R

as a linear combination of the current residual and all previous ™s. The approxi- ¨

¢ ¢

¡ ¡

"U

W

mate solution is updated by using (6.84). This is called the Generalized Conjugate Residual

(GCR) algorithm.

˜• £¤A ˜h¤ v ¢

¡¦

0 (%&$

'#

) 2

1 R

3

©V§ A¨g g ¨

²¬

¨m

¬ 1. Compute . Set . ¡

Qg ˜ ( g $v (Y ¬

2. For until convergence Do:

( (

y¢

P X G ¬ u

3.

PXXG

g u u w u © ¬ "U u ©

4. ¡

¢ u G ² § u ¬ vuR ² S u ¨ ¬ W "U u ¨

5. ¡

W

¬E P s P sXR X vu iR s X G u ¡

6. Compute , for ¡Y

( VQ(

(

y

£

u ¬ u

7. w

S R ¨

¡ ¡

W"U

"U

W

8. EndDo

¨ ¡¡µ£ „ ¢

§|5¥ yq¢| ¢ £ ¥§

„ 5| j C¦£¥

5§

9 C "–© ¡"

§

1B

§ §

vR S

u u© R

To compute the scalars in the above algorithm, the vector and the previous ™s ¨ ¡

are required. In order to limit the number of matrix-vector products per step to one, we

§ §

u© u

can proceed as follows. Follow line 5 by a computation of and then compute ¨ ¡

"U

W W"U

after line 7 from the relation

u

§ u R S £Rf

§ §¬

u u© R

w

¨

¡ ¡

"U

W ’U

W

§

R R

Both the set of ™s and that of the ™s need to be saved. This doubles the storage re-

¡ ¡

quirement compared with GMRES. The number of arithmetic operations per step is also

roughly 50% higher than GMRES.

The above version of GCR suffers from the same practical limitations as GMRES

and FOM. A restarted version called GCR(m) can be trivially de¬ned. Also, a truncation

§ R

of the orthogonalization of the ™s, similar to IOM, leads to an algorithm known as ¡

g

ORTHOMIN(k). Speci¬cally, lines 6 and 7 of Algorithm 6.20 are replaced by

¢ ²¬

i s

˜ E

²¬

¡

uR S

6a. Compute , for wW V(

(

s Gu P Xs

y

£ X X u¤G £R R ¡ vR P S

¬

u u¨ u

7a. .

w

¡

"U

W "U

W W"QH “

U

u

Another class of algorithms is de¬ned by computing the next basis vector as

¡

"U

W

u

r± „ ®

i

f

§¬

u wu R ¡ vR S

u

¡ ¡

"U

W R

§

vR S

u R

in which, as before, the ™s are selected to make the ™s orthogonal, i.e.,

T ¡

£T X