u £ T ¢ u £ S w £ u £ ¬ £

u£ £ ¨

±r — „ ® i

Uu£ Uu£

Xu£

¬£

S w u£¢ u£ S w £ £

¡ ¡

U U

The initialization is identical with that of Algorithm 7.5. The update of the approxi-

mate solution in (7.46) can now be split into the following two half-steps:

p±q— „ ®

°i

© ©¬

u£ u£ £u£ w u£

±r0 — „ ®

i

’U u £

W£

© ©¬ ¢

u£

u£ u£ w

U "U

W

¬ ¢

u£ u£

This can be simpli¬ed by de¬ning the vectors for odd as . Similarly, the

£ £

¢ !

"U

W

¬

u£ u£

sequence of is de¬ned for odd values of as . In summary,

¢ !

W"U

r±a — „ ®

i

¢!

£

¢ ¢

for odd de¬ne: W“¢

! !

¢

W“

With these de¬nitions, the relations (7.51“7.52) are translated into the single equation

©¦¬ © £

w

(

¢ ¢ ¢ ¢

W“ W“ W“

©

which is valid whether is even or odd. The intermediate iterates , with odd, which ¢

! !

are now de¬ned do not exist in the original CGS algorithm. For even values of the !

©

sequence represents the original sequence or iterates from the CGS algorithm. It is

¢

¥

convenient to introduce the matrix, !

¬ £ £

¤ (

(

$ &

¢ ¢

W“

and the -dimensional vector T

!

X

¬ ( V(

(

¢ ¢

W“

W

©

The general iterate satis¬es the relation

¢

± ¯ — „ ®

i

©¦¬ ©

w ¤

¢ ¢ ¢

±r— — „ ®

i

©¦¬ £

w

¢ ¢ ¢

W“ W“ W“

From the above equation, it is clear that the residual vectors are related to the -vectors £

¨ ¢

by the relations

r± — „ ®

i

t ¬

§²¨

¤

¨ ¢ ¢ ¢

±r — „ ®

i

¬ §W “ ² £

¨

¢ ¢ ¢

W“ W“

Next, a relation similar to the relation (6.5) seen for FOM and GMRES will be ex-

5 £ B¡ k| ¡ k¦¡B¦5

§ £ | § q| ¥ £§ ¤¢£

C£

¨ "

§

tracted using the matrix . As a result of (7.57), the following relation holds:

¤ ¢ T

X

¬R § ²

R R¨ y

£

¨

R

"U

W

$

Translated in matrix form, this relation becomes

a± — „ ®

i

¢

§ ¬

¤ ¢ ’U

¢ ¢ ¢

W

where

a± — „ ®

i

¢

¬

$ ¨ ( ¨ ( $¨

T ( &