1B

I

The ¬rst follows DQGMRES closely, de¬ning the least-squares solution progressively and $ ¢

© ©

exploiting the structure of the matrix to obtain a formula for from . Because ¢ ¢ ¢

¡ W“

of the special structure of , this is equivalent to using the DQGMRES algorithm with ¢

¢

¬ . The second way to proceed exploits Lemma 6.1 seen in the previous chapter. This

W

y

lemma, which was shown for the FOM/GMRES pair, is also valid for the CGS/TFQMR

pair. There is no fundamental difference between the two situations. Thus, the TFQMR

iterates satisfy the relation T

r±— „ ®

i

£ X

©² © ¬ © ² ¥©

¢ ¢ ¢ ¢

¢

W“ W“

where the tildes are now used to denote the CGS iterate. Setting T T

r± „ ®

i

X X

© ² ¥© ¬ ©² ©

! ¢£ y £

y ¢ ¢ ¢ ¢

W“ W“

¢ ¢

¢

W“ W“

£

¢¢ ! ¢¦

(

W“

©

the above expression for becomes

¢

r± „ ®

i

(¬ ©

© ¢£ ¢¦ w

¢ ¢

W“

©

Now observe from (7.55) that the CGS iterates satisfy the relation

¢¥

r± „ ®

i

© ¬ ¥© £

w

¢©

¥

¢ ¢ ¢

W“ W“ W“

From the above equations, a recurrence relation from can be extracted. The de¬nition

£ ¢

of and the above relations yield

£ ¢ T

X

© ² W “ ¥© w W “ T ¥© ² ¥© y ¬

£ ¢ ¢ ¢ ¢ ¢

W “T

¢

W“ eX £

X

© ² W “ © ² £ “ © ² W “T ¥© £ y w W “ ¬

£

¢ ¢ ¢ ¢ ¢

“

¢

W “

© ² W “ ¥ © W “ ² y w W “ ¬

X£ ¢

£

¢ ¢ ¢

“ ¢

W“

Therefore, T

£ X

² ¢¦

¢

¬ £ w £ y £W“

£ W“

¢ ¢ ¢

¢

W“ W“

T T

¢

W“

W“

£ £ X ¢

² Xy ²

²

The term is the squared tangent of the angle used in the

¢ !

¢ ¢

y W“ W“

rotation. This tangent will be denoted by , and we have ¢

W“ £

¢

£

¢ ¢ ¢¦ ¢

¢

¬ ¬ ¬ £ w¢

( £¢yw £ ¢£

(

¢ ¢

¢

¢ ¢

"U

W

¢

y

¢

The angle used in the -th rotation, or equivalently , can be obtained by examining the

!

5 £ B¡ k| ¡ k¦¡B¦5

§ £ | § q| ¥ £§ ¢¢£

¥£

¨ "

$

¡

matrix :

¢

X Y

Y

.

.

²

.

X X