"U ¢

W

W"U ¢ "U w ¢ W¦

"U

W

© (¬

©

W"U

13. ¢£

T "U !

W

"U

W

W’U

14. If is odd then X¨

¬ ¬

15. ;

¢

¢

¢ ¢¨

¢ S ¨(

W T¢ “ £

W“ "U ¢ £

W

’U ¢ ¥

W

"U

W ¬

16. ¢S w

X W “ ¢ S w ’U ¢ £ W"U ¢ ¦

W

§¬ §

17. £

¢S w ¢

¢¦

“

“ W“ W’U

"U

W

W

W

18. EndIf

19. EndDo

Notice that the quantities in the odd loop are only de¬ned for even values of . The ! !

©

residual norm of the approximate solution is not available from the above algorithm ¢

as it is described. However, good estimates can be obtained using similar strategies to

´ ¢£

5 £ B¡ k| ¡ k¦¡B¦5

§ £ | § q| ¥ £§ £

¨ "

those used for DQGMRES. Referring to GMRES, an interesting observation is that the

uA

recurrence (6.40) is identical with the recurrence of the scalars ™s. In addition, these two

sequences start with the same values, for the ™s and for the ™s. Therefore, X ¡

A S

¬

¡

¢A

¢T

’U

W X

¥$

Recall that is the residual for the least-squares problem

w!

¡

¢ !

"U

W y

5CI6

¡

²W £

¡X

42 ¢

Hence, a relation similar to that for DQGMRES holds, namely,

R±` „ ®

°i

ty¨

©§ ² w ¢A

¢ !

y

This provides a readily computable estimate of the residual norm. Another point that should

¢

be made is that it is possible to use the scalars , in the recurrence instead of the pair ¢ ¢

¢

¢ ( ¢ , as was done above. In this case, the proper recurrences are T

£ X

¬ £

¢ ¢ £ “¢

w¢

£ ¢ ¢ ¢

’U

W W

£X

£

¬ ¢X

¢ w ¢A

¢ ¢

’U

W ’U

W "U

W

£X £

¢ ¬ w ¢A

¢A ¢

’UW "U

W

¬ ¢

¢A

£ A

¢ ¢

"U

W "U

W

¬ ¢

¢¦ ¢

’U

W W"U

¥

¥¤A 7 §¨ ¦ ¥£

¤¢

Table 7.3 shows the results when TFQMR algorithm without precondi-

tioning is applied to three of the test problems described in Section 3.7.