’U

W W"U "U W

¥

¥¤A 7 §¨ ¦ ¥£

¤¢

Table 6.3 shows the results of applying the DQGMRES algorithm with no

preconditioning to three of the test problems described in Section 3.7.

Matrix Iters K¬‚ops Residual Error

F2DA 98 7216 0.36E-02 0.13E-03

F3D 75 22798 0.64E-03 0.32E-03

ORS 300 24138 0.13E+02 0.25E-02

¤ ¤A 7 §¡¤

¥

A test run of DQGMRES with no preconditioning.

See Example 6.1 for the meaning of the column headers in the table. In this test the number

¬

of directions in the recurrence is . Y

W W

y

It is possible to relate the quasi-minimal residual norm to the actual minimal residual

norm provided by GMRES. The following result was proved by Nachtigal (1991) [152] for

the QMR algorithm to be seen in the next chapter.

¨ ¡¡µ£ „ ¢

§|5¥ yq¢| ¢ £ ¥§

„ 5| j C¦£¥

5§

£C "–© ¡"

§

1B

cv ˜¤

¥¦ ¥¡ DA

C

' 1

Assume that , the Arnoldi basis associated with DQGMRES, is ¢

"U

W

of full rank. Let and be the residual norms obtained after steps of the DQGMRES

¡¨ (¢ ¨ !

¢

and GMRES algorithms, respectively. Then T

r± — „ ®

i

¨ £ £ £ (¢ ¨ X

¢

¢ £

¢

"U

W

£ ™£¤¢¡6

A

$

Consider the subset of de¬ned by

¢

¥ "UW

$ ¦ ¢ b¢¡² ¡ b¥ ¢

¤ 3

¬ ¨

¬ ¬ ¡ $

¨§ $

S t©

¨

§

¡ ² W"U ¡ S W ¡

T

¨ T ¢ b ¢X

¢b £ b¢ b ¬ ² X¬

Denote by the minimizer of over and , FS

¢ ¢

¡

! ¢ W"U W ¢ W! ¥

. By assumption, is of full rank and there is an w w!

¨

¢ ¢

y¤

W"U ¢ ¥ y

¬

nonsingular matrix such that is unitary. Then, for any member of ,

¢

¦ ¦

’U ( “ ¦ "U ¢ ¥

W

W ©¥

¨

¬ ¬ ¨

¦

W ¢

W"U "U

W

and, in particular,

r± — „ ®

i

£ ¢¨ £

£ $¢

W“¦

¢ b¢¡ b

²

£

Now is the minimum of the 2-norm of over all ™s and therefore,

FS

¡

¥W

¡ ¨

¢ £¨ ¤

¥ © ©

¬

£ ¨ ¢

¦ ¦

¢ ¢

¡ ¨

£ "U ¨ £

W "U

W

¤

( ¨ £ ¦¦ r± „ ®

i

£ T T

¬ X "U X

¢ ¢

¢ £ ¦£

The result follows from (6.59), (6.60), and the fact that .

W

— v

™ “gg• — l w g¢© ¦

“ ™

© ©

¡

The symmetric Lanczos algorithm can be viewed as a simpli¬cation of Arnoldi™s method

§

for the particular case when the matrix is symmetric. When is symmetric, then the Hes-

¡

senberg matrix becomes symmetric tridiagonal. This leads to a three-term recurrence

¢

in the Arnoldi process and short-term recurrences for solution algorithms such as FOM

and GMRES. On the theoretical side, there is also much more to be said on the resulting

approximation in the symmetric case.