left as an exercise.

¥

CDA 7 §¨ ¦ ¥£

¤¢

Table 7.1 shows the results of applying the BCG algorithm with no pre-

conditioning to three of the test problems described in Section 3.7. See Example 6.1 for the

meaning of the column headers in the table. Recall that Iters really represents the number

of matrix-by-vector multiplications rather the number of Biconjugate Gradient steps.

Matrix Iters K¬‚ops Residual Error

F2DA 163 2974 0.17E-03 0.86E-04

F3D 123 10768 0.34E-04 0.17E-03

ORS 301 6622 0.50E-01 0.37E-02

¤ DA 7 §¡¤

C

A test run of BCG without preconditioning.

¥y

Thus, the number 163 in the ¬rst line represents 81 steps of BCG, which require

˜

matrix-by-vector products in the iteration, and an extra one to compute the initial residual.

¢ £

¦ ¦¨©7¡63Q&

§¨6§ 6

8 I B E B b 7F 8

IB BF5 I H A B ©' & 8 & 8

3

$

The result of the Lanczos algorithm is a relation of the form

a±— w°„ ®

i

§ ¬

$ T ¢ ¢ ¢

"U

W

¥ Xy

in which is the tridiagonal matrix

w

¢ ! ! $

¬ ¢

¢

¢X

¢¡

"U

W

µ£ „ ¢

|5¥ „yq¢| ¢ £ ¥§

5| j C§¦£¥

5

¨

£ ¤£

C ¡

§ "–© ¡"

§

1B

I

Now (7.15) can be exploited in the same way as was done to develop GMRES. If is ¦

W

¬

de¬ned as a multiple of , i.e., if , then the residual vector associated with an ¨ ¨

¦ S

W

approximate solution of the form

b ¢ w

(¬ ©

©

is given by T

Xb

© § A¨ ¬ tA¨

² ©§ ² ¢ w

$ b ² ¨ ¬

§ ² ¦ S`¬ ¢

b$ ¢ ¢

r± w°„ ®

i

² "U S W ¬

W

b ¢ ¢

F

¡

"U

W

W

$

The norm of the residual vector is therefore

r± w°„ ®

i

²

¬ ty¨

b¢

©§ ² £

FS

¢ ¡

"U

W W

$

¬ t§ v¨

©²

If the column-vectors of were orthonormal, then we would have ¢

² $

£ b ¢ W"U

, as in GMRES. Therefore, a least-squares solution could be obtained from

FS

¡

²

£ b ¢

W b

the Krylov subspace by minimizing over . In the Lanczos algorithm, the FS

¡

W $

R¦™s are not orthonormal. However, it is still a reasonable idea to minimize the function T

²

! £ b ¢

Xb

¦ FS

¡

W

b b ¢ w

©

over and compute the corresponding approximate solution . The resulting so-

lution is called the Quasi-Minimal Residual approximation. $

T

Thus, the Quasi-Minimal Residual (QMR) approximation from the -th Krylov sub- ² FS !¬ £

Xb