¨ ¨ ty¨ ¬ ¨

©§ ²

1. Compute ; arbitrary;

3

¬

2. .

¨

¡

T

T

X ¨ ( u y $vu

(Y ¬

3. For , until convergence Do:

X¨ u 3

¨( T§ ¨( ¨T ¬

4. ¡

3

u § u ² u ¬ u ¢

5. ¨ ¡

u¢ § Xu¢(u¢ § ¬ u #

Xu 3

¢ W( §

6. 3

u ¢ u # w g u u w u © g ¬ u©

7. ¡

3

# ² u g ¤¬ "U u ¨ W

gu ¢ § u

8. ¢W

S ¢ i ¢ "U u ¡

3

T ¥ P S ¢ ¢ G ¬ S ¡

9.

¨ Xu

3

u # ² u u S w P u ¬ ¨G

§ u

10. ¡ ¡ ¡

W"U

"U

W

11. EndDo

¥

£¤A 7 §¨ ¦ ¥£

¤¢

Table 7.2 shows the results of applying the BICGSTAB algorithm with no

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

Matrix Iters K¬‚ops Residual Error

F2DA 96 2048 0.14E-02 0.77E-04

F3D 64 6407 0.49E-03 0.17E-03

ORS 208 5222 0.22E+00 0.68E-04

¤ ¤A 7 §¡¤

£

A test run of BICGSTAB with no preconditioning.

See Example 6.1 for the meaning of the column headers in the table. The number of

matrix-by-vector multiplications required to converge is larger than with BCG. Thus, us-

ing the number of matrix-by-vector products as a criterion, BCG is more expensive than

BICGSTAB in all three examples. For problem 3, the number of steps for BCG exceeds

the limit of 300. If the number of steps is used as a criterion, then the two methods come

very close for the second problem [61 steps for BCG versus 64 for BICGSTAB]. However,

BCG is slightly faster for Problem 1. Observe also that the total number of operations fa-

vors BICGSTAB. This illustrates the main weakness of BCG as well as QMR, namely, the

matrix-by-vector products with the transpose are essentially wasted unless a dual system

§

with must be solved simultaneously.

£ ¢ ¢

3Q© $ Q© P6G 5PF '# F C8 0A ¢

6

E3

A 3 I 553

I

§

The Transpose-Free QMR algorithm of Freund [95] is derived from the CGS algorithm.

© u

Observe that can be updated in two half-steps in line 6 of Algorithm 7.5, namely,

© ©¬ © ©¬ ¢

u£u w u u u u w i

and . This is only natural since the actual up-

i

u u

’U

W

U U

date from one iterate to the next involves two matrix-by-vector multiplications, i.e., the

µ£ „ ¢

|5¥ „yq¢| ¢ £ ¥§

5| j C§¦£¥

5

¨

¢£

£ ¡

§ "–© ¡"

§

1B

I

degree of the residual polynomial is increased by two. In order to avoid indices that are

multiples of , it is convenient when describing TFQMR to double all subscripts in the

W£

CGS algorithm. With this change of notation, the main steps of the Algorithm 7.5 (CGS)

become T T

± r¯¯ „ ® i

X X

¨ ¨ ( u £ ¨ ¨ ¨ (

¬ §

u£ u£

¡

±r— ¯ „ ® i

§£u u £ u £² u £ £ ¬ ¢

u£ u £T

£

¡

±r ¯ „ ® i

Xu£¢ w u£T

©£ £ ©

w ¦¬

u£ £

±r ¯ „ ® i

U u£ Xu£¢ w u£ T u£

§ X ¨ ² £ u u T ¬ £

¨ ¨

±r ¯ „ ® i

Uu £ X¨ u£

¨( ¨ ¨( U £¨ ¬

S