—™ ¡¦

' 7$

%# ) 2

1 3

£ $ ¨ ¬ S t²A¨ ¬ T ¨ $

3 3

© §X w

– ¨¬ ¦

1. Compute , , and

S ¡ ¡

¢ R ¢ © W vR § ¬ !¥ ¬

¢u ¢ u

2. De¬ne the matrix . Set .

$¢

Y

¢ ¢ ¢

! £

W

W "U

W y#V¡™( } ¬

3. For Do:

!( ( ˜

3

u ¦§(¬ u ¥ y

§T

4. Compute

X R (V( y v¬u vE

5. For Do:

Y

3 u¥ ¬ R

6. ¦(

£

3 R ¦ vR ² u ¬ u ¥

u

7. ¥ £

8. EndDo

¬ ! Y$¬ u ’U u u £ u u ¥ u ¬ u W’U u u 3

9. . If set and go to 12

£ £

W ¥¬

10. ¦

¤

£

$ W"U

"U

W

11. EndDo

£ b ¡ ² b ¢b ¢

w ©¦¬ ©

12. Compute the minimizer of and .

S ¢ ¢ ¢

F

¡

W

The second way to derive the GMRES algorithm is to use the equations (5.7) with

¥

§¬ . This is the subject of Exercise 4.

¢ ¢

E ' 7PP0¤ P5 £& 2HPP6¦2H 0A

¨ £

§¢ BF35 3 © ' 5F ' 5H

¢

The previous algorithm utilizes the Modi¬ed Gram-Schmidt orthogonalization in the Ar-

noldi process. Section 6.3.2 described a Householder variant of the Arnoldi process which

is numerically more robust than Gram-Schmidt. Here, we focus on a modi¬cation of GM- $

RES which retro¬ts the Householder orthogonalization. Section 6.3.2 explained how to get

$ ¡

u¦

the and the columns of at each step, from the Householder-Arnoldi algorithm. ¢

¡ "U

W

Since and are the only items needed to extract the approximate solution at the end

¢ ¢

of the GMRES process, the modi¬cation seems rather straightforward. However, this is

R¦

only true if the ™s are stored. In this case, line 12 would remain the same and the modi¬-

cation to the algorithm would be in lines 3-11 which are to be replaced by the Householder

variant of the Arnoldi process. It was mentioned in Section 6.3.2 that it is preferable not

R¦

to store the ™s because this would double the storage requirement. In this case, a formula

R¥

must be found to generate the approximate solution in line 12, using only the ™s, i.e., the

R

¥ ™s. Let T

X

¢b ¬ ( £ ¦¨( ¦ ¢ ¤( (

¦

WVT

TT

W

k´ C

³

| 7¥

§

©¬ ©

so that the solution is of the form . Recall that in the

¦ ¦ w w w ¦

¢ ¢¦ ¢

W Wu ¦ VWT

TT

Householder variant of the Arnoldi process, each is de¬ned by

¥ ¥ W ¦¬ u ¦ u ¡ u £

¥

Using a Horner-like scheme, we obtain

¦¬ ©

© w w £ ¡ £ T £ ¦ w ¡ T ¦ w ¥ £ T ¥ W ¥

¥ ¥¥

¢ ¢¦ ¢¡ ¢

WWW

W ™eX

XX

¥ ©¦¬ w w £ ¡ £ ¦ £ w ¡ ¦ ¥ ¥ ¥

w w

¢ ¢¦ ¢¡ ¢¡ ¢¦ ¢

W“ W“ W“

WW W

Therefore, when Householder orthogonalization is used, then line 12 of the GMRES algo-

rithm should be replaced by a step of the form

a±—0 „ ®i

3

¬ T

Y

a± 0 „ ®

i

X wuu u 3

¥¦¬

¬} ( ²

$( ! (V(

!

¡¦ V

a± 0 „ ®

i