W“

Note that is a scalar computed from the Lanczos algorithm, while results from the

¢S ¢¦

-th Gaussian elimination step on the tridiagonal matrix, i.e.,

!

r± „ ®

i

¢S

¬

© (

¢

¢¦

±r „ ®

i

W“¢

¬ ² ¢S¢©

¢¦

In addition, following again what has been shown for DIOM,

£ ¬ ¢ ¢

¥ W “¢ ¤ (

²¬ ¤ ¤ ¢©

in which . As a result, can be updated at each step as

©

¢ ¢ ¢

W“ ¤ w ¢© ¬ ¢©

¢ ¢

¡

W“

where is de¬ned above.

¢

¡

This gives the following algorithm, which we call the direct version of the Lanczos

algorithm for linear systems.

`g DCA ˜Q¤ v ¢

˜ ¡¦ ¢ © 76 9

DE@

' 7$

%# ) 2

1 3

£ ¨ ¬ $¬ ¤ t²A¨ ¬ ¨ 3 3 3

©§

S– ¨ ¬ ¦

1. Compute , , and

S

W Y`¬ Y ¬ $¬ ©

W

2. Set , S ¡

T W ( ¬! W

3. For , until convergence Do: ™

(˜

X 3

¦ S ² §§ ¬ ¥ y

¦( ¥ ¬¤

4. Compute and ¦ ¢ ¢ ¢ ¢ ¢

¤ ©² ¬ W“

¬©

i

5. If then compute and ! ¢ ¢ ¢ ¢

¡

W“ T y

¬

S ©²

6. ¢¦ ¢ ¢ ¢

X

S ²¤ ¦ W “ ¬

7. ¢ ¢ ¢ ¢

¢¦

¡ ¡

W“ ¦¬ ©

©

8. w

¢ ¢ ¢ ¢

¡

W“ ©

9. If has converged then Stop

¢

3

(¬ ¥

¦ ² ¥

10. ¢ ¢

¦£ ¥ ¬

S ¬

11. ,

¥ S

¢ ¢ ¢

"U

W

"U

W

"U

W

12. EndDo

¢b ¢ ¬

This algorithm computes the solution of the tridiagonal system pro- ¡S

W

gressively by using Gaussian elimination without pivoting. However, as was explained for

DIOM, partial pivoting can also be implemented at the cost of having to keep an extra

vector. In fact, Gaussian elimination with partial pivoting is suf¬cient to ensure stability

for tridiagonal systems. The more complex LQ factorization has also been exploited in this

context and gave rise to an algorithm known as SYMMLQ [159].

The two algorithms 6.15 and 6.16 are mathematically equivalent, that is, they deliver

the same approximate solution if they are both executable. However, since Gaussian elimi-

75 B1B¥ © £ 5 Ct|¦’¥ yC£¤¥ § ¤ l ¢ y5

£§ |5 |„

„ §

¦ C

¬ F$ b ¢

nation without pivoting is being used implicitly to solve the tridiagonal system , ¡S

W

the direct version may be more prone to breakdowns.

Observe that the residual vector for this algorithm is in the direction of due to ¦ ¢

"U

W

equation (6.66). Therefore, the residual vectors are orthogonal to each other as in FOM.

§

R

Likewise, the vectors are -orthogonal, or conjugate. These results are established in

¡

the next proposition.

Q¤ ¡ v A£

£¦ ¥ DA

C

' ' ©' 2 )

§)

t¢¨ ¬

©§ ² ¬

Let , , be the residual vec- ¨ ( Y

(

¢ ¢ !

y