¡¢¤ ¡

§ ¬ w¢

¤ ¢%¢

¢ ¢ ¢

"U

W W"U

¡

Here, the matrix is no longer Hessenberg, but band-Hessenberg, meaning that it has ¢

subdiagonals instead of only one. Note that the dimension of the subspace in which the

¡

solution is sought is not but . ! ¡ !

A second version of the algorithm uses a modi¬ed block Gram-Schmidt procedure

instead of the simple Gram-Schmidt procedure used above. This leads to a block general-

ization of Algorithm 6.2, the Modi¬ed Gram-Schmidt version of Arnoldi™s method.

‘ (¦

‘ m•

—

˜h¤ v ¢

¡¦ P¥ P76 8 G4E D

9EFD 3 ¢£¤A

B

G Q BD SQ B

£

0 (%&$

'#

) 2

1

¥¤

1. Choose a unitary matrix of size ¡

W

v

¬

2. For Do: !#V˜™(

( 3 (¥

y §¬ u u

3. Compute

v

¬

4. For do: E

(3 V(Q˜(

y vR ¡ u ¥ R

¬

u

5. ¡

3u¥ u¥

¬ ²

R uR

6.

7. EndDo

¡

u¥ ¬ u u u

8. Compute the Q-R decomposition

"U

W "U

W

9. EndDo

Again, in practice the above algorithm is more viable than its predecessor. Finally, a third

version, developed by A. Ruhe [170] for the symmetric case (block Lanczos), yields a vari-

ant that is quite similar to the original Arnoldi algorithm. Assume that the initial block of

orthonormal vectors, is available. The ¬rst step of the algorithm is to multiply ¦ ( ¦

(

¡

W X

§

by and orthonormalize the resulting vector against . The resulting vector

¦ ¥ ¦ ( ¦

(

W W X

§

£¦

is de¬ned to be . In the second step it is that is multiplied by and orthonormalized

¦

R ¦ W"X

U

against all available ™s. Thus, the algorithm works similarly to Algorithm 6.2 except for

§

a delay in the vector that is multiplied by at each step.

‘ A

˜h¤ v ¢

¡¦

¦

P¥ P76 8

9EFD 3 ¢£¤A

B

G Q BD &6 T¤94 ¢ 7 ¡

5

S @Q 8 @

¥

0 (%&$

'#

) 2

1

£R © R ¦ §

1. Choose initial orthonormal vectors .

¡

W X

v

¬ ¢¢¢

¡¡¡

2. For Do: w

"3

( $V(

!(

¢¡

¡

w ¡y

²z ¬

3. Set ;

W

3 y©

§¬

4. Compute ; ¥ ¦

T

H VQ˜

vR¬ E

5. For Do: (3 ( (

XR

y ¬

6. ¦( ¥

£

3

H ¥ (¬

² R¦ R£

7. ¥

H

8. EndDo 3 u £ ¥ 3

¬ ¬

£

u u¦

9. Compute and .

¥

£

H "U

W "U

W H ’U

W

10. EndDo

¬

Observe that the particular case coincides with the usual Arnoldi process. Also, the ¡

y

dimension of the subspace of approximants, is no longer restricted to being a multiple