P $"G6

This completes the proof.

³ ¨ ¡¡µ£ „ ¢

§|5¥ yq¢| ¢ £ ¥§

„ 5| j C¦£¥

5§

¡C

"–© ¡"

§

1B

An explicit expression for the coef¬cient and an approximation are

6¡ ¢ ) ¡ ¢ )

¬

readily obtained from (6.99-6.100) by taking :

¡

Y

£ £ H“

wH

w ¡ ¡ w ¡ ¡

² ²

¡ ¢ ) y y

¬

£ £

6¡ ¢ ) H“

wH

² ²

w 6¡ w 6¡

6¡ 6¡

y Gy

²

£c

H %££ wc

£

£

' "

£ T w ²

X £¢

Since the condition number of the matrix of eigenvectors is typically not ¤ ¤

known and can be very large, results of the nature of the corollary are of limited practical

T

interest. They can be useful only when it is known that the matrix is nearly normal, in

X

¢

¤£

which case, . '

y

¦ i¡}¤ ¤ i˜‘

— ©™

¢ 9¡

£

In many circumstances, it is desirable to work with a block of vectors instead of a single

vector. For example, out-of-core ¬nite-element codes are more ef¬cient when they are

§

programmed to exploit the presence of a block of the matrix in fast memory, as much as

possible. This can be achieved by using block generalizations of Krylov subspace methods,

§

for which always operates on a group of vectors instead of a single vector. We begin by

describing a block version of the Arnoldi algorithm.

‘

¡˜¤Q v ¢

¦ P¥ P7¤8

9FD 4C¤¤A

GE B Q BD6

3£

' 7$

%# ) 2

1

¥¤

1. Choose a unitary matrix of dimension .

¡

W

}

¬

2. For Do: #V¡™(

! ( (¡ ˜

y ¬ §

¬

uR E u u R

3. Compute VV˜ (

((

¡ Ry

£

u¥ ² §¬ u uR

4. Compute £R

¡

u¥u¥ ¬

W u u u

5. Compute the Q-R factorization of :

"U

W ’U

W

6. EndDo

The above algorithm is a straightforward block analogue of Algorithm 6.1. By con-

struction, the blocks generated by the algorithm are orthogonal blocks that are also orthog-

¥

onal to each other. In the following we denote by the identity matrix and use the W W

H

following notation:

¬ (8& ¢ ( £ ( T$

¤ (

¢

¡ X W¡ ¡

¬ ( ¢ ¢ u R ¢ uR Y`! u R

for w E

( (

¢

W y

¬ matrix of the last columns of

¢%

¡

«

³

yq¦" © ¡’¨ ¨ ¢ © j

„ 5| §

¦C

T£

¢ !

Then, the following analogue of the relation (6.4) is easily proved:

a± w°„ ®