‚ ‚

¥ ¡ ¢

£ £

¢

¡

£

For each new column generated in the block-Arnoldi process, rotations are required to ¡

¬ ¬

u

eliminate the elements , for down to . This backward order is w w

W W

¡

£

H y

important. In the above example, a rotation is applied to eliminate and then a second £

W R

£

rotation is used to eliminate the resulting , and similarly for the second, third step, etc. £

W

¨ ¡¡µ£ „ ¢

§|5¥ „yq¢| ¢ £ ¥§

5| j C§¦£¥

5

¢ £

"–© ¡"

§

1B

This complicates programming slightly since two-dimensional arrays must now be used

$

to save the rotations instead of one-dimensional arrays in the scalar case. After the ¬rst

¡

column of is processed, the block of right-hand sides will have a diagonal added under

¢

the diagonal of the upper triangular matrix. Speci¬cally, the above two matrices will have

the structure,

¡¢ ¡¢ ¡

¡¢¡¢ ¡

¡

¡¢ ¡¢

¢ ¡¢

„

„

$ $

¡¢ ¡

¡¡

¡¢

¡

¬ ¬

(

¦

¡

‚ ‚

where a represents a nonzero element. After all columns are processed, the following

least-squares system is obtained.

¢¡¢¡ ¡

¢ ¡¢ ¡ £

£

¢ ¡¢ £

¡ „

„

$ $

¢ £

¡¡

¢

£

£

¬ ¬

¦

£

‚ ‚

To obtain the least-squares solutions for each right-hand side, ignore anything below the

horizontal lines in the above matrices and solve the resulting triangular systems. The resid-

ual norm of the -th system for the original problem is the 2-norm of the vector consisting

E

of the components , through in the -th column of the above block of right-hand

w! Ew! E

y

sides.

Generally speaking, the block methods are of great practical value in applications in-

volving linear systems with multiple right-hand sides. However, they are not as well studied

from the theoretical point of view. Perhaps, one of the reasons is the lack of a convincing

analogue for the relationship with orthogonal polynomials, established in subsection 6.6.2

for the single-vector Lanczos algorithm. The block version of the Lanczos algorithm has

not been covered but the generalization is straightforward.

vcE—“˜lf¢ ™

—™ ™

1 In the Householder implementation of the Arnoldi algorithm, show the following points of detail:

¤¦

¥ ¤

¢# ¢ X#

is unitary and its inverse is .

v v

¤ ¢ # BBA v

`

¢ X# .

AA v

v

y £ w ¢ ¡y ¡|

|5 | §| C £

¦

¤¤ ¡ £ C £ v ¢ X #

•” 5