³ ¨ ¡¡µ£ „ ¢

§|5¥ yq¢| ¢ £ ¥§

„ 5| j C¦£¥

5§

¡C

"–© ¡"

§

1B

of the block-size as in the previous algorithms. The mathematical equivalence of Algo-

¡

rithms 6.22 and 6.23 when is a multiple of is straightforward to show. The advantage

! ¡

of the above formulation is its simplicity. A slight disadvantage is that it gives up some

§ u

potential parallelism. In the original version, the columns of the matrix can be com-

puted in parallel whereas in the new algorithm, they are computed in sequence. This can

be remedied, however, by performing matrix-by-vector products every steps. ¡ ¡

At the end of the loop consisting of lines 5 through 8 of Algorithm 6.23, the vector ¥

satis¬es the relation

u

f

² H ©§ ¬ (R¦ R

¥ ¦ £

H

£R

W

m

²¬ ¬ u u¦

where and are related by . Line 9 gives which results in

w ¥

W W ¡ £

H W"U "U

W

y

7QfH

XU

¬ H §§ R ¦ R

¦ £

H

£R

W

$

As a consequence, the analogue of the relation (6.5) for Algorithm 6.23 is

r± w°q ®

i

¡

§ ¬ ¢

¢ ¢

XU

$ T

¥¤

u u ¦ ( ¦

As before, for any the matrix represents the matrix with columns . The

X w $

¡ W

¥

matrix is now of size .

¢ ! !

¡

Now the block generalizations of FOM and GMRES can be de¬ned in a straightfor-

ward way. These block algorithms can solve linear systems with multiple right-hand sides,

r± r°w°q ®

i

¨ ¬ P R G t§ R

© 7E

¬

(P G ( "¢(

(¡

WVT y

TT

or, in matrix form

p±r°r°w°q ®

i

¬ ¤§ ©¢

(

R R

¥¤ ¨ ©

where the columns of the matrices and are the ™s and ™s, respectively.

R¢

¤ PG ¢P G

¡

¬˜E P ’©

Given an initial block of initial guesses for , we de¬ne the block of

G V(

( ¢

¡

y

initial residuals

£

¢

(8& X G ¨ ( P G ¨ ( eDG ¨ %!

P ( PW $

R R R

t§ ² ¬

©¨

where each column is . It is preferable to use the uni¬ed notation

P G ¨ P G PG

derived from Algorithm 6.23. In this notation, is not restricted to being a multiple of the !

R¦

block-size and the same notation is used for the ™s as in the scalar Arnoldi Algorithm.

¡

Thus, the ¬rst step of the block-FOM or block-GMRES algorithm is to compute the QR

factorization of the block of initial residuals:

¢ ¢

g

¬ & ¦ ( £ ¦ ( ¦ %

(

$

W

X ¢

’

¥

Here, the matrix is unitary and is upper triangular. This factorization

& ¦ V( $¦

( ¡ ¡

W¡

X

provides the ¬rst vectors of the block-Arnoldi basis.

Each of the approximate solutions has the form

r±Q0r°w°q ®

i

R

© R¬ PRG © R

( P G b ¢ w P G

Rb

PG©

and, grouping these approximations in a block and the in a block , we can

¤ PG

C³ C

³

yq¦" © ¡’¨ ¨ ¢ © j

„ 5| §

T£

¢ !

write

a±a r°w°„ ®

i

¤¥¬ ¤ w

¢