W

tion,

a± q ®

i

W"U u ¦¬ ¥¬ u ¦©§ u ¤ "U u ¬ § W"U u

u u u§

¥ ¥

¤ ¦

£

"U

W W

´C ¨ ¡¡µ£ „ ¢

§|5¥ yq¢| ¢ £ ¥§

„ 5| j C¦£¥

5§

"–© ¡"

§

1B

W¤(¡˜ w ¬

u u £R u

¥

Now observe that since the components of are zero, then for

(

£ £

any . Hence,

˜ w E

¥ ¢ ¥¬ v ( u ©§ ¤ ¬ u

¬¦

£ u

u ¥

#(V(

!

¢ ¢

£ £

W“

U y

This leads to the factorization,

r± w°„ ®

i

™& X §W§(T ( £ §W§( W ©’§ $¦ ¤

¬¦

¦(

¦ ( & ¢ £ (

(

$

¢ ¢ £ £

W

w ! ¤ & ¥$

where the matrix is and is upper triangular and is unitary. ¤

(

(

$ ¢ ¢

£ £

R Ty R

It is important to relate the vectors and de¬ned in this algorithm with vectors of ¦ £

¡ X

¥uy

the standard Arnoldi process. Let be the matrix obtained from the ¬rst w

¢ ! !

u

¤

¥ ¬

rows of the matrix . Since is unitary we have

w! W “u

¤ ¤ ¤

& ¢ £ (

(

$

! £

W W"U

y "U

W

W"U

and hence, from the relation (6.9)

u u

"U f u ’U f

W W u

¬ u ¦§§ ¬R

¡ uR vR

u R¡

¤ ¤

£ £

£R "U R

W W"U

W W

¤ ¤

¥ ¬ H¥

R¡ R R¡

where each is the -th column of the identity matrix. Since for ,

E E

¡ W

it is not dif¬cult to see that

±R%°t°q ®

i

¦¬ ¬

¥ ¥

R¡ u ( R ¦ R ¡ for

u w

¤ E

uW

W"U y

’U

W £

¬ u ©§ ¬$

R ¦ vR £ "U £R

u W

This yields the relation , for , which can be written in

¦ #(V(

!

y

W

matrix form as

¡

§ ¬ ¢

¢ ¢

’U

W

This is identical with the relation (6.5) obtained with the Gram-Schmidt or Modi¬ed Gram-

R¦

Schmidt implementation. The ™s form an orthonormal basis of the Krylov subspace ¢

R¦

and are identical with the ™s de¬ned by the Arnoldi process, apart from a possible sign

difference.

Although the Householder algorithm is numerically more viable than the Gram-

Schmidt or Modi¬ed Gram-Schmidt versions, it is also more expensive. The cost of each

of the outer loops, corresponding to the control variable, is dominated by lines 7 and 8.

}E

¬

R

¥

These apply the re¬‚ection matrices for to a vector, perform the matrix- YV(

(

Ry

§ ²z (t vE ¬

u© ¥

vector product , and then apply the matrices for to a vector. The

¦ (V(

Y