3

Assume, inductively, that the matrix has been transformed in successive steps into

0

@¥¡ £ § ¡ ¢¦£ ¨ £ ¡ £

n¥ ¢© ¥¥ ¡§

¡£ ¥ ¥

©¢

§ ©

the partially upper triangular form

344 99@8

9

60

E0

0 0 0

¦ ¦ ¡ ¦ ¦

§¥

¥¥ §¥

¥¥ §¥

¥¥

”” 6

44 E6

6 6

¦ ¦ ¦

99

¥§¥ ¥ ¥§¥ ¥ ¥§¥ ¥

44 ¢ ¦ ¦

99

¥§¥ ¥ ¥§¥ ¥ ¥§¥ ¥

”

44 .

.. . 99

. .

0 0 0 1) 0 99

44 §¥

¥¥ §¥

¥¥

.

' '

0 .

''

44 ¦

.

&

Y §¥

¥¥ 9

'" 0 4 ' 0 4' "

¦ ¦

¥§¥¥

”

. . .

. . .

5 A

. . .

'" "

¦ ¦

§¥

¥¥

‘ ”‘

q

3

This matrix is upper triangular up to column number . To advance by one step, it must

be transformed into one which is upper triangular up the -th column, leaving the previous

3

columns in the same form. To leave the ¬rst columns unchanged, select a vector

3

which has zeros in positions through . So the next Householder re¬‚ector matrix is

de¬ned as

t

G' '

3 ¨ 2 ¦ §¥

¦

'

'

in which the vector is de¬ned as £

6 £ t

¨ ¢ ¦ §¥

¦

'

£

where the components of the vector are given by

£

£

7 if

¥¤

q — @

t §¦¥

@

¦ ¨

£&

¦

if

'

¦

if

with

6 0

§‘' ( C ' ' 6

& t

¢

¦ B ©¦ & §¥

¨¦

¦ 0 %

( ¦

'

We note in passing that the premultiplication of a matrix by a Householder trans-

0

form requires only a rank-one update since,

b G 0

3

B 3

¢

where

0C G 0 G

Therefore, the Householder matrices need not, and should not, be explicitly formed. In

addition, the vectors need not be explicitly scaled.

3

Assume now that Householder transforms have been applied to a certain matrix

uY u d¥ £ ¡ ¨ ¢¤¥

n ¦¢¥

¤¤§ ¥ §¢

¡

£ ¡© ¥ ¡ © ¥ ¡ ¥ © ¥ ¥

¥ ¨

h˜