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.

Recall that our initial goal was to obtain a QR factorization of . We now wish to recover 0

0 '

the and matrices from the ™s and the above matrix. If we denote by the product

5

of the on the left-side of (1.22), then (1.22) becomes

t

5¡ ¨ ¢ & §¥

¦

0

¡

C ˜B

3

in which is an upper triangular matrix, and is an zero block.

5

Since is unitary, its inverse is equal to its transpose and, as a result,

” 11”6 0

5¡ 5¡

0

0 G

&

! ˜

If is the matrix of size which consists of the ¬rst columns of the identity

”¢

matrix, then the above equality translates into

5”¢G 0

0

The matrix represents the ¬rst columns of . Since

£GP

¢ G

” ” ¢ G G” ¢ 0 G 0

0 and are the matrices sought. In summary,

5

¤0 0

5

in which is the triangular matrix obtained from the Householder reduction of (see

5 0

(1.22) and (1.23)) and

1)#6 0 … 0 …

¤ ¤

0

” &

¨V¤ G ¥ ¤¢

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§ ¥¤

§©

QW F ©U

T

111)‘

3 i)1) 0 1¦ 0 ¦©

1. De¬ne

”

© ' '

2. For Do:

0 )1 6 ' 0 '

'

¦

3. If compute

& &

'

4. Compute using (1.19), (1.20), (1.21)

© ' '

' '

3

'''

5. Compute with G'

11 6 '¤'

) ' 0

6. Compute

7. EndDo

@

u d

u

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£¥

§

©

)

Note that line 6 can be omitted since the are not needed in the execution of the

0

next steps. It must be executed only when the matrix is needed at the completion of

G

—

the algorithm. Also, the operation in line 5 consists only of zeroing the components

˜ i1)1‘' ' '

and updating the -th component of . In practice, a work vector can be used for '

and its nonzero components after this step can be saved into an upper triangular matrix.

'

Since the components 1 through of the vector are zero, the upper triangular matrix 5

can be saved in those zero locations which would otherwise be unused.