R

£ £ ¤

a±!w°„ ®

i

£ £ £ £

¡ „

W R

¬ £ ¥

¤

¢ £ £ £ £

‚ ¥ R¤

R ¤R¤

R R

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£ £ £

R ¤¥ ¥¥ £

£

The Direct version of IOM is derived from exploiting the special structure of the LU fac-

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torization, , of the matrix . Assuming no pivoting is used, the matrix

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¢ ¢ ¢ ¢

is unit lower bidiagonal and is banded upper triangular, with diagonals. Thus, the ¤ ¢ W

above matrix has a factorization of the form

£

£ £ £

y£ ¢

™W

W ¥W£ `W£ R

£ £

£ £ £ ¤

¢

¡ „

„

W R

£y

¬ ¥ £ £ £

¥

¤

¢

¢

‚ ‚ ¥ R¤

R ¤R

R R

y¤ ¤¤ £ £

¢

R y ¥¥ £

¤¥

y

The approximate solution is then given by T

X

(¬ ©

© w £ W “¢ ¤

FS W “ ¢

¢ ¢ ¡

W

De¬ning

¥ ! ¤ W “¢

¢ ¢

and T

X

¬ £ FS W “ ¢ (

¢ ¡

W

the approximate solution is given by

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i

¬© © ¢¥

w ¢

¢

¥¤

Because of the structure of , can be updated easily. Indeed, equating the last

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¬¤¥

columns of the matrix relation yields

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¢

f

¬

¢R ( ¢R

£

¦

¡

¢ R

"UVH “

W

R

which allows the vector to be computed from the previous ™s and , with the help

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¢ ¢

¡ ¡

of the relation,

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W“f

¬ ² ¢

¢R R

£

¦

y

¢ ¢

¡ ¡

£

¢¢ ¢ £R

"QH “

WU

In addition, because of the structure of , we have the relation

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¢

W “¢ ¤ ¬ ¢

¥

´ ¨ ¡¡µ£ „ ¢

§|5¥ yq¢| ¢ £ ¥§

„ 5| j C¦£¥

5§

C

"–© ¡"

§

1B

in which

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¢ ¢ ¢

W“ W“

From (6.18),

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¢

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© ¥ w (¬ ¥ W “ ¢ ¤ & ¢

© W “

¥

w w

( ¢

¢$

¢ ¢ ¢ ¢

¡ ¡

W“ W“

W “ ¢ ¥ w © ¦¬

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Noting that , it follows that the approximation can be updated

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W“ W“