m¥ s®„DBD%C G

%¬¬¬

C ´ S

i„DB¬B% G

¨ %¬¬ 5 S £ 5 fS

for , in which initially and where is an element subtracted

C C

G

from a ¬ll-in element which is being dropped. It should be equal either to zero (no drop-

´ S ¨ S ´ ´

f S

ping) or to when the element is being dropped. At the end of the -th ¦

C

step of Gaussian elimination (outer loop in Algorithm 10.6), we obtain the -th row of ,

S´ 5S´ 6 v0¥ £

¦ ‘)

5S C C 'd

f

p¥¶

·

’ gc ”8

˜© ˜ ªpv

88 8k10 ³

¨¥ c© ¨ ¥£" Bc

$ ¡©

¢

¡

and the following relation is satis¬ed:

S

´ Cw

S

U5 C S £

5 S

%

5

C C

C

f

C

where is the row containing all the ¬ll-ins.

5S

C

The existence result which will be proved is valid only for certain modi¬cations of

the basic ILUT` strategy. We consider an ILUT strategy which uses the following

Y % "

a

modi¬cation:

© S

® w –

²

Drop Strategy Modi¬cation. For any , let be the element of largest

6 C £ sBBD%

%S£ ¡¥

®¬¬¬

modulus among the elements , in the original matrix. Then

C G

elements generated in position during the ILUT procedure are not subject to a S ¥ %0 `

the dropping rule.

This modi¬cation prevents elements generated in position from ever being dropped. a S ¥ %0 `

Of course, there are many alternative strategies that can lead to the same effect.

§ S ¥

A matrix whose entries satisfy the following three conditions:

F v) ¥ £

¦‘

E E

ss

® ”`3 G

²

for and

S 2¥S ¥

GI v0¥ £

¦ ‘)

E

3 s 2¥ £ ¢

¢

®s„¬DB¬D% G ¥ 0

%¬

for and

S % ¥

P W) ¥ £

¦‘

E

S ¥ ² ®³²‚`3 G for

%

'

S

f¦

¡

will be referred to as an matrix. The third condition is a requirement that there be at

least one nonzero element to the right of the diagonal element, in each row except the last.

The row sum for the -th row is de¬ned by s

”a 5 C S ¥ ` ¢ C ¬ C S ¥

„5 S ¥

C

f

§

¦

¡

A given row of an matrix is diagonally dominant, if its row sum is nonnegative. An

§

¦

¡ matrix is said to be diagonally dominant if all its rows are diagonally dominant. The

following theorem is an existence result for ILUT. The underlying assumption is that an

ILUT strategy is used with the modi¬cation mentioned above.

‚n)|¡w¤

¥¦ ¦¥ "¢ §

¦

´ ¡ E 5 S ´

If the matrix is a diagonally dominant matrix, then the rows

% 5 C S ´ E

H

¬BDB

¬¬ 5 C fS

de¬ned by (10.23) starting with and satisfy the

C£ 5S

%%%

%

C

G „DBD% G

%¬¬¬

following relations for

¦ ¤ v) ¥ £

‘

´ E

¢ C ¥

3 S

' v) ¥ £

¦‘

´ ´

C

f

E

a 5 S ` ¢ a 5 d S `0¢ %

EC C ) µ v) ¥ £

¦‘

´ ´ s s ¬E

¡‚

®²