p¥¶

·¤

’ ”8 ” &’ ¤ ¦§ ’ ”p¡ &ª8 ¤ ¥ c

$ © $ 8¥ ’

1 ¡¡© &

$

¢

¡

S

(this will be changed later) and a zero element has a level of ¬ll of . An element is

£

updated in line 5 of Algorithm 10.2 by the formula

P ¥v) ¥ £

¦‘

S S ¬ £ n S £ ¨

¯

£ £

S £ S

If is the current level of the element , then our model tells us that the size of the

£

updated element should be

£t ¡ ¢£t ¡ £t ¡ ¢£t ¡ £t ¡ ¥t ¡

¢ ¢ ¢ ¢

¡ ¡

¡ ¡

¤ ¤ ¤ ¤

6 ¦¨ 6 S £ ¯ 6 ¦¨ 6 ¬

£t ¡

¢

¡

SY£

Therefore, roughly speaking, the size of will be the maximum of the two sizes 6

£t ¡ £t ¡

¢ ¢

¡

¤ ¤

and , and it is natural to de¬ne the new level of ¬ll as,

6 &W

w

S £ % S £ S £ v( £

'%

& ¬

In the common de¬nition used in the literature, all the levels of ¬ll are actually shifted

¨

by from the de¬nition used above. This is purely for convenience of notation and to

E E

G S £ S

¢ £

conform with the de¬nition used for ILU(0). Thus, initially if , and

S £

otherwise. Thereafter, de¬ne recursively

&W

¦

w w£

S £ % S £ S £

&'%

v( G

¬

¥ ¦¦ £¤h¥ ¨

¥¨¨ 5"0§

§ ¢

§ ¡£

The initial level of ¬ll of an element of a sparse matrix is

S

de¬ned by U

¨ E E

&U

©

¢ S £ H

% ¥

S £

& ¬

!

Each time this element is modi¬ed in line 5 of Algorithm 10.2, its level of ¬ll must be

updated by &W

¦ ¤ ¥v) ¥ £

‘

w w£

S £ % S £ S £

&'%

v( G

¬

Observe that the level of ¬ll of an element will never increase during the elimination. Thus,

E

¢ S £

if in the original matrix , then the element in location will have a level of 0

% ¥

¬ll equal to zero throughout the elimination process. The above systematic de¬nition gives

rise to a natural strategy for discarding elements. In ILU` , all ¬ll-in elements whose level ¦

a

of ¬ll does not exceed are kept. So using the de¬nition of zero patterns introduced earlier,

the zero pattern for ILU( ) is the set &

S £

w¤

¡ a ¥ 07` F3

%(

%

S £

where is the level of ¬ll value after all updates (10.18) have been performed. The case

E

v

coincides with the ILU(0) factorization and is consistent with the earlier de¬nition.

In practical implementations of the ILU( ) factorization it is common to separate the

¦

symbolic phase (where the structure of the and factors are determined) from the nu-

merical factorization, when the numerical values are computed. Here, a variant is described

which does not separate these two phases. In the following description, denotes the -th £ 5S

S

row of the matrix , and the -th entry of . £ a ¥ 0 `

%

“™x•

"0§ ¡w¤h¨ n¦ ¤ £¢

¦¢

¢ !"

E

”a S £ ` £

S

1. For all nonzero elements de¬ne

£

®s„D¬BB% H H

%¬ ¬

2. For Do:

¶ 8˜ H ˜ ¤ ¨°¡ $ H&’ ”8 ” &’ ¤ ©¦§

©8 $ $

£¥

¢ ¡¨ ¨¦ ¤

© §¥ c#&) $

©1

¢