SS

£ E

§ §¦

¥

The result can be proved by induction on . It is trivially true for . To prove

that the relation (10.28) is satis¬ed, start from the relation

´ P5 C ´ S ¢¨ 5 C S ´ C

f 5 S

¨ 5S

C

p¶

· 8˜ H ˜ ¤ ¨°¡ $ H&’ ”8 ” &’ ¤ ©¦§

©8 $ $

£

¢ ¡¨ ¨¦ ¤

© §¥ c#&) $

©1

¢

¡

´% E ´ ´

C C

E E

S 3 S S

3

in which . Either is zero which yields , or 3 f S

S C

g

3

´ S

is nonzero which means that is being dropped, i.e., replaced by zero, and therefore f

´ S C

E E

again . This establishes (10.28). Note that by this argument except when

3 f S

´ ´ ´ S ¨

C

E E

S 3

the -th element in the row is dropped, in which case and . f S

C

¥ S

C

E

3 S

Therefore, , always. Moreover, when an element in position is not dropped, a ¥ 07`

%

then

´ ´ S ¢¨ S ´ ´

S S

3

C C

f

C

C

and in particular by the rule in the modi¬cation of the basic scheme described above, for

® ²” n¥

, we will always have for , S¥

qWµ v0¥ £

¦¥ ‘)

´ ´

S

3 fS

6C

6C

in which is de¬ned in the statement of the modi¬cation.

S¥

´

Consider the row sum of . We have f

5S

”a f 5 C S ´ ` ¢ C S ¡ªa 5 S ´ `0¢ ªa 5 C ´´ ` ¢ C

C C C C

a 5 S `

¨ C ¨ ¢

GGµ v0¥ £

¦ ‘)

S ¡ªa 5 S ´ `0¢

C

¨ C C `¢ a5

¦GGµ v0¥ £

µ‘)

´

C

a 5 S `0¢

C

w

which establishes (10.29) for . G ® ”

²

It remains to prove (10.30). From (10.29) we have, for ,

6 µ v0¥ £

¦ ‘)

´ C ´ ¨ s ´

f S S S ¡ f S ¡

f

s C

C f ´ C f ´ ¦GF µ v0¥ £

‘)

DB¬ ¡ 6 C S ¡ ¡ f 6 C S ´ ¡

¬¬ ¦GI µ v0¥ £

‘)

r¬¡ 6 C S ¡£ W¡ 6 C fS ¡

S ´

Note that the inequalities in (10.35) are true because is never dropped by assumption ¦

6C ¡ S ¡£

E¡

and, as a result, (10.31) applies. By the condition (10.27), which de¬nes matrices,

´ ¦s

6C

s

®

² #

®

is positive for . Clearly, when , we have by (10.34) . This completes

the proof.

The theorem does not mean that the factorization is effective only when its conditions are

satis¬ed. In practice, the preconditioner is ef¬cient under fairly general conditions.

¦

¨2 ¡3¢6

§ 2 ¡ T SQY SR rb¨Q3WIhID¤ f P

W UP Y 9 Y H f H T

BP9 H

A poor implementation of ILUT may well lead to an expensive factorization phase, and

possibly an impractical algorithm. The following is a list of the potential dif¬culties that

may cause inef¬ciencies in the implementation of ILUT.

Generation of the linear combination of rows of (Line 7 in Algorithm 10.6).

¦

Selection of the largest elements in and .

Need to access the elements of in increasing order of columns (in line 3 of