versions of GE may be different.

It is helpful to interpret the result of one incomplete elimination step. Denoting by , S5

´ ¦

, and the -th rows of , , and , respectively, then the -loop starting at line 2

5 S££

5S

5S ´

of Algorithm 10.3 can be interpreted as follows. Initially, we have . Then, each £ 5S

elimination step is an operation of the form

¬ 5 ´ S ¡¨ 5 S ´ 5 S ´

However, this operation is performed only on the nonzero pattern, i.e., the complement of

. This means that, in reality, the elimination step takes the form

E

´ 5S ´ ´ S ¢“5 S

Cw

¨

% 5 8S

5

E 8 ´ S

¦ ¦

C

in which is zero when and equals when . Thus, the row

a ¥ 0 ` a ¥ % 7`

%

S

E 8 ´ S

C

cancels out the terms that would otherwise be introduced in the zero pattern. In

5S

the end the following relation is obtained:

´ S £

S

E 8

f'd

P5 S £ 5 S ´ C

¨ ¬ ¤¥5 S ¨“5

f E 8

¦ C

E

S

Note that for . We now sum up all the ™s and de¬ne

¡ 0 `

a% 5S

C f'd S E 8 R6 ¥v) ¥ £

¦‘

C

¬

5S 5S

f

C

The row contains the elements that fall inside the pattern at the completion of the

5S

™ SS

-loop. Using the fact that , we obtain the relation,

G

S

F ¥v) ¥ £

¦‘

¬i5 S ¨5 ´ S C

5 S £

f

Therefore, the following simple property can be stated.

‚£

¦ ¦¦ E¤¨ ¤¦ n¦

¥¨ £ £ t¶"0§

¢ ¦

Algorithm (10.3) produces factors and such that

§

¦

¢

¨

¥

¤

¨

in which is the matrix of the elements that are dropped during the incomplete elimina-

¤

¦ S C ££

¨

tion process. When , an entry of is equal to the value of obtained at ¤

a ¥ % 7` S

¶ 8˜ H ˜ ¤ ¨°¡ $ H&’ ”8 ” &’ ¤ ©¦§

©8 $ $

¡¥

¤ ¡¨ ¨¦ ¤

© §¥ c#&) $

©1

¢

¡

S C

the completion of the loop in Algorithm 10.3. Otherwise, is zero.

§¥¥¦£¤¢@ T P ¡@ T P

W P0(T T P

53¨3¢6

42 §2 ¡ U IH3£

e E

The Incomplete LU factorization technique with no ¬ll-in, denoted by ILU(0), consists of

taking the zero pattern to be precisely the zero pattern of . In the following, we denote

®

by the -th row of a given matrix , and by , the set of pairs § a(§`

5S % a ¥ % 7` 3 ¥ 0 3

%

C E G

S

¢ C

such that .

¦

¦

¶t)0d

¢§

¤¥£ ¢

¡ ¥

The ILU(0) factorization for a ¬ve-point matrix.

The incomplete factorization ILU(0) factorization is best illustrated by the case for

which it was discovered originally, namely, for 5-point and 7-point matrices related to ¬nite

difference discretization of PDEs. Consider one such matrix as illustrated in the bottom

¯ p ®

left corner of Figure 10.2. The matrix represented in this ¬gure is a 5-point matrix of

Q

H