®„BDB% G ¥

%¬¬¬

Finally, we establish that is nonnegative by examining for . f'd f

f d f

f f

For , it is clear that because of the structure of . For the case f ¤ f i

¥

f f

G

¢

, (10.10) can be exploited to yield

¥

G 'd ¬E

¢ df f'd „f'd

f

f

f

d

Therefore, all the columns of are nonnegative by assumption and this completes the

f

f

proof.

x®4“a G x®`

¨`¯ ¨

Clearly, the matrix obtained from by removing its ¬rst row and ¬rst a

f

G

¡

column is also an -matrix.

Assume now that some elements are dropped from the result of Gaussian Elimination

outside of the main diagonal. Any element that is dropped is a nonpositive element which

is transformed into a zero. Therefore, the resulting matrix is such that £

f

w

f C

£ x¤

%

f

S C % E E

SS

where the elements of are such that . Thus,

¤

3f £

f

¡

and the off-diagonal elements of are nonpositive. Since is an -matrix, theorem £

f f

¡

1.18 shows that is also an -matrix. The process can now be repeated on the matrix

£

® H% ® H` f

, and then continued until the incomplete factorization of is obtained. The

£

a

¡

above arguments shows that at each step of this construction, we obtain an -matrix and

that the process does not break down.

The elements to drop at each step have not yet been speci¬ed. This can be done stat-

ically, by choosing some non-zero pattern in advance. The only restriction on the zero

pattern is that it should exclude diagonal elements because this assumption was used in the

¶ 8˜ H ˜ ¤ ¨°¡ $ H&’ ”8 ” &’ ¤ ©¦§

©8 $ $

£¥

¢¤ ¡¨ ¨¦ ¤

© §¥ c#&) $

©1

¢

¡

¡

above proof. Therefore, for any zero pattern set , such that

&

qG¥v0¥ £

¦¥ ‘)

£ ¢ ¡ a % 7` 8

¢ ®

3 ¥ 0`3 %(

%

¥ ¥

G

¥¦

¤

an Incomplete LU factorization, , can be computed as follows.

“™x© ! ¨¨0 % `£ £ ¨ % ¨ ˜ ¤ D%0 © 0 ¢ §5"0§ wE¤©n¦§¥£¢

• ¢ ¡ ¨¦ ¤¢

!

¦ ¡ % 7` G ¨ s®% s®% ¬Bw B¬D¬ % G ª

1. For Do:

2. For and if Do:

a

£ w G S £ S £

3.

¦ a %0 ` ®%„BD¬B% m

¬¬

4. For and for Do:

¥ ¥

£ ¨ S £ ¨ G S £ S £

§

5.

6. EndDo

7. EndDo

8. EndDo

w

©n¥

s®% BBD% G

¬¬¬

The For loop in line 4 should be interpreted as follows: For and only for

those indices that are not in execute the next line. In practice, it is wasteful to scan

¥ ¥

w

®

from to because there is an inexpensive mechanism for identifying those in this set

G

that are in the complement of .

Using the above arguments, the following result can be proved.

t¶"0§ “n)|w¤

¥¦ ¦¥¡

¢ ¡

Let be an -matrix and a given zero pattern de¬ned as in

(10.11). Then Algorithm 10.1 is feasible and produces an incomplete factorization,

Gq¥v0¥ £

¦ ‘)

¦

¢

¨

¥

¤

which is a regular splitting of .

§©¨¥¦£

§

At each step of the process, we have

£

w