£ ¤ %

d

f

@ s h

¨

©

where, using to denote a zero vector of dimension , and to denote the vector

C

S£

s®„D¬BD% ±ª% C

%¬ ¬

of components ,

7

˜ dA @ w

¨ 8

¬ ¡ % ®

E G8 `(

a

£ G

From this follow the relations

£

w w

¬

£ ¤ ¤

fd

¨

®

Applying this relation recursively, starting from up to , it is found that

sG s Gw Gq¥v0¥ £

¦µ ‘)

f ¬BD¬ « BD¬

w w w

s s s

«ds

¬ ¬¬ BD¬

¬¬ ¬ 'd

£ ¤ ¤ d ¤

'd

f fd d

f f f f

Now de¬ne

` DB¬ ¦

s

¬ fd s

¬¬ £

(f'd a

%

f'd f

¶

’ ”8 ” &’ ¤ ¦§ ’ ”p¡ &ª8 ¤ ¥ c

$ © $ 8¥ ’ ¨¥

§¤

1 ¡¡© &

$

¢

¡

Then,

d

¦ w

f ¢

with

« DB¬

w w

¬ d s ¤ w « d s

s s

¬¬ DB¬

¬¬

¤ ¤ f'd

¢

'd

f f f

¨ o®` ¯¡ ¨ o®`

Observe that at stage , elements are dropped only in the lower part of

Y

a a

. Hence, the ¬rst rows and columns of are zero and as a result

¤

¤ f DB¬ ¤ f BD¬

s s

¬¬ ¬¬

'd

f d

f

so that can be rewritten as

¢

« BD¬

s w

BD¬ w ¤ w

qa d s

¬¬ ¬¬ « ¬f

f ¤c

¤

`

¢

fd

If denotes the matrix

¤

w

DB¬ w ¤ w s

3

¬¬ «

f© ¤

¤ ¤ %

'd

f

then we obtain the factorization

§

¦

3

¨ x¤

%

`

¦ ¦

where is a nonnegative matrix, is nonnegative. This completes the

¤

d a

f d

f d

f

proof.

Now consider a few practical aspects. An ILU factorization based on the form of Al-

w

®

gorithm 10.1 is dif¬cult to implement because at each step , all rows to are being G

modi¬ed. However, ILU factorizations depend on the implementation of Gaussian elimi-

nation which is used. Several variants of Gaussian elimination are known which depend on

the order of the three loops associated with the control variables , , and in the algorithm.

¥

Thus, Algorithm 10.1 is derived from what is known as the variant. In the context of 0I

%% ¥

Incomplete LU factorization, the variant that is most commonly used for a row-contiguous

data structure is the variant, described next for dense matrices.

Q 0

%% ¥

‘© % £ ¦ ¦ ¡ %

£¢• ¡© £ ¨ % ¨ © % £ %

t¶"0§ ¡w¤h¨ n¦ ¤ £¢