w

h S ¨S d ¨ SS

d¢

d

f f

05)0d ¥ ¤¥£ ¢

¢ § ¢ §

¡

Stencil associated with the product of the

¦

and matrices whose stencils are shown in Figure 10.9.

As before, the ¬ll-in elements are represented by squares and all other elements are

¬lled circles. A typical row of the matrix associated with the above stencil has nine nonzero

elements. Two of these are ¬ll-ins, i.e., elements that fall outside the original structure of

¦

the and matrices. It is now possible to determine a recurrence relation for obtaining

¦

the entries of and . There are seven equations in all which, starting from the bottom,

are

h S ¢S c

S

d h S0¢ S ¨ w h S E

¨S

d

f

dw

f

S ¢e S

2¥ S

S S

f'd w ¦ f'd w ¨

w y

i

2¥ S ¨ S S S e

S¢ S

S S

S ¦S ¨ w S ¨ S ¨

f

f fE

h S ¦S w h S

¥

fd

'd ¦

f

h S

¬

hS

¶

’ ”8 ” &’ ¤ ¦§ ’ ”p¡ &ª8 ¤ ¥ c

$ © $ 8¥ ’ ¡¥

1 ¡¡© &

$

¢

¡

¦

This immediately yields the following recurrence relation for the entries of the and

factors:

h S ¢ c

S S

h dS ¨ S h S¢

— S ¨

¨

d

f d

f

¨ iS 2¥ S 2$` S

S S0¢ a

f'd ¨ d ¦

f

y

¨ ¨ ªS0¢ ¨

Y¥ S ¨ S S

S

SS S

S ¦ S! S ¨

f S ¨

¨ ¨f f

h S ¦ Se

U¨— 'd h ¦ 2¥

fS d

f

h S h S

¬

In proceeding from the nodes of smallest index to those of largest index, we are in effect

performing implicitly the IKJversion of Gaussian elimination. The result of the ILU(1)

obtained in this manner is therefore identical with that obtained by using Algorithms 10.1

and 10.3.

¡@ T P

¥

I5p PcXU f

3¨23¢6

2§ ¡ R HP R @ TP f

¡

In all the techniques thus far, the elements that were dropped out during the incomplete

elimination process are simply discarded. There are also techniques which attempt to re-

duce the effect of dropping by compensating for the discarded entries. For example, a

popular strategy is to add up all the elements that have been dropped at the completion of

¦

the -loop of Algorithm 10.3. Then this sum is subtracted from the diagonal entry in .

This diagonal compensation strategy gives rise to the Modi¬ed ILU (MILU) factorization.

´

Thus, in equation (10.14), the ¬nal row obtained after completion of the -loop of

5S

Algorithm 10.3 undergoes one more modi¬cation, namely,

¨ SS ´ C SS ´ C

a 5S `

˜ a „¬BD¬B% % ` C

%¬

in which . Note that is a row and is the sum of the elements 5S 5S

´

GG

G

in this row, i.e., its row sum. The above equation can be rewritten in row form as 5S

˜S a 5 S C ` ¨ 5 S ´

and equation (10.15) becomes

S

¬W5 S ¨ ˜S a 5 S C ` ui v) ¥ £

¦¥ ‘

´ S C

w

5 S £

5

f

Observe that

S S

¨ ˜S a 5 S C `

´ S 5 ´ S

f'd