h S ¦

¨ S

Observe that the elements and are identical with the corresponding elements of

f

the matrix. The other values are obtained from the following recurrence:

c

S

iS h S¢

d

S

ie

S S¢

¬ S ¦ S ¨ ¨

¨ S f'd S y

S¢

S

The above recurrence can be simpli¬ed further by making the observation that the quan-

tities and need not be saved since they are scaled versions of the corre-

h S ¢ 0

S SS

d ¢

d f

sponding elements in . With this observation, only a recurrence for the diagonal elements

is needed. This recurrence is:

S¢

¨

0¥v) ¥ £

¦' ‘

S c

SS S

¨ Sy ¨ s® DBD% G @

¬¬¬

S¢ %h %„

%

S¢ S0¢

d

f d

¢

with the convention that any with a non-positive index is replaced by and any other ¥

G

element with a negative index is zero. The factorization obtained takes the form

) v) ¥ £

¦‘

(¡

¨ ¨

`

¨ ` d a !

¨f ¨ ©

a

¨ ¨

in which is the strict lower diagonal of , is the strict upper triangular part of ,

!

and is the diagonal obtained with the above recurrence. Note that an ILU(0) based on

¨

the IKJversion of Gaussian elimination would give the same result.

For a general sparse matrix with irregular structure, one can also determine a pre-

¡

conditioner in the form (10.20) by requiring only that the diagonal elements of match

those of (see Exercise 10). However, this will not give the same ILU factorization as the

one based on the IKJvariant of Gaussian elimination seen earlier. Why the ILU(0) factor-

ization gives rise to the same factorization as that of (10.20) is simple to understand: The

¦

product of and does not change the values of the existing elements in the upper part,

except for the diagonal. This also can be interpreted on the adjacency graph of the matrix.

This approach can now be extended to determine the ILU(1) factorization as well as

¦

factorizations with higher levels of ¬ll. The stencils of the and matrices in the ILU(1)

factorization are the stencils of the lower part and upper parts of the LU matrix obtained

from ILU(0). These are shown in Figure 10.9. In the illustration, the meaning of a given

stencil is not in the usual graph theory sense. Instead, all the marked nodes at a stencil

based at node represent those nodes coupled with unknown by an equation. Thus, all

the ¬lled circles in the picture are adjacent to the central node. Proceeding as before and

combining stencils to form the stencil associated with the LU matrix, we obtain the stencil

shown in Figure 10.10.

¶ 8˜ H ˜ ¤ ¨°¡ $ H&’ ”8 ” &’ ¤ ©¦§

©8 $ $

¡¥

¡¨ ¨¦ ¤

© §¥ c#&) $

©1

¢

¡

¦

h SY¥ h S

d

f

¨ S

S S¢ f

G

¨

S S

·t)0d

¢§ ¦

¤¥£ ¢

¡ ¥

Stencils associated with the and factors of

the ILU(0) factorization for the matrix associated with the sten-

cil of Figure 10.8.

h S ¦S

w

h S ¥ h S ¦ f'd

'd

f

h S ¥S

«d

w¦ w

w

S S S ¨ e 0¢ ¨

S

S 2¥ S

S

S ¦S ¨ w S ¨

¡

f

f

w

SS S S

'd ¢ d ¥

f f

h S ¨S ¨

¢

h S ¢S

d h