¯¡ ®

¨

size corresponding to an mesh. Consider now any lower triangular

" ¦

matrix which has the same structure as the lower part of , and any matrix which has

the same structure as that of the upper part of . Two such matrices are shown at the top of

§

¦

Figure 10.2. If the product were performed, the resulting matrix would have the pattern

shown in the bottom right part of the ¬gure. It is impossible in general to match with

¦

this product for any and . This is due to the extra diagonals in the product, namely, the

¶

’ ”8 ” &’ ¤ ¦§ ’ ”p¡ &ª8 ¤ ¥ c

$ © $ 8¥ ’ ¡¥

¤

1 ¡¡© &

$

¢

¡

wp

¨p® P¨

®

diagonals with offsets and . The entries in these extra diagonals are called

G G

¬ll-in elements. However, if these ¬ll-in elements are ignored, then it is possible to ¬nd

¦

and so that their product is equal to in the other diagonals. This de¬nes the ILU(0)

¦

factorization in general terms: Any pair of matrices (unit lower triangular) and (upper

§

¦

#

¨

triangular) so that the elements of are zero in the locations of . These ab `

constraints do not de¬ne the ILU(0) factors uniquely since there are, in general, in¬nitely

¦

many pairs of matrices and which satisfy these requirements. However, the standard

ILU(0) is de¬ned constructively using Algorithm 10.3 with the pattern equal to the zero

pattern of .

¡ “™x• "¢0§ ¡w¤h¨ n¦

¤ £¢ ¦

¢

®s„D¬B¬B% H H

%¬

1. For Do:

¦ ¡ % 7` ™%„D¬BB% G

¨ ¬¬

b`

2. For and for Do:

a

a

G£ | S £

£ S w

3. Compute

¦ a ¥ %0 ` s®„D¬B¬B% m¥

%¬

abx`

4. For and for , Do:

£ S £ ¨ S £ S £G

5. Compute .

6. EndDo

7. EndDo

8. EndDo

In some cases, it is possible to write the ILU(0) factorization in the form

I ¥v) ¥ £

¦‘

¡ ¨ ¨

`

¨ ` 'd a !

¨f ¨ %„

a

¨ ¨

where and are the strict lower and strict upper triangular parts of , and is a

¨

!

certain diagonal matrix, different from the diagonal of , in general. In these cases it is

suf¬cient to ¬nd a recursive formula for determining the elements in . A clear advantage ¨

is that only an extra diagonal of storage is required. This form of the ILU(0) factorization is

equivalent to the incomplete factorizations obtained from Algorithm 10.4 when the product

of the strict-lower part and the strict-upper part of consists only of diagonal elements

and ¬ll-in elements. This is true, for example, for standard 5-point difference approxima-

tions to second order partial differential operators; see Exercise 4. In these instances, both

the SSOR preconditioner with and the ILU(0) preconditioner can be cast in the form

¡ G &

(10.16), but they differ in the way the diagonal matrix is de¬ned. For SSOR(¡ ), ¨

G

is the diagonal of the matrix itself. For ILU(0), it is de¬ned by a recursion so that

¨

the diagonal of the product of matrices (10.16) equals the diagonal of . By de¬nition,

¦

together the and matrices in ILU(0) have the same number of nonzero elements as the

original matrix .

t¶"0#¥¨¥

¢§ ©§ Table 10.2 shows the results of applying the GMRES algorithm with

ILU(0) preconditioning to the ¬ve test problems described in Section 3.7.

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