“incomplete” factorization replaces (12.21) by

An j

a S I

˜

˜ c

) # } ) £ v ) „ ) 5 ) x #c r )

D

in which is the matrix of the elements that are dropped in this reduction step. Globally,

)#

the algorithm can be viewed as a form of incomplete block LU with permutations.

Thus, there is a succession of block ILU factorizations of the form

¥

) £a ) „

u˜) 2 ) w'2 D

)

¦) ))5 ¥ ¥ ¥

)

¢ ¢ ¢

)£

)„

¢

D

˜

c t

¢ ¢

)5 ¦ c r) ¦)#

¦

v) „

˜ ˜

with de¬ned by (12.22). An independent set ordering for the new matrix will

) )1

cr cr

then be found and this matrix is reduced again in the same manner. It is not necessary to

˜

save the successive matrices, but only the last one that is generated. We need also to

)

save the sequence of sparse matrices

¥

An¦ j

)£

)„ I

˜

) Dcr c

¢

v) „ ) 5 ¦

which contain the transformation needed at level of the reduction. The successive per- H

mutation matrices can be discarded if they are applied to the previous matrices as

)2 C

soon as these permutation matrices are known. Then only the global permutation is needed,

which is the product of all these successive permutations.

An illustration of the matrices obtained after three reduction steps is shown in Figure

12.7. The original matrix is a 5-point matrix associated with a grid and is therefore ¢

¢X ¢e

¡

¡e

of size . Here, the successive matrices (with permutations applied) are shown C

¡

dd D

Q

˜

together with the last matrix which occupies the location of the block in (12.23). ¢

)

™

}|{ p w— x {— o wp7k…

{7 { | { |} #

£¤¢

¡ e

“

#

¡

¤ ¤

% # 3¡

!

Illustration of the processed matrices obtained

from three steps of independent set ordering and reductions.

We refer to this incomplete factorization as ILUM (ILU with Multi-Elimination). The

preprocessing phase consists of a succession of applications of the following three ˜{i

R

steps: (1) ¬nding the independent set ordering, (2) permuting the matrix, and (3) reducing

it.

’u % # ¡v¤r …¨ ¦3¥¢

¦¤ ¡

£8 4 ! ! &$

¢

6 )¡§ ©¨

¦)6 §¡ 6

!0

d

&

6

¨

˜¢ ˜

1. Set .

D

2. For Do:

9H

D {i2¢i¢’ei

˜R t h h h t t

e

˜

3. Find an independent set ordering permutation for ; )2 )

)˜

4. Apply to to permute it into the form (12.20);

)2

5. Apply to ;

)2 )

ihii¢t c

t hh

¢hihi¢t

6. Apply to ;

)2 2 )2

th c

v

˜

7. Compute the matrices and de¬ned by (12.22) and (12.23).

) )

cr cr

8. EndDo

In the backward and forward solution phases, the last reduced system must be solved but

not necessarily with high accuracy. For example, we can solve it according to the level of

tolerance allowed in the dropping strategy during the preprocessing phase. Observe that

if the linear system is solved inaccurately, only an accelerator that allows variations in

the preconditioning should be used. Such algorithms have been discussed in Chapter 9.

Alternatively, we can use a ¬xed number of multicolor SOR or SSOR steps or a ¬xed

polynomial iteration. The implementation of the ILUM preconditioner corresponding to

—™

—t