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structure as the -part of , the question is whether or not it is possible to ¬nd and

that yield an error that is smaller in some sense than the one above. We can, for example,

¦§

¨

try to ¬nd such an incomplete factorization in which the residual matrix has zero

elements in locations where has nonzero entries. This turns out to be possible in general

¦

and yields the ILU(0) factorization to be discussed later. Generally, a pattern for and

¦

can be speci¬ed and and may be sought so that they satisfy certain conditions. This

leads to the general class of incomplete factorization techniques which are discussed in the

next section.

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§5"0# ¥§¨

¢§ ©

Table 10.1 shows the results of applying the GMRES algorithm with

SGS (SSOR with ) preconditioning to the ¬ve test problems described in Section

¡ G

3.7.

Matrix Iters K¬‚ops Residual Error

F2DA 38 1986 0.76E-03 0.82E-04

F3D 20 4870 0.14E-02 0.30E-03

ORS 110 6755 0.31E+00 0.68E-04

F2DB 300 15907 0.23E+02 0.66E+00

FID 300 99070 0.26E+02 0.51E-01

¤ 5)0§ ¥¥©

§ ¢

A test run of GMRES with SGS preconditioning.

See Example 6.1 for the meaning of the column headers in the table. Notice here that the

method did not converge in 300 steps for the last two problems. The number of iterations

for the ¬rst three problems is reduced substantially from those required by GMRES with-

out preconditioning shown in Table 6.2. The total number of operations required is also

reduced, but not proportionally because each step now costs more due to the precondition-

ing operation.

““nfh…tv"g—P!U fE–”“ •— “X “™x•

˜ • • • … …

¡

¥¦u¥¦

®%„BDB% G ‘¥ 0% S £

¬¬¬ %

Consider a general sparse matrix whose elements are . A general

Incomplete LU (ILU) factorization process computes a sparse lower triangular matrix

¨ ¦§ ¤

¦

and a sparse upper triangular matrix so the residual matrix satis¬es cer-

tain constraints, such as having zero entries in some locations. We ¬rst describe a general

¡

ILU preconditioner geared toward -matrices. Then we discuss the ILU(0) factorization,

the simplest form of the ILU preconditioners. Finally, we will show how to obtain more

accurate factorizations.

p¡¶

·

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W

A general algorithm for building Incomplete LU factorizations can be derived by perform-

ing Gaussian elimination and dropping some elements in predetermined nondiagonal posi-

¡

tions. To analyze this process and establish existence for -matrices, the following result

of Ky-Fan [86] is needed.

‚n)|¡w¤

¥¦ ¦¥ 5"¢ §

§

¡

Let be an -matrix and let be the matrix obtained from the

f

¡

¬rst step of Gaussian elimination. Then is an -matrix.

f

£

§ §¦

¥

Theorem 1.17 will be used to establish that properties 1, 2, and 3 therein are

satis¬ed. First, consider the off-diagonal elements of :

f

£ S£

S

¨ ¬

¡£ £ ff

fS

ff £ E

£ S S ¢

3 fS

Since are nonpositive and is positive, it follows that for .

£ f ¤% f £ %

£ £ ¥

ff

Second, the fact that is nonsingular is a trivial consequence of the following stan-

f

dard relation of Gaussian elimination

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f ¢ 5D §s

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DBD% £ « % f C

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where

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f f

ff