73¡3¢6

62 2 ¡ UT RP e Y

T9W 8

9

BH P

£

¢

Consider a block-tridiagonal matrix blocked in the form

£¤ ¨¦

§

«

¨ !

¤ §

f

¤ §

« « £!

¨

¤ §

R!P v) ¥ £

¦6 ‘

.. .. ..

3 ¬

. . .

¥ ©

h h¨ h!

d

f d h

f h¨

One of the most popular block preconditioners used in the context of PDEs is based on

this block-tridiagonal form of the coef¬cient matrix . Let be the block-diagonal matrix ¨

consisting of the diagonal blocks , the block strictly-lower triangular matrix consisting

¦S ¨

of the sub-diagonal blocks , and the block strictly-upper triangular matrix consisting

S

of the super-diagonal blocks . Then, the above matrix has the form

S!

¦w

w

3

¬

¨

8˜ H ˜ ¤ ¨°¡ $ H&’ ”8 ” &’ ¤ ©¦§

©8 $ $

0¨

¢§ ¡¨ ¨¦ ¤

© §¥ c#&) $

©1

¢

¡

A block ILU preconditioner is de¬ned by

GF P v0¥ £

¦ ‘)

` ¦w

w

¡ a `

f'd %a

¦

where and are the same as above, and is a block-diagonal matrix whose blocks

S

are de¬ned by the recurrence:

GI P v0¥ £

¦ ‘)

¦

¨ S ¨ ª(

S S S S !

%

f'd

¦

in which is some sparse approximation to . Thus, to obtain a block factorization, 'd

f

approximations to the inverses of the blocks must be found. This clearly will lead to S

dif¬culties if explicit inverses are used.

An important particular case is when the diagonal blocks of the original matrix are S¨

tridiagonal, while the co-diagonal blocks and are diagonal. Then, a simple recur- S

S!

rence formula for computing the inverse of a tridiagonal matrix can be exploited. Only the

tridiagonal part of the inverse must be kept in the recurrence (10.76). Thus,

GP P v0¥ £

¦ ‘)

f¨ %

f E £8 ¦ ¦ ¤ P v0¥ £

‘)

¨ ±s% DBD% G @

¬¬¬

S¨

S S %S ! %

S

d

f

E £8 ¦

where is the tridiagonal part of .

d

f

E8¦ £

Sa S a fd

™ ¡

¨ ¬G

for

` 3 ¢¥

¡

C ` C

The following theorem can be shown.

“n)|w¤

¥¦ ¦¥¡ "0§

¢

Let be Symmetric Positive De¬nite and such that

E E

© 3 S

®s„DBD% G @

%¬¬¬ ¢

, and for all .

£ £

SS %

¥

© The matrices are all (strict) diagonally dominant.

S¨

¡

Then each block computed by the recurrence (10.77), (10.78) is a symmetric -matrix.

S

¡

In particular, is also a positive de¬nite matrix.

We now show how the inverse of a tridiagonal matrix can be obtained. Let a tridiagonal

matrix of dimension be given in the form

g ¨ f

£¤ ¨¦ §

«

¤ §

¤ §

« « ¨

£ ¨

¤ §

.. .. ..

. . .