¥ ©¡

fd ¨

¨

¡ ¡

'd

f¡

¨ ¡

and let its Cholesky factorization be

˜

¨

%

with X &

y

% &

¨ (S

’ ”8 ” ’ ¤ §

$ © } ”vG

’ ˜8’ ’ ”p¥ &)0©

8 1 D¨

§§

¡¡© &

$ ¨' %©

¥ $ &

$

¢

¡ ¡

and

£¤ §¦

¤ §

¨G

¤ §

«¨

¤ §

G

.. ..

¬

. .

¨

¥ ©

¨ ¡

'd

f ¨G

¨ ¡

G

˜d ˜

The inverse of is . Start by observing that the inverse of is a unit upper

fd ¨ fd ´ S

triangular matrix whose coef¬cients are given by

DB¬ « S ¨ f S ¨ S ´ ¨ ¨

¬¬ q¬ ¥ ”

²²

for 3

fd G

˜ d ¨

¥ `

¨

As a result, the -th column of is related to the -st column by the very

¨

a

¥

d

f

G

simple recurrence,

H

¨ ¨©w

for

¨ % ¥

'd

f

starting with the ¬rst column . The inverse of becomes

¨

f f ¡

˜ d Pfd ¬ ˜ ¨ ¨ G y ' P v) ¥ £

¦‘

f'd ¨

f'd

f

See Exercise 12 for a proof of the above equality. As noted, the recurrence formulas for

computing can be unstable and lead to numerical dif¬culties for large values of .

'd

f

D!¥ CbQgf S¨eIHcIA8

53 3¡¢6

42 2 WH

T9

BH Pe Y9

A general sparse matrix can often be put in the form (10.74) where the blocking is ei-

ther natural as provided by the physical problem, or arti¬cial when obtained as a result of

RCMK ordering and some block partitioning. In such cases, a recurrence such as (10.76)

can still be used to obtain a block factorization de¬ned by (10.75). A 2-level precondi- ¦

tioner can be de¬ned by using sparse inverse approximate techniques to approximate . S

These are sometimes termed implicit-explicit preconditioners, the implicit part referring to

the block-factorization and the explicit part to the approximate inverses used to explicitly

approximate .

'd S

f

fh”—H‘—wznu…—% ““nfht…vg—P!U

˜ • …“ ’ ˜ • • "

v¦

¥

£

¢

When the original matrix is strongly inde¬nite, i.e., when it has eigenvalues spread on both

sides of the imaginary axis, the usual Krylov subspace methods may fail. The Conjugate

Gradient approach applied to the normal equations may then become a good alternative.

Choosing to use this alternative over the standard methods may involve inspecting the spec-

trum of a Hessenberg matrix obtained from a small run of an unpreconditioned GMRES

algorithm.

7¨

¶§ 8˜ H ˜ ¤ ¨°¡ $ H&’ ”8 ” &’ ¤ ©¦§

©8 $ $

¡¨ ¨¦ ¤

© §¥ c#&) $

©1

¢

¡

If the normal equations approach is chosen, the question becomes how to precondition

the resulting iteration. An ILU preconditioner can be computed for and the precondi-

tioned normal equations,

˜ ˜ da ˜ ¢ fd a ˜ da