H

®

number of operations (additions and multiplications) of this procedure is for (1), a ! 0

`

˜

H

®H

w

®

for (2), for (3), and for (4). The cost of the preconditioning operation by

` a

!

®

, i.e., operations, must be added to this, yielding the total number of operations:

d ¨

f

˜

Hw

Hw

w Hw

®w® ®

® ¤¢

a ! 0

`

a c

!`

˜

Hw

w

w

i® ®

a ! ` (` $`

a!

a #

"

Hw

i

® ¬u x

b`

a #

"

H

x

For the straightforward approach, operations are needed for the product with ,

aD `

¶ ’ ”p£¡¨8 ³¡© &’ ”8 ” ’ ¤ § ¡¨8 ¨¦˜

8¥© $ $ ©

¡¡

¢ ¤ © §¥ &

$

˜! Hw®

H

for the forward solve, and for the backward solve giving a total of

(c

a !` a `

”a ˜ !`0 H w ® w a !`0 H w bx`0 H @®°@abx`0 Q

¬¨

a

Thus, Eisenstat™s scheme is always more economical, when is large enough, although

the relative gains depend on the total number of nonzero elements in . One disadvantage

of this scheme is that it is limited to a special form of the preconditioner.

¥

· §5

¥§¨

©

®¡

x

For a 5-point ¬nite difference matrix, is roughly , so that with b`

a

®¢G

the standard implementation operations are performed, while with Eisenstat™s imple-

W" G

®

mentation only operations would be performed, a savings of about . However, if the f£ E

® ¤GH

other operations of the Conjugate Gradient algorithm are included, for a total of about

®¢

operations, the relative savings become smaller. Now the original scheme will require

®i" H

operations, versus operations for Eisenstat™s implementation.

ªm¢ " nfhr…vxf¦

˜ • • "

¥¦v

In the case of GMRES, or other nonsymmetric iterative solvers, the same three options for

applying the preconditioning operation as for the Conjugate Gradient (namely, left, split,

and right preconditioning) are available. However, there will be one fundamental difference

“ the right preconditioning versions will give rise to what is called a ¬‚exible variant, i.e.,

a variant in which the preconditioner can change at each step. This capability can be very

useful in some applications.

¥ ¦

D3ehA8IHcXrbY PcRcXC¥ 3C E F¡DT

Y H

6 3¨3

2 §2 WU He

R W UP

BH f

As before, de¬ne the left preconditioned GMRES algorithm, as the GMRES algorithm

applied to the system,

¦ ¤ v' £

‘

Bd ¤ufd ¡

¬ f ¡

The straightforward application of GMRES to the above linear system yields the following

preconditioned version of GMRES.

¢ ¦©'© £ ¨ £ & © £ 0I! ¨ £0 ¨ £ # ªm¢

˜

" · wE¤©n¦§¥£¢

¡ ¨¦ ¤¢ £ ¤

u§ C z° a “ ` 'd ¢ C C

£

« § ¨ f ¡

1. Compute , and 7

f

±„¬BDB% m¥

% ¬¬

2. For Do:

£ d ¢ G

¡

3. Compute f

¬ ¬ ¬ ª

¥ %„DBD%

4. For , Do:

£ % ` G 2¥

5. S

aS

£ 2¦¨ C

6. S S¥

C

¶

’ ”8 ”#’ ¤ ¦§

$ © Q¡

§

c© ¡ ¡©