WP W

¡

In many cases, is the result of a factorization of the form

§

¦

¡ ¬

Then, there is the option of using GMRES on the split-preconditioned system

¬ ´ d ¦ – ( 'd ´ fd ¦ 'd

%f f

f

In this situation, it is clear that we need to operate on the initial residual by at the start d

f

¡

¦

of the algorithm and by on the linear combination in forming the approximate h h

'd

f ‚ ` 'd

¨ f

solution. The residual norm available is that of . ah

A question arises on the differences between the right, left, and split preconditioning

options. The fact that different versions of the residuals are available in each case may

affect the stopping criterion and may cause the algorithm to stop either prematurely or with

¡

delay. This can be particularly damaging in case is very ill-conditioned. The degree

of symmetry, and therefore performance, can also be affected by the way in which the

preconditioner is applied. For example, a split preconditioner may be much better if

is nearly symmetric. Other than these two situations, there is little difference generally

between the three options. The next section establishes a theoretical connection between

left and right preconditioned GMRES.

¥ ¦

Y ¡DT RcW 9 CA8 Cd U WUpB Ce 9 XC¥

H

3¨3

2 §2 fU

P

Ya Pe

¡

¦8 5AXU P Y cAW¤¥ 33

PR U He

WP W

When comparing the left, right, and split preconditioning options, a ¬rst observation to

¦

¡ 'd ¡ 'd

make is that the spectra of the three associated operators , , and f d

f f 'd

f

are identical. Therefore, in principle one should expect convergence to be similar, although,

as is known, eigenvalues do not always govern convergence. In this section, we compare

the optimality properties achieved by left- and right preconditioned GMRES.

¶ ’ ”p£¡¨8 ³¡© &’ ”8 ” ’ ¤ §

8˜ 8¥© $ $ ©

¡¨ ¨¦ ¤

¡

© §¥ &

$

For the left preconditioning option, GMRES minimizes the residual norm

uH 'd ¡ ‚ f'd ¡ §

«§ f ¨ %

among all vectors from the af¬ne subspace &W

Ge¥v' £

¦I ‘

h

¡ BBD% P d ¡% P # w ¡ w

h

¬¬¬ f ( P d

D d ` %

af

% f

C

¡ P

in which is the preconditioned initial residual . Thus, the approximate

P df

solution can be expressed as

¡ fd h £fd ¡ w h

P bqf'd `

a ¢

¨±

where is the polynomial of degree which minimizes the norm

h

¢ d

f G

«u‚ P b fd ¡` ¢¤ 'd ¡ P P §¨

§a f

¨ ¢3±

among all polynomials of degree . It is also possible to express this optimality

¢

C G

condition with respect to the original residual vector . Indeed,

¡aDqfd ¡ ` ¥U C ¡ fd ¡ P bud ¡` ¤q'd ¡ P P ¢ C 'd

a f ¢ f ¨

¢¨ ¬

f

A simple algebraic manipulation shows that for any polynomial , ¢

Ge¥v' £

¦P ‘

C C

¡b 'd ¡` ¢ ¡ x` ¢ d ¡

af % a f'd

d

f f

from which we obtain the relation

¦ ¤ ¥v' £

‘

C¡ ¢ C a d

u ¡` ¢¦qd ¡ ¨ ¡ £x` £f'd ¡ ¨

¡¢

f

¬

P P b d

af fd f

Consider now the situation with the right preconditioned GMRES. Here, it is necessary

´

to distinguish between the original variable and the transformed variable related to

´ ´

¡

by . For the variable, the right preconditioned GMRES process minimizes

C d

f ´f ¨ ´&

W 'd ¡ “

the 2-norm of where belongs to

¡£x` % BDB C f'd ¡ % C # w C ´ §h w ´ G' ¥v' £

¦‘

C h

¬¬¬% ( d !daf

f

%

´f

C

'd ¡ ‘¨ –

in which is the residual . This residual is identical to the residual