w P ˜ f

9.

f

10. EndDo

Not shown here are the right and split preconditioned versions which are considered in

Exercise 3.

nu…

u…t%¢

•

S¢

v ¡

When the matrix is nearly symmetric, we can think of preconditioning the system with the

symmetric part of . This gives rise to a few variants of a method known as the CGW

method, from the names of the three authors Concus and Golub [60], and Widlund [225]

who proposed this technique in the middle of the 1970s. Originally, the algorithm was not

w ` f« ¡

¡ 3

¨

viewed from the angle of preconditioning. Writing , with , ab

the authors observed that the preconditioned matrix

¡ ¨ 8 ¢ fd ¡

d

f

is equal to the identity matrix, plus a matrix which is skew-Hermitian with respect to the

¡ -inner product. It is not too dif¬cult to show that the tridiagonal matrix corresponding to

¡

the Lanczos algorithm, applied to with the -inner product, has the form

£¤ ¨¦

§

« ¨ G

¤ §

¤ §

£ ¨ « ¦ ¤ v' £

‘

¢ G¬

¬ ¬ ¬

h

¥ ©

¨

h h

fd hG

G

¶

”}$

8’

¨(c2 ¤ ¡" ¡© ¨¡

§

©

$ ¥ © c"

©

As a result, a three-term recurrence in the Arnoldi process is obtained, which results in a

solution algorithm that resembles the standard preconditioned CG algorithm (Algorithm

9.1).

A version of the algorithm can be derived easily. From the developments in Section

6.7 relating the Lanczos algorithm to the Conjugate Gradient algorithm, it is known that

can be expressed as

f

w

¬

f

The preconditioned residual vectors must then satisfy the recurrence

¡ ¨

fd

5

P P

f

E

a P % f'd ¡ ¨

$`

¡

and if the ™s are to be -orthogonal, then we must have .

P P

As a result,

C

a P % P ` aP% `

a P % 'd ¡` ¬

a P% `

f

Also, the next search direction is a linear combination of and ,

P

f f

w

¬

P

f f

Thus, a ¬rst consequence is that

u ¡ ` a P % x` % fd x a % ud ¡` a 'd fd ¨ a % `

f

f

d ¡

because is orthogonal to all vectors in . In addition, writing that is

f d

f f

'd ¡

¡

-orthogonal to yields

f

a 'd ¡% f $P` ¨—

¬ a f fd ¡% `

$` ¢

P ¡f

¨ d ¡ a

¨

Note that and therefore we have, just as in the standard PCG

P

f f

algorithm,

C P

a f P % f $P` a f C % f $`

a P % P ` ¬

a % $`

P

ª“E˜ “

˜ •

– q–R$P

¤ ¤

1 Let a matrix and its preconditioner be SPD. Observing that is self-adjoint with