T

¡

X

§ ¬ ˜

²

$ R¦( u for wW

Y E t

¡

y

¡¬

u

For the case (full orthogonalization) the ™s are semi-conjugate, i.e.,

T

W ¡

¬ XR

§ (u for

Y E t

¡ ¢

¡

m( •

—™

¡

The Generalized Minimum Residual Method (GMRES) is a projection method based on

¬ £ ¨ ¨ #

§¬ ¬

taking and , in which is the -th Krylov subspace with ¦

¢ ¢ ¢ !

W

. As seen in Chapter 5, such a technique minimizes the residual norm over all

w©

vectors in . The implementation of an algorithm based on this approach is similar

¢

to that of the FOM algorithm. We ¬rst describe the basic idea and then discuss a few

practical variations.

!$ £

¢ I0A B ©' & 8 ¡5 QI D7F P% 5 H A

)B 8

F3

H3

¢

There are two ways to derive the algorithm. The ¬rst way exploits the optimality property

w© ©

and the relation (6.5). Any vector in can be written as ¢

a±!w°„ ®

i

(b¢ w

© ¢©

¬

b

where is an -vector. De¬ning T T

!

a± 0 „ ®

i

© A¨ ¬ £ ty¨ ¬ X b X

(£ b ¢ w

©§ ²

§²

¦

the relation (6.5) results in T

Xb

A¨ ¬ V§ A¨

§² © ² © ¢ w

$

b §t ¨¬

² ¢

¡

b$ ² W £S ¬

¦ ¢ ¢

R±`0 „ ®

°i

b ¡ ² "U S

W

¬

¢ ¢

F

¡

W "U

W

$

Since the column-vectors of are orthonormal, then

T T

¢

"U

W a±00 „ ®

i

© § y¨ £ Xb ¢ w

¡

Xb

£ b ¢

² ¬ ²W

!

¦ FS

¡

´ ¨ ¡¡µ£ „ ¢

§|5¥ yq¢| ¢ £ ¥§

„ 5| j C¦£¥

5§

C "–© ¡"

§

1B

w §© $

The GMRES approximation is the unique vector of which minimizes (6.20). By ¢

T

¢b ¢ w

©¬ ©

(6.19) and (6.22), this approximation can be obtained quite simply as

¢

£ b ¢

¡

¬Xb

¢b ²W

where minimizes the function , i.e., ¦ FS

¡

a±a0 q ®

i

$

( ¢b ¢ w

©¬ © where

¢

± ¯ 0 „ ®

i

£

¡

FS ¡FI¦ ¤ ¢ b b¢

¬ ²W

C4 6 H T

¡

X

¢b !¥

The minimizer is inexpensive to compute since it requires the solution of an w!

y

least-squares problem where is typically small. This gives the following algorithm. !