¢ ¢

¨

UG ¤ £

¢

¤

W PG

The result follows immediately by using the well known result of Theorem 6.4 from ap-

proximation theory. This gives the polynomial which minimizes the right-hand side. ¨

A slightly different formulation of inequality (6.102) can be derived. Using the rela-

tion, T

£

¢ ¢

£ £

X “

¬ ² ²

§w §w

w

y

¢

) ¥

˜ y y

¢

£

²

§w

…

y

˜ y

then T T

¢

£X

X

²

§ w ¦˜ w

w w

¦˜ ¦˜

y

¢

) Ty

y y y¢

˜ X

w ¦ ¦ §˜ w ¦˜ w

y

˜ y y

³C

£ | ¢ Cv¥ ¡y¤t ¢

| §|

¥

" © £

¢ £

¢

Now notice that T £

a±a w°„ ® i

¬

X

w ¦ ¦ §˜ w ¦˜ w w¦§ w¦

y y y

£ w R ¢ © ± ¯ w°„ ®

i

R ¢ ©

R ¢ ©1² R « ¢ © ¬

« R ¢ © w R !¢ © a±— w°„ ® i

« R ¢ © ² R !¢ © ¬

« w¢ a± w°„ ®i

²¢ ¬

y

y

¢ R ¢ Q R !¢ ¢

¬

in which is the spectral condition number .

©

©

«

Substituting this in (6.102) yields,

²w ¢¢ £ ¢

a± w°„ ®

i

© ² ¨ © ©

© ²¨

˜ ¦

y

¢

¥y

This bound is similar to that of the steepest descent algorithm except that the condition

§

number of is now replaced by its square root.

¡63 I ' 22P5 P0¥©' )

35¤E

#P"

!! 5)E

F5

¢

T

§

We begin by stating a global convergence result. Recall that a matrix is called positive X

T

§ w §

de¬nite if its symmetric part is Symmetric Positive De¬nite. This is equivalent ˜

X

W©(V§© ©

to the property that for all nonzero real vectors . Y

Av ¡˜¤

¥¦ ¥ A

' P

1

§

If is a positive de¬nite matrix, then GMRES(m) converges for any

.

!

y

£ 6 A

£

This is true because the subspace contains the initial residual vector at each ¢

restart. Since the algorithm minimizes the residual norm in the subspace , at each outer ¢

iteration, the residual norm will be reduced by as much as the result of one step of the

Minimal Residual method seen in the previous chapter. Therefore, the inequality (5.18) is

satis¬ed by residual vectors produced after each outer iteration and the method converges.

Next we wish to establish a result similar to the one for the Conjugate Gradient

method, which would provide an upper bound on the convergence rate of the GMRES

iterates. We begin with a lemma similar to Lemma 6.5.

’w

¢¥ A

P#

1 1

©

Let be the approximate solution obtained from the -th step of the

¢ !

V§ A¨ ¬

©² ©T