0A B ©' & 8 Q5H A 'Q5 2P5 P0¤¥©' )

¨#P!$ ¢

§ ! 35 E

)E

)

IH 3

©

As usual, denotes the norm de¬ned by

T

¥¢ X

£

W© t§ ©

©¬ W (

The following lemma characterizes the approximation obtained from the Conjugate Gra-

dient algorithm.

´A ’ w

¢¥

P#

1 1

©

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

¢ !

©ym§© ¬ £

²¨ © ©

CG algorithm, and let where is the exact solution. Then, is of the ¨

¢ ¢ ¢

form T

¨ X § ¢ w © ¬ ©

¢ ¢

²!XT

¢

where is a polynomial of degree such that

T T

T

¢

FC4I6 y ¬ £ eX § ¢ ²

£ eX

X

§

§ ¢ §² i U

¢

¤

³C ¨ ¡¡µ£ „ ¢

§|5¥ yq¢| ¢ £ ¥§

„ 5| j C¦£¥

5§

£ "–© ¡"

§

1B

6£ ™£¤¢¡

A

© §

This is a consequence of the fact that minimizes the -norm of the error in ¢

T

w w© ©

the af¬ne subspace , a result of Proposition 5.2, and the fact that is the set of ¢ ¢

X

§ ¢£ ²

¢

all vectors of the form , where is a polynomial of degree . ¨ !

y

From this, the following theorem can be proved.

cv ˜¤

¥¦ ¥¡ A

' 1

©

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

¢ !

©

Conjugate Gradient algorithm, and the exact solution and de¬ne ¨

p±q w°q ®

°i

R¢©

¬ «²

R

¦

¢ ©1 R ¢ ©

«

Then,

²T

© c¨ © r±0 w°q ®

i

© ²f§©

¨ (X

¢

w ¦˜

¢

) y

in which is the Chebyshev polynomial of degree of the ¬rst kind.

¢ !

)

£ ² ¨ ©

A™¤ ¢¡6 T

£

© §

From the previous lemma, it is known that minimizes -norm of

¢

X

the error over polynomials which take the value one at , i.e., ¨ YT

© ² ¨ © X

5I6 8¢ ¬

£ § ¨ C4

¢

¢U ¤

W PG

¤WV(( y Er( R ©

¬ r¤(( y rE( R F §

¬

If are the eigenvalues of , and the components of the

initial error in the eigenbasis, then

£ T T T TT

«f ¬

£ £X £X £ eX £

£X

XR

§ R F R © ¨R ©

£

© ¨ P HR 6

¨

TT

£R

£ £ eX

X

W

£

QX H s ¡ ©U

P 6 ¨ ©

¨

¤¢

Therefore, T

© ² ¨ © X

£ 0 © 0¨

FI6

C4 QX H s 6 ©U

P