©¬ © FS W “ ¢

¢ ¡

W

´ CDA ˜h¤ v ¢ v

V

—

¡¦

89H576 QC 8CD § ¥ 7¦ §5 ¢ § 7C@ ¥US ¢

¤5

D E6

DS

@

0 (%&$

'#

) 2

1 3 ¢

£ ¨ ¬ S ©V§ A¨ ¨ 3 3

²¬

–Q ¬ ¦

1. Compute , , and

S ¨

W v

¬

2. For Do:

!#V˜™(

((

u ¦ u S ² u ©§yT ¬ u ¥

v W “ X u ¦ ( u¦ ¥ ¬ u

¬

3. (If set )

Y` ¦ S

!W

y

4. 3 u ¦ u ² u ¬ u ¥

5. ¥

£u ¥ ¬ 3

¬ ! Y`¬ W"U u u "U u u S

uS

6. . If set and go to 9

S ¥ ¬ W ¦

7.

T "U "U

W

W

8. EndDo

X R R RT £

rX ( ( S ¦ 54 4 ¤ ¬

¦ V( ¦ %¬

9. Set , and .

H

S

( $ &¢

¢ ¢

w © ¬ © "U W

bW ¬b

10. Compute and

S W“

¢ ¢ ¢ ¢

F

¡

¢

W

Many of the results obtained from Arnoldi™s method for linear systems are still valid. For

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example, the residual vector of the approximate solution is such that ¢

a± „ ®

i

¢b ¢¡

² ¬ V§ A¨

©² ¦

¢!

S

¢ ¢

W"U "U

W

The Conjugate Gradient algorithm can be derived from the Lanczos algorithm in the

same way DIOM was derived from IOM. In fact, the Conjugate Gradient algorithm can be

§

viewed as a variation of DIOM(2) for the case when is symmetric. We will follow the

same steps as with DIOM, except that the notation will be simpli¬ed whenever possible. ¬

First write the LU factorization of as . The matrix is unit lower ¤¢£ £

¢ ¢ ¢ ¢

bidiagonal and is upper bidiagonal. Thus, the factorization of is of the form

¤ ¢ ¢

£S

¦

£y© W £¦ S

¬ „

„

R¦

y© ¥

¤

S

¢

‚ ‚

R R

y© ¤ ¤ ¥

S

¦

y© ¥ ¥

¦

y

The approximate solution is then given by, T

X

(¬ ©

© w £ W “¢ ¤

FS W “ ¢

¢ ¢ ¡

W

Letting

¥ ! ¤ W “¢

¢ ¢

and

¬ £ ( ¡FS W “ ¢

¢

W

¨ ¡¡µ£ „ ¢

§|5¥ yq¢| ¢ £ ¥§

„ 5| j C¦£¥

5§

¡ C

"–© ¡"

§

1B

then,

¬© © ¢¥

w ¢

¢

R

¥

As for DIOM, , the last column of , can be computed from the previous ™s and ¦

¢ ¢ ¢

¡ ¡

by the simple update

¬ ²

¢ $¦ W ¤¦

“¢ ¢S &

¢ ¢