H

W

¥

£ 0 "X “ H © 01 ' (

)

WU

²

H ¡

y

where is some constant. The convergence factor of a sequence is the limit

)

¢

£ HW

553¬

H 642

£T @8H

97

X

¦ ¢

¬

It follows from the above analysis that . The convergence rate is the (natural)

A

logarithm of the inverse of the convergence factor

² ¬A 5B

C2

©

The above de¬nition depends on the initial vector , so it may be termed a speci¬c

convergence factor. A general convergence factor can also be de¬ned by

¢

£ HW%

H ¨QISTR " 56F4E¬ D

£ X P $H V 6 @GH2

WU 97

µ£ „ ¢

|5¥ j qz A C¡y 5

„ 5| | 5£§|

¡C ¢ ¡

@¡

¨§ £ "!

This factor satis¬es

¢

HW%

£ H ¦ P H 6 D

" 55E¬

642

X $VS ¡

T £ WU @78H

9

¢ X

¦ ¬ H W H ¦ 55E¬ 642

@8H

97

Thus, the global asymptotic convergence factor is equal to the spectral radius of the it-

eration matrix . The general convergence rate differs from the speci¬c rate only when

¦

the initial error does not have any components in the invariant subspace associated with

the dominant eigenvalue. Since it is hard to know this information in advance, the general

convergence factor is more useful in practice.

¥

CDB©7 ©¦§¥¢£

A9 ¨ ¤

Consider the simple example of Richardson™s Iteration, T

p±qa i ¯®

°

X

© (¬

© tA¨

©§ ²

w

(

"QH

WU H H

where is a nonnegative scalar. This iteration can be rewritten as

T

r±0 a i ¯®

© X § ² ¬ W"QUH © ¨

– w

T

H X

§ ² ¬ R ¦ §²

Thus, the iteration matrix is and the convergence factor is .

¤r(( vEe( ©

¬

Assume that the eigenvalues , are all real and such that,

R© Ry ©

R ©

¢ ¢

« ¦

R

Then, the eigenvalues of are such that

R R ©²

R © ² T

¢ ¢

« y

y

X

R¢©

In particular, if and , at least one eigenvalue is , and so R ©

¦

Y Y

¢

« y y

for any . In this case the method will always diverge for some initial guess. Let us assume

«R

that all eigenvalues are positive, i.e., . Then, the following conditions must be

© Y

¢

satis¬ed in order for the method to converge:

² R¢© (

²«

² yy y

R ¢ ©

y

R !¢ ˜

The ¬rst condition implies that , while the second requires that . In

©

Y

other words, the method converges for any scalar which satis¬es

˜

Y R

¢ ©

T $ #

The next question is: What is the best value for the parameter , i.e., the value of "

X X

which minimizes ? The spectral radius of is

T¦ ¦

X

¬ ² ²

0v(0 R ¢ © 0 R ¢ ©

0§PH ¥

6 ©

¦

«

y y

This function of is depicted in Figure 4.4. As the curve shows, the best possible is