reached at the point where the curve with positive slope crosses the curve 0 R !¢ ©

0

y

² 0R

with negative slope, i.e., when

10

© ¢

«

y

² ²y ¬ R¢©

R !¢ © w

«

y

| ¢ Cv¥ ¡At ¢

| §| ¡C

¡¨

²

0 R ¢ ©1 0

y

1 ² 0R

10

© ¢

«

y

$ " ¥¢£¡

¤ W

X s W ¡

X

T

X

vAB©7 ¡¤4 ¤2

9 9 65 3

The curve as a function of .

¦

This gives

a±aa i ¯®

˜

¬ $X" R

¢ © w R ¢ ©

«

Replacing this in one of the two curves gives the corresponding optimal spectral radius

² R¢©

R ¢ ©

¬ $X" «

R

¢ © w R ¢ ©

«

This expression shows the dif¬culty with the presence of small and large eigenvalues. The

convergence rate can be extremely small for realistic problems. In addition, to achieve

good convergence, eigenvalue estimates are required in order to obtain the optimal or a

T

near-optimal , and this may cause dif¬culties. Finally, since can be very large, the R ¢ ©

X

curve can be extremely sensitive near the optimal value of . These observations

¦

are common to many iterative methods that depend on an acceleration parameter.

¦

¨6¨

§ § 63

5 6E 0A A B & F C9&

38

F B

¥ ¤h

©' $ 2¨) 2 DB9

CA

§ ) )§

¦

§ ¬ § ²

Let be three given matrices satisfying . The

( (

§

pair of matrices is a regular splitting of , if is nonsingular and and are

(

W“

nonnegative.

With a regular splitting, we associate the iteration

± ¯ a i ¯®

w

© ¬ © ¨

–W “

W “

W"VH

U H

The question asked is: Under which conditions does such an iteration converge? The fol-

lowing result, which generalizes Theorem 1.15, gives the answer.

T

Av ¡˜¤

¥¦ ¥

X

¤9

£A

' P

1

§

Let be a regular splitting of a matrix . Then if

( W “

y

§ §

and only if is nonsingular and is nonnegative. W“

µ£ „ ¢

|5¥ j qz A C¡y 5

„ 5| | 5£§|

¡C ¢ ¡

@¡

¨§ £ "!

T

£ X

™£¤¢¡6

A

¬

De¬ne . From the fact that , and the relation

¦ ¦

W “ T

y r±— a i ¯®

X

¬§ ² ¦

T T

§

it follows that is nonsingular. The assumptions of Theorem 1.15 are satis¬ed for the

X X

T