¬ –w

(

¨ ¢t§

¬©

or . This answers question (a). Next, we focus on the other two questions.

22P5 3P0¤¥©' ) & 8 PEP5 §¡63

¦F5

!$6§

35

5E

5)E ¥&

A

² ©©

If is nonsingular then there is a solution to the equation (4.29). Subtracting (4.29)

¨

¦

from (4.28) yields T T

r± a i ¯®

X X

¬ §© ² H © ¦4V§© ² "QUH ©

¬¨ W ¬ §t©

¨© ²

¨ ’UQH 4

¦

W

VVT

TT

Standard results seen in Chapter 1 imply that if the spectral radius of the iteration matrix ¦

¨© ²H©

is less than unity, then converges to zero and the iteration (4.28) converges toward

the solution de¬ned by (4.29). Conversely, the relation T T T

¬ VTTWT ¬ X W “ H © ² H © §¬ H © ² W"QUH © ©X ¦ ² ² X

H¦

¦

T

§©

shows that if the iteration converges for any and then converges to zero for any

¦H ¦

X

vector . As a result, must be less than unity and the following theorem is proved:

¦ ¦

T

cv ˜¤

¥¦ ¥¡ X

DB9

CA

' 1

²

Let be a square matrix such that . Then is nonsin-

¦ ¦ ¦

T y

§©

gular and the iteration (4.28) converges for any and . Conversely, if the iteration (4.28)

X

©

converges for for any and , then .

¦

y

Since it is expensive to compute the spectral radius of a matrix, suf¬cient conditions that

T

guarantee convergence can be useful in practice. One such suf¬cient condition could be X

obtained by utilizing the inequality, , for any matrix norm. ¦ ¦

´ ¡C

| ¢ Cv¥ ¡At ¢ ¡¨

| §|

v

¦

CDB9 ¢

¡¦¢

A

0

' ## '

Let be a square matrix such that for some matrix norm

¦ ¦

y

¦² ©©

. Then is nonsingular and the iteration (4.28) converges for any initial vector .

Apart from knowing that the sequence (4.28) converges, it is also desirable to know

² H (¬

© §©

how fast it converges. The error at step satis¬es ¨

£ W

H

¬

£H 4 £

¦

H

¬

The matrix can be expressed in the Jordan canonical form as . Assume for W “ ¨¦§¥ ¦

¤¤

¦

simplicity that there is only one eigenvalue of of largest modulus and call it . Then ©

¦

¦ H

¬ £ W “ ¤

£ ¤H

©

©

H

A careful look at the powers of the matrix shows that all its blocks, except the block ¦

©

associated with the eigenvalue , converge to zero as tends to in¬nity. Let this Jordan © W

block be of size and of the form ¡

¬}

( % w ¦

©

$¬ X

where is nilpotent of index , i.e., . Then, for ,

% % Y W

¡ ¡

T T R

W “ fX &R

%

X

X W

¬ ¬ ¬

#H ©! H % W “ © w H H % w ©

H ¦ © “© %

" $E

£R

If is large enough, then for any the dominant term in the above sum is the last term,

©

W

i.e.,

W

W"X “ H © (H ¦ W“X %

'

U

²

¡

y

¬

Thus, the norm of has the asymptotical form

£