¬ § « ¤ —3 § f

§ §

¤

As a result, when the algorithm converges, then it does so quadratically.

§©¨¥¦£

§

De¬ne for any ,

«

w

` ¤ ¤

”a

¨ ¤a

`

G

a W` ¤

§ B

§

Recall that achieves the minimum of over all ™s. In particular,

a ` ¤ '% &

¤§ B § §f

§

GF I v0¥ £

¦ ‘)

§ « ¤ §& D«§ a G ` ¤ g3

§

¬ p ¤ g3

§§

This proves quadratic convergence at the limit.

For further properties see Exercise 16.

¦¦ §

©

¥

9

2 ¡3¢6

2 ¡ 9 RI3XcY

HeU UC

e DIIIb 5dbQ9gf P

B H B e H WP H Y

¡

A notable disadvantage of the right or left preconditioning approach method is that it is

¡

dif¬cult to assess in advance whether or not the resulting approximate inverse is non-

¦

singular. An alternative would be to seek a two-sided approximation, i.e., a pair , , with

¦

lower triangular and upper triangular, which attempts to minimize the objective func-

tion (10.45). The techniques developed in the previous sections can be exploited for this

purpose.

¦

In the factored approach, two matrices and which are unit lower and upper trian-

gular matrices are sought such that

¦

£

¢

¨

¦

where is some unknown diagonal matrix. When is nonsingular and , then

¨ ¨ ¨

d ¨ ¦

¦

are called inverse LU factors of since in this case . Once more, the

% f'd f

matrices are built one column or row at a time. Assume as in Section 10.4.5 that we have

the sequence of matrices

7 £

A

f

f

¦ F

s

in which . If the inverse factors are available for , i.e.,

%

¦

%¡ ¨ R

’ ”8 ” ’ ¤ %©Y¡Y"¡ $ "p ‚ r ¦¨¥

$ © § © ©8¥ ’ §§ ¤ £

¢

¡¡© &

$

¢

¡

¦%

then the inverse factors for are easily obtained by writing

f7

f f¦

7 7 7

E E

£ I I v) ¥ £

¦‘

§¨

P

E E¨

AG

A A A

y

¨ f f

G

y

in which , , and are such that

P f

P I v) ¥ £

¦‘