H

HH

¬ ²

¤ £ ££ ¨

y H H

¬ F4 ¡ £ ¨

C

H H

At each step the reduction in the residual norm is equal to the sine of the acute angle

§

between and . The convergence factor is therefore bounded by

¨ ©

¨ T

X

v

¬ VW§(©

©

¡

QHI6

P 54

C (

R X W U R

T T

X X

tW§("©

© © V§

© t’§(©

©

in which is the acute angle between and . The maximum angle is

§

¡

guaranteed to be less than when is positive de¬nite as the above results show. ˜¢

¦¨©7¡63

¢ ¨6§

§ BF5 I6©2E & 8

3' 6P2G¡5 A ¡6 5P0A F

5

AE5)F © F5

§

In the residual norm steepest descent algorithm, the assumption that is positive de¬nite

§

is relaxed. In fact, the only requirement is that is a (square) nonsingular matrix. At

§¬¥ ¨§¬

each step the algorithm uses and , giving the following sequence of ¦ ©

¦

operations:

§ ££ ¬ ¦ ( t££A² ¨ ¨

© § (¨

± ¯ 0 „— ®

i

( §§ ¦

¦

w© ©

¤¦

However, an algorithm based on the above sequence of operations would require three

matrix-by-vector products, which is three times as many as the other algorithms seen in

this section. The number of matrix-by-vector operations can be reduced to two per step by

computing the residual differently. This variant is as follows.

µ£ „ ¢

|5¥ qzl 5 ¢ ¤ ¤¥

„ 5| | §

¥C ¡

§ "!

˜Q¤ v ¢ 4¤

—

´¡¦ B @ ¡ ¥ Q ¢ 0 CvA

5 8D &95©E ¢ 5$ S ¢ R¢ HUS

5 55

S6

39

' 7$

%# ) 2

1

3

©V§ ²A¨ ¬ ¨

1. Compute

2. Until convergence, Do: 3

¨ §¬¦

3. £ § £ ¦

£ £

3

§§

¬ §

4. Compute and

¦ ¦

3

¦ w © ¬ ©

5. 3

¦§§ g¬ ¨

²¨

6.

7. EndDo

T

V§ ²f¨ ¬ X © £

© ¢²

Here, each step minimizes in the direction . As it turns out, £

¡

this is equivalent to the steepest descent algorithm of Section 5.3.1 applied to the normal

¨ § ¬ ©V§ § §§ §

equations . Since is positive de¬nite when is nonsingular, then,

§

according to Theorem 5.2, the method will converge whenever is nonsingular.

cv f ¢ ¢© cv ¢ ¢©z¦ vv ™ ¢ “© v

™ “ “ “ – “

—™——™

S

¡

We begin by considering again the block relaxation techniques seen in the previous chapter.

¬ r¤

To de¬ne these techniques, a set-decomposition of is considered as the ˜ ( §

© ((

¦

y

de¬nition of subsets of with ((

¦

¦ ¦

¡

W X

¬

(¦ R¦ ¦ R ¦

£R

X W

¤ R R¦ R¦

Denote by the size of and de¬ne the subset as

T T T