§

It is worthwhile to point out that need not be nonsingular in the above proposition. When

§ ©

is singular there may be in¬nitely many vectors satisfying the optimality condition. ¥

¨6¨ ¢ 73 ' A ) 5 £ P6 QF 6P0E B ©DC77A 63 P5 A E B

§§ 3

E 'B A 8 5

I35A

'

'3

F

We now return to the two important particular cases singled out in the previous section,

¦ § ¬

¬

namely, the cases and . In these cases, the result of the projection process

can be interpreted easily in terms of actions of orthogonal projectors on the initial residual

or initial error. Consider the second case ¬rst, as it is slightly simpler. Let be the initial ¨

tyc¨ ˜ ¨

©§ ² ¬ © § c¨ }¥ ¨

²¬

residual , and the residual obtained after the projection process

¥

§ ¬

with . Then, T

a± q— ®

i

t ¨ ¬ X

© y¨ ˜¥ ¨

§² ¬ §²

X X

w

§

§²

In addition, is obtained by enforcing the condition that be orthogonal to .

X X

¨

§

Therefore, the vector is the orthogonal projection of the vector onto the subspace

X ¨

§ . This is illustrated in Figure 5.2. Hence, the following proposition can be stated.

Q¤ ¡ v A£

£¦

´ vA

9

' ' ©' 2 )

§)

©

Let be the approximate solution obtained from a projection pro-

¥

§ ¬ © § ¨ X ¬˜¥ ¨

²

cess onto orthogonally to , and let be the associated residual. Then, ¦

¥

T

a±!w°„— ®

i

¥ ¨

¬ ¥² ¨

(

§

¥

where denotes the orthogonal projector onto the subspace .

A result of the proposition is that the 2-norm of the residual vector obtained after one

projection step will not exceed the initial 2-norm of the residual, i.e.,

¨ £ ¥ ¨

(£

a result which has been established already. This class of methods may be termed residual

projection methods.

µ£ „ ¢

|5¥ qzl 5 ¢ ¤ ¤¥

„ 5| | §

£C ¡

§ "!

¨

§

¢¬ X §

¥ ¨

¡

O

´

85¤4 2

76 3 ¤A

£

Interpretation of the projection process for the

§ ¬

case when .

(¨ ©

¬ § ¬

Now consider the case where and is Symmetric Positive De¬nite. Let £

˜¥© ² ¨ ©

be the initial error, where denotes the exact solution to the system and, similarly,

© ² ¨ ¦¬

© w ¦¬ ¥ ©

©

let where is the approximate solution resulting from the projection X

£ ¥

step. Then (5.9) yields the relation T

X

(©¬}¥ £ §

§ ¬ ¥¨ ² (X

£

§ ¨

²

where is now obtained by constraining the residual vector to be orthogonal to :

X X

T

¡ ¥

¬X

§ t ¨

² X ¥( Y`

(

The above condition is equivalent to T T

¡ ¥

XX

¬X

t £ §

² ¥( ¡Y`

(

T

§

Since is SPD, it de¬nes an inner product (see Section 1.11) which is usually denoted by

X