I T

(

¡ ¥

¬y X

² X

£ £(

¥ (`

Y

§

The above condition is now easy to interpret: The vector is the -orthogonal projection

X

of the initial error onto the subspace . £

h¤ ¡ v A£

£¦ ´A ´

' ' ©' )$ )

§

©

Let be the approximate solution obtained from an orthogonal ¥

© ²fT §© }¥ £

¨¬

projection process onto and let be the associated error vector. Then, ¥

X

(t £ ¥ ² }£ ¬¥

¥

where denotes the projector onto the subspace , which is orthogonal with respect to

§

the -inner product.

§§

A result of the proposition is that the -norm of the error vector obtained after one projec-

tion step does not exceed the initial -norm of the error, i.e.,

£ ¥ £

(

³£C

© ¡C|¥

|„5 £§|

¡!t

§

w ©©

§

which is expected because it is known that the -norm of the error is minimized in .

This class of methods may be termed error projection methods.

¦

¢ ¨6§

§ ' % ©#3 3PQ& 8 P5 EP5

3

5

3' E

©

©

If no vector of the subspace comes close to the exact solution , then it is impossible

© ¥©

to ¬nd a good approximation to from . Therefore, the approximation obtained by

any projection process based on will be poor. On the other hand, if there is some vector

©

in which is a small distance away from , then the question is: How good can the

approximate solution be? The purpose of this section is to try to answer this question.

¤

§ © ¦ ¢ © ( ¡ © ¨¦¢

£

£© ²

§ © §£ ² © ( ¢© §£

¡ ©

¨¦

§

¥£ © ¤¢ ¡

£ ©

´

7 ¡¤4 ¤2

65 3 ¤A

¥

Orthogonal and oblique projectors.

¦

§£

©¢

£

Let be the orthogonal projector onto the subpace and let be the (oblique)

projector onto and orthogonally to . These projectors are de¬ned by

§ © §£ £©¦ ¢ ² © ( ¢© §£ ¤¦ ¢

¡£ (

² © ( ¢©

( §© ¡

§

and are illustrated in Figure 5.3. The symbol is used to denote the operator

¢

£ ¢ §£ ¦

¬§ § (

¢

¬ ©

and it is assumed, without loss of generality, that . Then according to the property Y

(1.54), the approximate problem de¬ned in (5.5 “ 5.6) can be reformulated as follows: ¬nd

¢¥ ©

¡ such that T

§£ ¦

¬ X ¦A¨

¥© § ² (`

Y

µ£ „ ¢

|5¥ qzl 5 ¢ ¤ ¤¥

„ 5| | §

¥C ¡

§ "!

or, equivalently,

¢¥ © –(¨ §£ ¦ ¬ ¥ © §

¡

¢

¤

Thus, an -dimensional linear system is approximated by an -dimensional one. !

The following proposition examines what happens in the particular case when the

§

subspace is invariant under . This is a rare occurrence in practice, but the result helps

in understanding the breakdown behavior of the methods to be considered in later chapters.

h¤ ¡ v A£

£¦ ´ A

' ' ©' )$ )

§

¬ © § ¨

Assume that is invariant under , , and belongs to Y

. Then the approximate solution obtained from any (oblique or orthogonal) projection

method onto is exact.

£ A™¤ ¢¡6

£

©¦

An approximate solution is de¬ned by

¥ T

§£

¬ X ¦A¨

¥© § ² Y`

(