nonsingular. This, along with (5.8), shows that is nonsingular. ¢

µ£ „ ¢

|5¥ qzl 5 ¢ ¤ ¤¥

„ 5| | §

£C ¡

§ "!

§

Now consider the particular case where is symmetric (real) and an orthogonal pro-

§¬

jection technique is used. In this situation, the same basis can be used for and , which

are identical subspaces, and the projected matrix, which is , is symmetric. In ¢

§

addition, if the matrix is Symmetric Positive De¬nite, then so is . ¢

7v © AcvAz•

™ ™™ ¡

¢£

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This section gives some general theoretical results without being speci¬c about the sub-

spaces and which are used. The goal is to learn about the quality of the approximation

obtained from a general projection process. Two main tools are used for this. The ¬rst is

to exploit optimality properties of projection methods. These properties are induced from

those properties of projectors seen in Section 1.12.4 of Chapter 1. The second tool consists

of interpreting the projected problem with the help of projection operators in an attempt to

extract residual bounds.

§¡63

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£A

¢

"6¨ ¢

! § A B & 8 I C7 ' '

BA F b&

A

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In this section, two important optimality results will be established that are satis¬ed by the

§

approximate solutions in some cases. Consider ¬rst the case when is SPD.

h¤ ¡ v A£

£¦ ´ £¤A

' ' ©' )$ )

§

¬(

§

Assume that is Symmetric Positive De¬nite and . Then

w ©

¥©

a vector is the result of an (orthogonal) projection method onto with the starting vector

© §

if and only if it minimizes the -norm of the error over , i.e., if and only if

T T

¬ X ¥© X

©

5I6 QR

C 4 S U

% % (

R

¥U

¤

where T T T

¥¢ X

£

© t§©r( X © l¨ © § ! X © %

²¨ ² W

T

£ X

™£¤¢¡6

A

¥© ©

As was seen in Section 1.12.4, for to be the minimizer of , it is necessary %

T § T ¥ © ²f¨ ©

and suf¬cient that be -orthogonal to all the subspace . This yields

¡Y ¬ ¦ ( ¥ © t§© §

XX

(¡ ²¨

¦ (

or, equivalently, T

¡Y ¬ X ¦ ’¦y¨

(¡ ( ¥© § ²

¦ (

which is the Galerkin condition de¬ning an orthogonal projection process for the approxi-

©

mation . ¥

§ ¬

We now take up the case when is de¬ned by .

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|„5 £§| £C

¡!t

§

§ ¥¤A Q¤ ¡ v A£ £¦

´ ' ' ©' 2 )

§)

¬

§

Let be an arbitrary square matrix and assume that .

¥©

Then a vector is the result of an (oblique) projection method onto orthogonally to

©

with the starting vector if and only if it minimizes the -norm of the residual vector

˜

w © ¢© ©ty¨§²

¡

over , i.e., if and only if T T

X X

¢ ¢

¬ ¥© ©

FI6 R

C 4 S U (

R

T

¤¥U

t²A¨

X

¢

© ©§ £

where .

! T

£ X

6 A ¢

£

© ©b

As was seen in Section 1.12.4, for to be the minimizer of , it is necessary

¥

b

© § w¨

² §¬

and suf¬cient that be orthogonal to all vectors of the form , where belongs

¦

¦

¥

to , i.e., T

§ ¡ ¦

¬ X ¦ ¦c¨

( ¥© § ² ¡$

(Y (

©