¨ ¬ §£

© ¨

where is a nonzero vector in . The right-hand side is in , so we have .

¥ ¦

© § ¥ © "y¦§

§ ¬ ¥©

Similarly, belongs to which is invariant under , and therefore, . Then

¥

the above equation becomes

¬ ¦A¨

¥© § ² ¡Y

(

©

showing that is an exact solution.

¥

¥

¬©

The result can be extended trivially to the case where . The required assumption in Y

A¨ ¨

²¬ V§

©

this case is that the initial residual belongs to the invariant subspace .

T

An important quantity for the convergence properties of projection methods is the

§© X ¤¢

¨£

² ©

£

distance of the exact solution from the subspace . This quantity plays ¨

T

¨©

a key role in the analysis of projection methods. Note that the solution cannot be well

© X £ ¢ ² £

approximated from , if is not small because ¨ T

² £ ¨ X © £ ² ¢ ¥ © T ¨

© X ¤¢

£ £

²

£ ¨© £ ¨©

The fundamental quantity is the sine of the acute angle between the

©

solution and the subspace . The following theorem establishes an upper bound for the

¨

§

residual norm of the exact solution with respect to the approximate operator . ¢

T

CDA ´ ¥cv ˜¤

¦ ¥¡ ² § ¨¦ ¬ £X

§£ ¤¢

£

' 1

¨

Let and assume that is a member of and

¡

Y ¬ © ¨©

. Then the exact solution of the original problem is such that T

p±r°w°„— ®

i

£ §© ¨

£

§© ¤¢

¨£

X

¨ §² ²

¡

¢

¡ ¨ ™A 6 £

£¤¢¡

Since , then T

§£ ¦

§© £©¢ § A¨ T X

t¨ © § ²A¨

¬ ²

¨

¢

§£ ¦ £ ¢ ² ¨ tT § X

¬ ©

¨© §

§£ ¦ £©¢ ²f¨ © T §

X

¬ §©

¨

§£ ¦ X£¢

¬ ² §

¨©

w | ¢ § ¥ l 5 ¢ ¤ ¤¥ © £ l BCzEe|t

| | § | q

¤¥ C

C

¤¢

£

²

Noting that is a projector, it follows that T T

§£ ¦ £ £

§ £¢T £¢

X X

¬ © § y¨

² ² T § ² ©

¨ ¨

¢

§£ ¦ £ X ¤¢ £

£

¤¢

£ X

² § ² © ¨ (

which completes the proof.

It is useful to consider a matrix interpretation of the theorem. We consider only the

¬

particular case of orthogonal projection methods ( ). Assume that is unitary, i.e.,

¥

¬ ¬¨ ¨

that the basis is orthonormal, and that . Observe that .

¦§ © ¢ ¦ ((

W

Equation (5.11) can be represented in the basis as T T

²y¨ £ £

§© ¤¢

¨£

X X

§ §© ²

¨ ¡