is symmetric, it is unitarily similar to a diagonal matrix, , and

¢ ¤

T T T

¬ X ©r"© W “ ¢ X r©©z¢ r©X( ˜¤ W “ # ¤ X T r©X (˜¦# ¤ T

X

©¤

©

(

(

¬ ˜¨©˜¤ W •# §(¦#

© ¤( “ © ¤ © ¤

T

£b

X b

b (( b

¢¥¬

¬©¤ ¬ R ST

Setting , and , note that

R

«

W

« ¬ X b( b # ! © f

R ©R S

R

r¤X V(T ( yW vEe( R © T X T

¬X

is a convex combination of the eigenvalues . The following relation holds,

b ©!¬ W© “ ¢ r"¢ ©( W ©( ©

©

with

« f ¬ X b( b W “ #T ¬ X bT

Ry R S

© ©

£R

T

W

Xb

T T

a©Q© y (

Noting that the function is convex, is bounded from above by the linear curve

©

X X

that joins the points and , i.e.,

©

« Q y ( « © X b T

©

yW W

©

²

wy © y

© © ©©

« «

W

W

Therefore, T T T

X b

X X ©

W©¢

(© ¬ r© © ²

wy

¢ ! W “

©©

© (

y

© © ©©

« «

W W

w | ¢ § ¥ l 5 ¢ ¤ ¤¥ © £ l BCzEe|t

| | § | q

¢¥ C

¥

£© R©

© © ©

«

W

T

X

¬

The maximum of the right-hand side is reached for yielding,

w

© W£ T ©

« £ ©X

W

T

T

T

w © X b X

X

©

W© z

© ¬ r© © «

" W “ ¢ ( ¢

©©

W£ (

©©

«

W

which gives the desired result.

This lemma helps to establish the following result regarding the convergence rate of

the method.

4 Av ¡˜¤

´ ¥¦ ¥ ¤A

£

' P

1

§ §

Let be a Symmetric Positive De¬nite matrix. Then, the -norms of

© ² ¨ (¬

©

the error vectors generated by Algorithm 5.2 satisfy the relation

£

H H

² ± ¯ w°„— ®

i

R ¢ ©1 R ¢ ©

«

£ £

(

R ¢ © w R ¢ ©

’QH

WU H

«©

and Algorithm 5.2 converges for any initial guess .

T T

£ £ X X

6 A

£

§¬ ¬

Start by observing that and then by

W"QH £ £ £( £(

¨

U

W"UQH W"UQH "QH

WU "QH

WU

simple substitution, T