¨©§’( ¨§§ ² ¨ ² ¨ (

§

¨ ¨

H H HH

HH

H

H

H

©§ ² ¨

By construction, the new residual vector must be orthogonal to the search ¨

H

H

H

§

direction , and, as a result, the second term in the right-hand side of the above equation

©

¨

H

vanishes and we obtain T

£

X

§HX ² H ¬

¨T ¨ T( ©

£

¨ ¨

"QH

WU H TH

X

T

§ ² H £¨ ( H ¨ ¬ ¨ ( ¨©

XH H T H X

§ ² £ ¨ ¬ §

¨( © T X ¨( § p±q0 „— ®

°i

¨

¨ X

HHT

HH

’§ §

¨© ( ¨§ £ X ¨ ( ¨

£H

y £ H H HT H

§ ² £ ¨ ¬ £¨

£ H £ XH ¨ ( H § ¨

§

£ ¨© ¨( ¨

H

y

HH

H

T

From Theorem 1.19, it can be stated that

X

W©tT§

(© r±0 0 „— ®

i

(Y X

W©©

(

´¥C

w | ¢ § ¥ l 5 ¢ ¤ ¤¥ © £ l BCzEe|t

| | § | q

T

X

§£ w § § « £ ¢ © ©§ ¬

R

where . The desired result follows immediately by using the in-

˜

£

equality .

¨

¨

H H

T T

There are alternative ways of obtaining inequalities that prove convergence. For ex- X X

§

ample, starting from (5.21), (5.22) can be used again for the term and ¨( ¨

¨( §

¨

H H H H

similarly, we can write T T T

X rX

X

VX© § W “ ’§T( V§ ¬ X r©(V§ T

©

© § §

wW“ “

R¢© Y

(

VW§"t§

© (©

VW§"t§

© (© «

˜

§

since is also positive de¬nite. This would yield the inequality T T

W“

a±a0 „— ®

i

£ £

£ ¨

§ X § ² X

(£

¨

T T W“

X ’QH w

WU H

y

R¢X ¬

in which . ¢ ¢ ¢

© ˜

«

Another interesting observation is that if we de¬ne T

X

H ¨ ( £ H §§

¨

¡

¬

¤ (£

¨§§ ¨

H

H H

then (5.21) can be rewritten as T T

£ X X

§² §

¨ ( §T X ¨ ( § T

£

£ ¨

¨

¬

£ XH H

¨ ¨ HH

’§ § ¨( ¨ ©( © ¨

¨

W"VH

U £H y