¢ ¢ ¢ ¢ ¢

W“

"U

W y

R¨ R¦

Recall that the vectors ™s are multiples of the Lanczos vectors ™s. As a result, should ¡

¢

be the inverse of the scalar of the Lanczos algorithm. In terms of the -vectors this ¨

¢

T

means

X

¢¨( ¢¨ T

§¬

¡

X

¢

¢¨( ¢©

¨

Equating the inner products of both sides of (6.75) with , and using the orthogonality ¨ ¢

W“

of the -vectors, gives the following expression for , after some algebraic calculations,

¨ T ¢

£ X

T a± „ ®

i

W“

¡

¨ ¢¨( ¢

¢

²y ¬

¥ W “y ¢ X W “

¢ ¡

¨“ ¢¨(

¢ ¢

W“ W

The recurrence relation for the approximate solution vectors can be extracted from

T T T

the recurrence relation for the residual vectors. This is found by starting from (6.74) and

² y X ¬ T X ¨ ¢

X X

using the relation between the solution polynomial and the ¢

¢ ¢ ¢

W“ T W“

residual polynomial . Thus, ¨ ¢

T

X

² X

¨ ¢

y¬

¢ W"U

T T

¢

T T

X

X

² ²T y X ²

w T X ¢ T¨ ¢

¨

¢ ¨

¢

² T y T ¬ W“

¡

¢ ¢

XeX y

² X W “ ¢ ¬

¢X ¢ X

² £

w ¡

¢¨ ¢

¢ ¢ ¢

“

y

This gives the recurrence, T

T

a± „ ®

i

w X ¢¨ ¢ ©X ²

© ¬ ²©

¡

¢ ¢

¢ ¢ ¢

W“

"U

W y

All that is left for the recurrence to be determined completely is to de¬ne the ¬rst two

§©

iterates. The initial iterate is given. The ¬rst vector should be of the form

t© ¬ W ©

² ¨

¡

(

to ensure that is orthogonal to . This means that the two-term recurrence can be started

¨

¨

W

g

¬ ©

with , and by setting . Putting these relations and de¬nitions together gives

$!

Y

W“

y

the following algorithm.

¡V

• `

˜

¡˜¤h v ¢

¦ © HY8C¦ © 845

¤

55 ©79V¤8 ¡ E©0

5

5 E 6 58 &C@ ¨9@

S 6 Q8

3 CDA

0 (%&$