©

a± a „ ®

i

“

© W

u #W

uS w uDu X

¬ ² ¡W “u u

©

y W“ “ W“

De¬ne, T

T

D

X

X

¬ § §

u¨ ¨ T u © T u

(

X uX u

¬ §¡§

u p ¨ ©

¡

According to the above formulas, these vectors can be updated from a double recurrence

u uS

provided the scalars and were computable. This recurrence is T T

a± ¯ „ ®

i

§X u u # ² u ¨ T X u § u # ² u ¬ Xu

u¨ ¡

W"U u

§ ² S w "U ¬ u

¨

¡ ¡

W"U W

Consider now the computation of the scalars needed in the recurrence. According to

¬

u T u T u S T

the original BCG algorithm, with T T

X ¨ X u W"U X u D £X

D uD X

¨ ¨ ( ¨

¬ § § ¬ §

u

¨

¨ (

T T T

u

Unfortunately, is not computable from these formulas because none of the vectors

¨ ¨ X £X

uD uD uD

¨X

§ § § u

, or is available. However, can be related to the scalar ¨

T T T

uD u © (p ¨ X ¨ ¨ X X

¬ § §

u ¥

which is computable via T T T T T T T

uD u DX T u ©

X X X X X X

¨ ¨ ¨ ¨ ¨ ¨ ¨ ( u ¨

¬ § § ¬ § § ¬

u ¥ u © (p ¨ (

¨ ¨ X u © §

u

u ¥

To relate the two scalars and , expand explicitly in the power basis, to

obtain T

T

T

X uD

u

u

uX uX

¨ ¨ ¨ ¨ W

¬ § § §

u ¥ P G£ ¦ w w

P G P( ¨ “

¦

T T W

uD ¨X ¨ ¨ H X

§ §

Since is orthogonal to all vectors , with , only the leading power is W u

T

relevant in the expansion on the right side of the above inner product. In particular, if ¡

PG

X u D W

is the leading coef¬cient for the polynomial , then

T

T u u

% X u D P ¤¦ X uD

G P ¤¦G

¬ § § ¬

u ¥ u

u W p ¨

¨

uW

( " ¡ ¡

PG PG

DW W

u u

When examining the recurrence relations for and , leading coef¬cients for these

© "U

W ’U

W

polynomials are found to satisfy the relations

u

u

u u

( P ¤¦ u # ²¬ ²¬ (P G u

¡ ¡

r"U ¦

PW G

G

r"U G

PW

W

W

W W

µ£ „ ¢

|5¥ „yq¢| ¢ £ ¥§

5| j C§¦£¥

5

¨

¤£

C ¡

§ "–© ¡"

§

1B

I