and as a result

u#

u ¥ u

¬

"u U

W "u U

W

u

¥

uS

which yields the following relation for :

p±° ¯ „ ®

i

u ¥ u

¬ ¥

uS "u U

W u#

¥

uT T

Similarly, a simple recurrence formula for can be derived. By de¬nition,

T

D

X¨ X T uD X

§ §

(t ¨ T u T

¨X

§ § ¬

u

u (t ¨ X u

X¨

§¡

¡ ¨

and as in the previous case, the polynomials in the right sides of the inner products in both

T T

the numerator and denominator can be replaced by their leading terms. However, in this

uD X X

¨ T ¨ ¨ ¨

§ § T u D¡

case the leading coef¬cients for and are identical, and therefore, T

DT XTu ¨ ¨ X X

Tu

§ §Tu D

(¨

§ ¡ D T§ T ¬

u X ¨ ¨ X § X

Tu (p ¨

T u © (p ¨ X X¨ X

§T¬ ¨§

Tu

X¨X

u © (T ¨ X

Tu

§ ¡ §T ¨X ¨ §

¨X T u DX

X

X

§ §©¬ Tu ¨(

X¨

§¡§ § u© u

T T ¨ (p

¨

§ u ¡X § X

¬

u u

Since , this yields,

© ¨

¡

T r±0 ¯ „ ®

i

u ¥

§¬

u X¨ u

¨(

¡

u#

Next, the parameter must be de¬ned. This can be thought of as an additional free

u#

parameter. One of the simplest choices, and perhaps the most natural, is to select to

T

achieve a steepest descent step in the residual direction obtained before multiplying the

X

T T T

u# u#

X §u# ²

residual vector by in (7.40). In other words, is chosen to minimize the 2-

u DX X

§ u© § ² §

norm of the vector . Equation (7.40) can be rewritten as ¨

T

"U

W

uX

§u # ² ¬

u¨ ¢

"U

W

in which

§u ²

u ! u u

¢ ¨ ¡

u#

Then the optimal value for is given by T

Xu u

§T ¬ r±a ¯ „ ®

i

¢¢ (

u# Xu

’§ ¢ § (u ¢

© ©

u u

Finally, a formula is needed to update the approximate solution from . Equa-

"U

W

tion (7.40) can be rewritten as

§ u # ² u § u ² u ¬ u ¢ § u # ² u ¢¤¬

u¨ u

¢

¨ ¡

"U

W

which yields

u u # w u ¡u w u

© ¬ "U u © ¢

W

³ ¤£

5 £ B¡ k| ¡ k¦¡B¦5

§ £ | § q| ¥ £§ C

¨ "

After putting these relations together, we obtain the ¬nal form of the BICGSTAB

algorithm, due to van der Vorst [210].

fmm“˜‘ A ˜h¤ v ¢

‘ ©—• ¡¦

0 (%&$

'#

) 2

1 3