T

2. Run steps of the nonsymmetric Lanczos Algorithm, i.e.,

!

¬ X W¥ (¥ ( W ¦ (

3

–Q ¨ ¬ ¦

3. Start with , and any such that

S ¥

W W y

4. Generate the Lanczos vectors , ¦ ¢ ¦ ( ¥

( ¢

T W W

5. and the tridiagonal matrix from Algorithm 7.1. ¢

¬ X 3

¢b ¢b ¢

(¬ ©

©

6. Compute and . w

FS W “ ¢ ¢

¡

W

Note that it is possible to incorporate a convergence test when generating the Lanczos

vectors in the second step without computing the approximate solution explicitly. This is

due to the following formula, which is similar to Equation (6.66) for the symmetric case,

a± q ®

i

¬ £ u t²A¨ u ¦ 0 u b u ¡

©§ (£

u X0

’U

W "U

W

and which can be proved in the same way, by using (7.3). This formula gives us the residual

norm inexpensively without generating the approximate solution itself.

—y © m’g• ( v Vm‘ }©

“ • ™ ¢

£

¡

The Biconjugate Gradient (BCG) algorithm can be derived from Algorithm 7.1 in exactly

the same way as the Conjugate Gradient method was derived from Algorithm 6.14. The

algorithm was ¬rst proposed by Lanczos [142] in 1952 and then in a different form (Con-

jugate Gradient-like version) by Fletcher [92] in 1974. Implicitly, the algorithm solves not

¨ ¬ ¨© § ¨

"¬ t§

¨© §

only the original system but also a dual linear system with . This

dual system is often ignored in the formulations of the algorithm.

µ£ „ ¢

|5¥ „yq¢| ¢ £ ¥§

5| j C§¦£¥

5

¨

¡¤£

C ¡

§ "–© ¡"

§

1B

I

¦

$6§ 6

! £ ©' DB % 0A

5H

E) 0A B ©' & 8 A P5 B© 8 3 078

5A

E

IH 3

The Biconjugate Gradient (BCG) algorithm is a projection process onto

’§( VTWTT ( W ©W§( W ¦ ¤ c ¢¤¬ ¢

¦ ¦W “

¢

§ ©

¡

W

orthogonally to T

¢X

’§( W ¥ ¤ c £¬ §

¢ ¥ ¥“

( ( T

¢ § ©

WW

¡

VWT W

TT

X ¥

¬ ¬

£

taking, as usual, . The vector is arbitrary, provided , but it

¨ Q ¨

¦ ¥ ¥( ¦ Y

¨ ¬ ¨

W W W W

© ¨§ ¨ §

is often chosen to be equal to . If there is a dual system to solve with , ¦

¨

W © ² ¨

§

then is obtained by scaling the initial residual .

¥

W

Proceeding in the same manner as for the derivation of the Conjugate Gradient al-

gorithm from the symmetric Lanczos algorithm, we write the LDU decomposition of ¢

as

r± w°„ ®

i

¬ ¤¢£

¢ ¢

and de¬ne

p±r°w°„ ®

i

¬¥ ¤ W “¢

¢ ¢

The solution is then expressed as T

XT

¬© © w

FS W “ ¢

¢ ¢ ¡

W X

©¬ FS W “ ¢ T W “ ¢

w ¤ £

¢ ¡

WX

©¬ ¥

w £

FS W “ ¢

¢ ¡

W

© ©