¨u ¨

W“ u¨

Gradient algorithm. Like the Conjugate Gradient algorithm, the vectors and are in the

u¦ u¥

same direction as and , respectively. Hence, they form a biorthogonal sequence.

"U

W ’U

W

De¬ne similarly the matrix

r±Q0w°„ ®

i

¬¨¥

¥

!£ ¢

“¢

¢

¨R ¨¢ R¡

¥ ¥

Clearly, the column-vectors of and those of are A-conjugate, since,

T ¢

¡

“¢

X

¨¢

¥

¬ §§ § ¢ W !£ ¬ ¬

¥ ¥ W 1£ ¤ ¤

“¢ W “¢ W “¢

¢ ¢

¢

Utilizing this information, a Conjugate Gradient“like algorithm can be easily derived from

the Lanczos procedure.

‘ ¤¥A ˜Q¤ v ¢ • Vma‘©

• T

¡¦

&95 P¥ V2 5U7C ¡ ¤ P©ETQ

6D

S@

S 6 Q @8

' 7$

%# ) 2

1 R

3

¬ X ¨ ¨ ¨

3 ¨ ¨ ¥

©¨ ¨¬ ¬

V§ ²A¨ ¨ ¬ ¨

1. Compute . Choose such that .

( `

Y

3 3

2. Set, ,

T ¨ T

¡ ¡

X ¨ ( u y $}u

(Y ¬

3. For , until convergence Do:

X ¨u u 3

§ u¨( ¨ ¬ u

4. ( ¡ ¢

¡

3

©

u u wu© ¬

5. ¡

3

u § u ² u ¨ ¬ "U u ¨ W

6. ¡

3

¨ u T § u ² ¨ u ¨ T ¬ "U ¨ u ¨

W

7. ¡

X ¨u

X ¨u u 3

¨ ( u ¨ ¬ "U u S W

8.¨( ¨

"U

W

’U

W

y5 B!¥ © £ § t¡ ”¥ ¢ ¢y5

£ j |„

„ §

¡¤£

CC

u 3

¬

¨3 u

u ¡u S w

9. ¡

¨ u ¡ u S w "U ¨ u W"U ¨ u

W¨ ¬

10. ¡

W"U

"U

W

11. EndDo

¨ ¨ ¨ ¨

§ ¬

If a dual system with is being solved, then in line 1 should be de¬ned as 3

© § ² ¨¨ ¨ ¨u u w ¨u ¨u

© ©¬

and the update to the dual approximate solution must ¡

’U

W

beinserted after line 5. The vectors produced by this algorithm satisfy a few biorthogonality

properties stated in the following proposition.

Q¤ ¡ v A£

£¦ £¤A

' ' ©' 2 )

§)

The vectors produced by the Biconjugate Gradient algorithm sat-

isfy the following orthogonality properties: T

a±Qaw°q ®

i

¬ XX ¨ ¨ R ¨ ( u u ¨ ¥

¬E

for

T ¡Y

( t

(

± ¯ w°q ®

i

¥

§ ¬ ¬E

for

R ¢( (¡Y t

¡ ¡

£ 6 A

£

The proof is either by induction or by simply exploiting the relations between the

¨u ¨ u ¨ ¨u ¨¢

¥

¥ ¢¥

u

vectors , , , , and the vector columns of the matrices , , , . This is ¢ ¢