relation (7.1). Denote by the tridiagonal matrix ¢

£S

W£ X £ S

a±0 q ®

i

¬ „

R

¢

‚ ¢S

¢X ¢

W“ W “¢ X ¢

uS uX( X

u

If the determinations of of lines 7“8 are used, then the ™s are positive and

T

"U

W ’U

W

£¬

uS uX

. X

T

§

R¦ u¥

Observe from the algorithm that the vectors belong to , while the ™s ¦(

¢

X

§ W

are in . In fact, the following proposition can be proved.

¥(

¢

W

¤Q ¡ v A£

£¦ DA

C

' ' ©' 2 )

§)

If the algorithm does not break down before step , then the !

¬Ee( R ¦ ¬X } ( u ¥T

vectors , and , form a biorthogonal system, i.e.,

#V(

!( $V(

!(

V

y y

¬

R ¥( u¦ vX

uR T (E !

y X

T

§

£ £

R © R ¦ § £R © R ¥ §

Moreover, is a basis of and is a basis of

¦(

¢ ¢ ¢

X

§ ¡¢¡¢¡¢ W W ¢¢¢ W

¡¡¡

and the following relations hold,

¥( ¢

W a±a q ®

i

§ ¬ ¢X

w¢ ¦ ( ¢¡

¢ ¢ ¢

"U

W "U

W

µ£ „ ¢

|5¥ „yq¢| ¢ £ ¥§

5| j C§¦£¥

5

¨

£ ¡

§ "–© ¡"

§

1B

I

± ¯ „ ®

i

P

¥

w(¢¥

¬¬ § § ¥

¢S ( ¢¡

¢ ¢

¢

±r— „ ®

i

"U

W ’U

W

¥

¢

¢

¢

£ A™¤T ¢¡6

£

R ¥(R ¦

The biorthogonality of the vectors will be shown by induction. By assump-

¬ XW u ¦ ( ¦ u ¥ (

tion . Assume now that the vectors and are biorthogonal,

¥( ¦ ¥

T

W u ¥ ( W ¥

W

y

u ¦ ( ¦

and let us prove that the vectors and are biorthogonal.

X

E ’U $ T W R ¥ ( u ¦

W WT

"U E

W

¬ v

¬

First, we show that for . When , then T

Y T

"U X

W

Xu u u¦ u S Xu ¥(u¦ u X

¬ § ² ²

u ¥( u¦ & u ¥(

¥ ( © $ W “u X

¦

W“

W"U

"U

W T

The last inner product in the above expression vanishes by the induction hypothesis. The Xu u