$ ¢

! I H BA ©' & 8 50A

H

3

¢

To introduce the Lanczos algorithm we begin by making the observation stated in the

following theorem.

qzzA y5

5| |„ £©

75 B1B¥ © £

„ §

¦ ¡¢

¢ B ¥C

§

Av ¡˜¤

¥¦ ¥

¤A

£ ' P

1

Assume that Arnoldi™s method is applied to a real symmetric matrix

§ uR

. Then the coef¬cients generated by the algorithm are such that

£

R±` „ ®

°i

¬ u vuRu

²

¤¦¤ £¡Y

¢( E (

£

a±0 „ ®

i

y y

¬ v

¬

u R( u !#(V™(

(˜

£ £

’U

W "U

W y

¡

In other words, the matrix obtained from the Arnoldi process is tridiagonal and sym-

¢

metric.

£ ¡

6 A

£

¬ ¡§

The proof is an immediate consequence of the fact that is a

¢ ¢

¢

symmetric matrix which is also a Hessenberg matrix by construction. Therefore, must

¢

be a symmetric tridiagonal matrix.

The standard notation used to describe the Lanczos algorithm is obtained by setting

u ( uu uS u (u

! !

£ £

W “

¡

and if denotes the resulting matrix, it is of the form,

¢ ¢

£S

W£ S £ S

a±a „ ®

i

¬ „

R

¢

‚ ¢S

¢S ¢

W“ W “¢ S ¢

This leads to the following form of the Modi¬ed Gram-Schmidt variant of Arnoldi™s

method, namely, Algorithm 6.2.

˜h¤ v ¢

¡¦ ¦ ¤¦

S ¨8 D B

7 © RtCDA

5 ¢ § 7C@

D E6 Q

39

0 (%&$

'#

) 2

1

1. Choose an initial vector of norm unity. Set ¦ ¡Y ! `!

¦ S ( Y

W W

v

¬

2. For Do: !#V˜™( 3

((

u ¦ Ty

² §¬ u

u¦uS

3. ¥ X u u§ 3 u

W“ ¬¬

4. ¦( ¥ 3

² u

u¥

u¦u

5. ¥ u u 3

¬ ¬

£¥

uS

6. . If then Stop

S $

Y

u – u 3 "U u W

’U

W ¬

7. ¦ S¥

"U

W

"U

W

8. EndDo

It is rather surprising that the above simple algorithm guarantees, at least in exact

¬yeE( R ¦

arithmetic, that the vectors are orthogonal. In reality, exact orthogonality ˜ (

((

y R¦

of these vectors is only observed at the beginning of the process. At some point the ™s

start losing their global orthogonality rapidly. There has been much research devoted to

¬nding ways to either recover the orthogonality, or to at least diminish its effects by partial

or selective orthogonalization; see Parlett [160].

¡

The major practical differences with Arnoldi™s method are that the matrix is tridi- ¢

agonal and, more importantly, that only three vectors must be saved, unless some form of

reorthogonalization is employed.

¨ ¡¡µ£ „ ¢

§|5¥ yq¢| ¢ £ ¥§