r©( W “ R © u

¬ u R © W “

§

H H ¢¢¢ W

¡¡¡

then this process is mathematically equivalent to the computation of the factorization

¬ ¤H £

H H

of the moment matrix , in which is upper triangular and is lower triangular. Note ¤ £

H H H

vR !

u

that is a Hankel matrix, i.e., its coef¬cients are constant along anti-diagonals, i.e.,

¤ c¢

¤ }H

¬ ¡

for .

wE

Because T T T

u ¢( ¦ X X X

¬ § §

u ¢( u u u

¡

¥ (

¡ ¡ ¡ ¡

W W

we observe that there is a serious breakdown at step if and only if the inde¬nite norm of

u

the polynomial at step vanishes. If this polynomial is skipped, it may still be possible

¡

u

to compute and continue to generate the sequence. To explain this simply, consider

T T T T

¡

"U

W £

u¢

X X X X

(¬

© ©¦¬

¢

u u u

and

¡ ¡

W“ W“

"U

W

¢ ¢ £ u ¡ (

u u

Both and are orthogonal to the polynomials . We can de¬ne (some- (

¡

“

u ¡ W’U W

¬ ¢ ¢

u u u

what arbitrarily) , and then can be obtained by orthogonalizing against ¡

"U

W W"U

u u

and . It is clear that the resulting polynomial will then be orthogonal against all

¡ ¡

W“

polynomials of degree ; see Exercise 5. Therefore, the algorithm can be continued

from step in the same manner. Exercise 5 generalizes this for the case where poly-

w W

y

nomials are skipped rather than just one. This is a simpli¬ed description of the mechanism

which underlies the various versions of Look-Ahead Lanczos algorithms proposed in the

literature. The Parlett-Taylor-Liu implementation [161] is based on the observation that

the algorithm breaks because the pivots encountered during the LU factorization of the

moment matrix vanish. Then, divisions by zero are avoided by performing implicitly a

H”˜

v

¥ ¥

pivot with a matrix rather than using a standard pivot.

˜

y y

The drawback of Look-Ahead implementations is the nonnegligible added complexity.

Besides the dif¬culty of identifying these near breakdown situations, the matrix ceases ¢

to be tridiagonal. Indeed, whenever a step is skipped, elements are introduced above the

superdiagonal positions, in some subsequent step. In the context of linear systems, near

breakdowns are rare and their effect generally benign. Therefore, a simpler remedy, such

as restarting the Lanczos procedure, may well be adequate. For eigenvalue problems, Look-

Ahead strategies may be more justi¬ed.

³ £

y5 B!¥ © £ ¢ £© y5

„ § |„ ky5„" ©£ vE ©

| § |

!–¡

§

—y ¢—z — v™}• ’i

© m’g• — l z k }©

“ ™ ™© “

££

¢¡

We present in this section a brief description of the Lanczos method for solving nonsym-

metric linear systems. Consider the (single) linear system:

a± q ®

i

¨ ¬ t§

©

¤ y¤ §

¥ §©

where is and nonsymmetric. Suppose that a guess to the solution is available

c¨ z ¨

²¬ V§

©

and let its residual vector be . Then the Lanczos algorithm for solving (7.8)

can be described as follows.

w

˜ } V

—

˜h¤ v ¢

¡¦ § ¤¦ ¤¥US ¢

S ¨8 D B

¥ C¥ Q ¢ D$ © R£¤A

5 ¢ § 7C@

D E6 Q CD

8 976 Q

8@5 5

0 (%&$

'#

) 2

1 3 ¢

¨

3

©V§ A¨ ¨

²¬ ¬ £