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In exact arithmetic, the core of Algorithm 6.14 is a relation of the form

² u ©§ ¬ ²

uS u¦ u¦u uS u¦

¦

W“ W“

"U

W "U

W

This three-term recurrence relation is reminiscent of the standard three-term recurrence

relation of orthogonal polynomials. In fact, there is indeed a strong relationship between

the Lanczos algorithm and orthogonal polynomials. To begin, recall that if the grade of ¦

T T

W

is , then the subspace is of dimension and consists of all vectors of the form¢

!X X!

¢

¤¦¦ ¥

¡ ¡¡

§ ²

¢ ¢

, where is a polynomial with . In this case there is even an

¦ !

W y ²

isomorphism between and

, the space of polynomials of degree , which

¢ ¢ !

W“ y

is de¬ned by T

¢ X

¡ W¦ § ¢ ¬ ©

¡¡ ¢

¢

¢

W“

Moreover, we can consider that the subspace is provided with the inner product

¢TT T

W“X

§ ¬i ± ¯ „ ®

i

¢ X X

§

¢

(¦ ¦

(

¡ ¡

W W

This is indeed a nondegenerate bilinear form under the assumption that does not exceed

!

R¦T

, the grade of . Now observe that the vectors are of the form

¦

W X

¬ R¦ §

¢ R ¦

W“ W

R¦

and the orthogonality of the ™s translates into the orthogonality of the polynomials with

respect to the inner product (6.64). It is known that real orthogonal polynomials satisfy a

three-term recurrence. Moreover, the Lanczos procedure is nothing but the Stieltjes algo-

rithm; (see, for example, Gautschi [102]) for computing a sequence of orthogonal poly-

nomials with respect to the inner product (6.64). It is known [180] that the characteristic

polynomial of the tridiagonal matrix produced by the Lanczos algorithm minimizes the

norm over the monic polynomials. The recurrence relation between the characteris-

i

T

tic polynomials of tridiagonal matrices also shows that the Lanczos recurrence computes

X

§

the sequence of vectors , where is the characteristic polynomial of .

¦ ¢

¡ ¡

W

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The Conjugate Gradient algorithm is one of the best known iterative techniques for solving

T

sparse Symmetric Positive De¬nite linear systems. Described in one sentence, the method X

’§t ¨

is a realization of an orthogonal projection technique onto the Krylov subspace ( ¢

where is the initial residual. It is therefore mathematically equivalent to FOM. How-

¨

§

ever, because is symmetric, some simpli¬cations resulting from the three-term Lanczos

recurrence will lead to more elegant algorithms.

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We ¬rst derive the analogue of FOM, or Arnoldi™s method, for the case when is sym-

§© ¨ ¬ t§

©

metric. Given an initial guess to the linear system and the Lanczos vectors

Ee( R ¦

¬ together with the tridiagonal matrix , the approximate solution ob-

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!( ¢

y

tained from an orthogonal projection method onto , is given by T

¢

a±— „ ®

i

¬ X