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[N (z) âˆ’ C(z)]âˆ’1 âˆ’ [W (z) âˆ’ C(z)]âˆ’1

= [N (z) âˆ’ C(z)]âˆ’1 (W (z) âˆ’ N (z))[W (z) âˆ’ C(z)]âˆ’1

= qC [pN qC âˆ’ qN pC ]âˆ’1 (qN pW âˆ’ pN qW )[qC pW âˆ’ pC qW ]âˆ’1 qC

âˆ’1

Ë™C Ë™

= z âˆ’â€˜âˆ’m qC (z)qâˆ’1 pC qC (z):

Similarly, we Ã¿nd that

âˆ’1

Ë™C Ë™

[E(z) âˆ’ C(z)]âˆ’1 âˆ’ [S(z) âˆ’ C(z)]âˆ’1 = z âˆ’â€˜âˆ’m qC (z)qâˆ’1 pC qC (z)

and hence (3.59) is established in its operator form. Complex multipliers are not used in it, and so

(3.59) holds as stated.

An important consequence of the compass identity is that, with z = 1, it becomes equivalent to the

vector epsilon algorithm for the construction of E(1) as we saw in the scalar case. If the elements

sj âˆˆ S have representation (3.14), there exists a scalar polynomial b(z) of degree k such that

f (z) = a(z)=b(z) âˆˆ Cd [[z]]: (3.60)

If the coe cients of b(z) are real, we can uniquely associate an operator f(z) with f (z) in (3.60),

and then the uniqueness theorem implies that

( j)

= f (1) (3.61)

2k

and we are apt to say that column 2k of the epsilon table is exact in this case. However, Example 3.2

indicates that the condition that b(z) must have real coe cients is not necessary. For greater gener-

ality in this respect, generalised inverse, vector-valued PadÃƒ approximants (GIPAs) were introduced

e

[n=2k]

(z) âˆˆ Cd [z] and a real scalar denominator

[22]. The existence of a vector numerator polynomial P

polynomial Q[n=2k] (z) having the following properties is normally established by (3.47) and (3.48):

deg{P [n=2k] (z)} = n; deg{Q[n=2k] (z)} = 2k;

(i) (3.62)

Q[n=2k] (z) is a factor of P [n=2k] (z):P [n=2k]âˆ— (z);

(ii) (3.63)

(iii) Q[n=2k] (0) = 1; (3.64)

f (z) âˆ’ P [n=2k] (z)=Q[n=2k] (z) = O(z n+1 );

(iv) (3.65)

where the star in (3.63) denotes the functional complex-conjugate. These axioms su ce to prove

the following result.

72 P.R. Graves-Morris et al. / Journal of Computational and Applied Mathematics 122 (2000) 51â€“80

Theorem 3.10 (Uniqueness [24]). If the vector-valued PadÃƒ approximant

e

R[n=2k] (z) := P [n=2k] (z)=Q[n=2k] (z) (3.66)

of type [n=2k] for f (z) exists; then it is unique.

Proof. Suppose that

Ë† Ë† Ë†

R(z) = P(z)=Q(z); R(z) = P(z)= Q(z)

are two di erent vector-valued PadÃƒ approximants having the same speciÃ¿cation as (3.62) â€“ (3.66).

e

Ë†

Let Qgcd (z) be the greatest common divisor of Q(z); Q(z) and deÃ¿ne reduced and coprime polyno-

mials by

Ë† Ë†

Qr (z) = Q(z)=Qgcd (z); Qr (z) = Q(z)=Qgcd (z):

From (3.63) and (3.65) we Ã¿nd that

Ë†âˆ—

Ë†

Ë† Ë† Ë†

z 2n+2 Qr (z)Qr (z) is a factor of [P(z)Qr (z) âˆ’ P(z)Qr (z)] Â· [P âˆ— (z)Qr (z) âˆ’ P (z)Qr (z)]: (3.67)

The left-hand expression of (3.67) is of degree 2n+4k âˆ’2:deg{Qgcd (z)}+2. The right-hand expression

of (3.67) is of degree 2n + 4k âˆ’ 2:deg{Qgcd (z)}. Therefore the right-hand expression of (3.67) is

identically zero.

Ë† [n=2m] (z) = b(z):bâˆ— (z) and P [n=2m] (z) = a(z)bâˆ— (z), the uniqueness theorem shows that the

Ë†

By taking Q

generalised inverse vector-valued PadÃƒ approximant constructed using the compass identity yields

e

f (z) = a(z)bâˆ— (z)=b(z)bâˆ— (z)

exactly. On putting z =1, it follows that the sequence S, such as the one given by (3.14), is summed

exactly by the vector epsilon algorithm in the column of index 2k. For normal cases, we have now

outlined the proof of a principal result [37,2].

Theorem 3.11 (McLeodâ€™s theorem). Suppose that the vector sequence S satisÃ¿es a nontrivial re-

cursion relation

k k

Ã¿i si+j = Ã¿i s âˆž ; j = 0; 1; 2; : : : (3.68)

i=0 i=0

with Ã¿i âˆˆ C. Then the vector epsilon algorithm leads to

( j)

= sâˆž ; j = 0; 1; 2; : : : (3.69)

2k

provided that zero divisors are not encountered in the construction.

The previous theorem is a statement about exact results in the column of index 2k in the vector

epsilon table. This column corresponds to the row sequence of GIPAs of type [n=2k] for f (z),

evaluated at z = 1. If the given vector sequence S is nearly, but not exactly, generalized geometric,

we model this situation by supposing that its generating function f (z) is analytic in the closed unit

disk D, except for k poles in D := {z: |z| Â¡ 1}. This hypothesis ensures that f (z) is analytic at

z = 1, and it is su ciently strong to guarantee convergence of the column of index 2k in the vector

P.R. Graves-Morris et al. / Journal of Computational and Applied Mathematics 122 (2000) 51â€“80 73

epsilon table. There are several convergence theorems of this type [28â€“30,39]. It is important to

note that any row convergence theorem for generalised inverse vector-valued PadÃƒ approximants has

e

immediate consequences as a convergence result for a column of the vector epsilon table.

A determinantal formula for Q[n=2k] (z) can be derived [24,25] by exploiting the factorisation prop-

erty (3.63). The formula is

0 M01 M02 ::: M0; 2k

M10 0 M12 ::: M1; 2k

. . . .

. . . .

Q[n=2k] (z) = ; (3.70)

. . . .

M2kâˆ’1; 0 M2kâˆ’1; 1 M2kâˆ’1; 2 ::: M2kâˆ’1; 2k

z 2k z 2kâˆ’1 z 2kâˆ’2 ::: 1

where the constant entries Mij are those in the Ã¿rst 2k rows of an anti-symmetric matrix M âˆˆ R(2k+1)Ã—(2k+1)

deÃ¿ned by

ï£±

ï£´ jâˆ’iâˆ’1

ï£´

ï£´ H

câ€˜+i+nâˆ’2k+1 Â· cjâˆ’â€˜+nâˆ’2k for j Â¿ i;

ï£´

ï£´

ï£²

l=0

Mij =

ï£´ âˆ’Mji for i Â¡ j;

ï£´

ï£´

ï£´

ï£´

ï£³

0 for i = j:

As a consequence of the compass identity (Theorem 3.9) and expansion (3.16), we see that entries

in the vector epsilon table are given by

( j)

= P [ j+2k=2k] (1)=Q[ j+2k=2k] (1); j; kÂ¿0;

2k

From this result, it readily follows that each entry in the columns of even index in the vector epsilon

table is normally given succinctly by a ratio of determinants:

0 M01 ::: M0; 2k 0 M01 ::: M0; 2k

M10 0 ::: M1; 2k M10 0 ::: M1; 2k

. . . . . .

( j)

. . . . . .

= Ã· :

. . . . . .

2k

M2kâˆ’1; 0 M2kâˆ’1; 1 ::: M2kâˆ’1; 2k M2kâˆ’1; 0 M2kâˆ’1; 1 ::: M2kâˆ’1; 2k

sj sj+1 ::: s2k+j 1 1 ::: 1

For computation, it is best to obtain numerical results from (3.4). The coe cients of Q[n=2k] (z) =

2k [n=2k] i

i=0 Qi z should be found by solving the homogeneous, anti-symmetric (and therefore consistent)

linear system equivalent to (3.70), namely

M q = 0;

[n=2k]

where qT = (Q2kâˆ’i ; i = 0; 1; : : : ; 2k).

74 P.R. Graves-Morris et al. / Journal of Computational and Applied Mathematics 122 (2000) 51â€“80

4. Vector-valued continued fractions and vector orthogonal polynomials

(0)

The elements 2k lying at the head of each column of even index in the vector epsilon table are

values of the convergents of a corresponding continued fraction. In Section 3, we noted that the

entries in the vector epsilon table are values of vector PadÃƒ approximants of

e

f (z) = c0 + c1 z + c2 z 2 + Â· Â· Â· (4.1)

as deÃ¿ned by (3.16). To obtain the continued fraction corresponding to (4.1), we use Viskova-

tovâ€™s algorithm, which is an ingenious rule for e ciently performing successive reciprocation and

re-expansion of a series [2]. Because algebraic operations are required, we use the image of (4.1)

in A, which is

f(z) = c0 + c1 z + c2 z 2 + Â· Â· Â· (4.2)

with ci âˆˆ V. Using reciprocation and re-expansion, we Ã¿nd

C

J âˆ’1

z J cJ z 1J ) zÃ¿1J ) z 2J ) zÃ¿2J )

( ( ( (

i

f(z) = ci z + Â·Â·Â· (4.3)

1âˆ’1âˆ’1âˆ’1âˆ’1âˆ’

i=0

with i( J ) ; Ã¿i( J ) âˆˆ A and provided all i( J ) = 0; Ã¿i( J ) = 0. By deÃ¿nition, all the inverses implied in

(4.3) are to be taken as right-handed inverses. For example, the second convergent of (4.3) is

J âˆ’1

(J)

âˆ’ zÃ¿1J ) ]âˆ’1 ]âˆ’1

(

ci z i + z J cJ [1 âˆ’ z

[J + 1=1](z) = 1 [1

i=0

and the corresponding element of the vector epsilon table is

(J)

= [J + 1=1](1);

2

where the type refers to the allowed degrees of the numerator and denominator operator polynomials.

The next algorithm is used to construct the elements of (4.3).

Theorem 4.1 (The vector qd algorithm [40]). With the initialisation

Ã¿0J ) = 0;

(

J = 1; 2; 3; : : : ; (4.4)

(J) âˆ’1

= cJ cJ +1 ; J = 0; 1; 2; : : : ; (4.5)

1

(J) (J)

the remaining i ; Ã¿i can be constructed using

( J +1)

(J)

+ Ã¿mJ ) =

( ( J +1)

+ Ã¿mâˆ’1 ; (4.6)

m m

(J)

Ã¿mJ )

( ( J +1) ( J +1)

= Ã¿m (4.7)

m+1 m

for J = 0; 1; 2; : : : and m = 1; 2; 3; : : : :

Remark. The elements connected by these rules form lozenges in the âˆ’ Ã¿ array, as in Fig. 8.

Rule (4.7) requires multiplications which are noncommutative except in the scalar case.

P.R. Graves-Morris et al. / Journal of Computational and Applied Mathematics 122 (2000) 51â€“80 75

Fig. 8.

Proof. First, the identity

C + z [1 + zÃ¿Dâˆ’1 ]âˆ’1 = C + z âˆ’ z 2 Ã¿[zÃ¿ + D]âˆ’1 (4.8)

(J)

= âˆ’Ã¿1J ) ; Ã¿ = âˆ’

( (J)

is applied to (4.3) with = cJ ; Ã¿ = âˆ’ 1, then with 2, etc. We obtain

J

z J +1 cJ 1J )

(

z 2 Ã¿1 J ) 2 J )

( (

i

f(z) = ci z + Â·Â·Â· : (4.9)

1 âˆ’ z( 1J ) + Ã¿1J ) ) âˆ’ 1 âˆ’ z( 2J ) + Ã¿2J ) ) âˆ’

( ( ( (

i=0

( J +1)

Ã¿ = âˆ’Ã¿1J +1) , then with

(

Secondly, let J â†’ J + 1 in (4.3), and then apply (4.8) with =âˆ’ ;

1

= âˆ’ 2J +1) ; Ã¿ = âˆ’Ã¿2J +1) , etc., to obtain

( (

J

z 2 1J +1) Ã¿1J +1)

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