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ï£° q1 ï£»
âˆ’1 âˆ’1
cj+2 âˆ’ cj+1 c0 c1 Â·Â·Â· c2j+2 âˆ’ cj+1 c0 cj+1
e2j+1
I
Not all the elements of the matrix in (3.33) are vectors. An inductive proof that e2j+1 is a vector
(at least in the case when the cj are real vectors and the algebra is a division ring) was given by
Salam [43,44] and Roberts [41] using the designant forms of Sylvesterâ€™s and Schweinsâ€™ identities.
We next construct the numerator and denominator polynomials of the OPAs of f(z) and prove
(3.31) using Berlekampâ€™s method [3], which leads on to the construction of vector PadÃƒ approxi-
e
mants.

DeÃ¿nitions. Given the series expansion (3.22) of f(z), numerator and denominator polynomials
Aj (z); Bj (z) âˆˆ A[z] of degrees â€˜j ; mj are deÃ¿ned sequentially for j = 0; 1; 2; : : : ; by
âˆ’1
Aj+1 (z) = Aj (z) âˆ’ zAjâˆ’1 (z)ejâˆ’1 ej ; (3.34)

âˆ’1
Bj+1 (z) = Bj (z) âˆ’ zBjâˆ’1 (z)ejâˆ’1 ej (3.35)
in terms of the error coe cients ej and auxiliary polynomials Dj (z) which are deÃ¿ned for j=0; 1; 2; : : :
by
Ëœ
ej := [f(z)Bj (z)Bj (z)]j+1 ; (3.36)

Ëœ âˆ’1
Dj (z) := Bj (z)Bjâˆ’1 (z)ejâˆ’1 : (3.37)

These deÃ¿nitions are initialised with
A0 (z) = c0 ; B0 (z) = I; e 0 = c1 ;
Aâˆ’1 (z) = 0; Bâˆ’1 (z) = I; eâˆ’1 = c0 : (3.38)

Example 3.5.
âˆ’1 âˆ’1
A1 (z) = c0 ; B1 (z) = I âˆ’ zc0 c1 ; e1 = c2 âˆ’ c1 c0 c1 ;

D1 (z) = c1 âˆ’ z c1 câˆ’1 c1 :
âˆ’1
Ëœ Ëœ0 âˆ’1 (3.39)

Lemma 3.6.
Bj (0) = I; j = 0; 1; 2; : : : : (3.40)
68 P.R. Graves-Morris et al. / Journal of Computational and Applied Mathematics 122 (2000) 51â€“80

Proof. See (3.35) and (3.38).

Theorem 3.7. With the deÃ¿nitions above; for j = 0; 1; 2; : : : ;
f(z)Bj (z) âˆ’ Aj (z) = O(z j+1 ):
(i) (3.41)
Ëœ
(ii) â€˜j := deg{Aj (z)} = [ j=2]; mj := deg{Bj (z)} = [(j + 1)=2]; deg{Aj (z)Bj (z)} = j: (3.42)
Ëœ Ëœ
(iii) Bj (z)Bj (z) = Bj (z)Bj (z) âˆˆ S[z]: (3.43)

(iv) ej âˆˆ V: (3.44)
C

Ëœ
(v) Dj (z); Aj (z)Bj (z) âˆˆ V[z]: (3.45)
C

f(z)Bj (z) âˆ’ Aj (z) = ej z j+1 + O(z j+2 ):
(vi) (3.46)

Proof. Cases j=0; 1 are veriÃ¿ed explicitly using (3.38) and (3.39). We make the inductive hypothesis
that (i) â€“ (vi) hold for index j as stated, and for index j âˆ’ 1.
Part (i): Using (3.34), (3.35) and the inductive hypothesis (vi),
f(z)Bj+1 (z) âˆ’ Aj+1 (z) = f(z)Bj (z) âˆ’ Aj (z) âˆ’ z( f(z)Bjâˆ’1 (z) âˆ’ Ajâˆ’1 (z))ejâˆ’1 ej = O(z j+2 ):
âˆ’1

Part (ii): This follows from (3.34), (3.35) and the inductive hypothesis (ii).
Part (iii): Using (3.27) and (3.35), and hypotheses (iii) â€“ (iv) inductively,
Ëœ Ëœ Ëœ ËœËœ
Bj+1 (z)Bj+1 (z) = Bj (z)Bj (z) + z 2 Bjâˆ’1 (z)Bjâˆ’1 (z)|ej |2 |ejâˆ’1 |âˆ’2 âˆ’ z[Dj (z)ej + e j Dj (z)] âˆˆ S[z]
Ëœ Ëœ
and (iii) follows after postmultiplication by Bj+1 (z) and premultiplication by [Bj+1 (z)]âˆ’1 , see
[37, p. 45].
Part (iv): By deÃ¿nition (3.36),
2mj+1
ej+1 = cj+2âˆ’i Ã¿i ;
i=0

Ëœ
where each Ã¿i = [Bj+1 (z)Bj+1 (z)]i âˆˆ S is real. Hence
ej+1 âˆˆ V:
C

Part (v): From (3.35) and (3.37),
Ëœ ËœËœ
âˆ’1 âˆ’1
Dj+1 (z) = [Bj (z)Bj (z)]ej âˆ’ z[e j Dj (z)ej ]:
Using part (v) inductively, parts (iii), (iv) and Lemma 3.3, it follows that Dj+1 (z) âˆˆ V[z].
C
Using part (i), (3.40) and the method of proof of part (iv), we have
Ëœ Ëœ
Aj+1 (z)Bj+1 (z) = [f(z)Bj+1 (z)Bj+1 (z)]j+1 âˆˆ V[z]:
C
0

Part (vi): From part (i), we have
j+2
+ O(z j+3 )
f(z)Bj+1 (z) âˆ’ Aj+1 (z) = j+1 z

for some âˆˆ A. Hence,
j+1
Ëœ Ëœ Ëœ
j+2
Bj+1 (z) + O(z j+3 ):
f(z)Bj+1 (z)Bj+1 (z) âˆ’ Aj+1 (z)Bj+1 (z) = j+1 z

Using (ii) and (3.40), we obtain = ej+1 , as required.
j+1
P.R. Graves-Morris et al. / Journal of Computational and Applied Mathematics 122 (2000) 51â€“80 69

Corollary. The designant of a Hankel matrix of real (or complex) vectors is a real (or complex)
vector.

Proof. Any designant of this type is expressed by e2j+1 in (3.31b), and (3.44) completes the proof.

The implications of the previous theorem are extensive. From part (iii) we see that
Ëœ
Qj (z) : I := Bj (z)Bj (z) (3.47)
deÃ¿nes a real polynomial Qj (z). Part (iv) shows that the ej are images of vectors ej âˆˆ Cd ; part (vi)
(0)
justiÃ¿es calling them error vectors but they are also closely related to the residuals b âˆ’ A 2j of
Ëœ
Example 3.1. Part (v) shows that Aj (z)Bj (z) is the image of some Pj (z) âˆˆ Cd [z], so that
d d
Ëœ
Aj (z)Bj (z) = [Re{Pj }(z)]i Ei + [Im{Pj }(z)]i Fi : (3.48)
i=1 i=1

From (3.17) and (3.18), it follows that
Ë†
Pj (z) Â· Pjâˆ— (z) = Qj (z)Qj (z); (3.49)
Ë† Ë† Ëœ
where Qj (z) is a real scalar polynomial determined by Qj (z)I = Aj (z)Aj (z). Property (3.49) will later
be used to characterise certain VPAs independently of their origins in A. Operator PadÃƒ approximants
e
were introduced in (3.34) and (3.35) so as to satisfy the accuracy-through-order property (3.41)
for f(z). To generalise to the full table of approximants, only initialisation (3.38) and the degree
speciÃ¿cations (3.42) need to be changed.
For J Â¿ 0, we use
J
A( J ) (z) B0J ) (z) = I;
(
e0J ) = cJ +1 ;
(
ci z i ;
=
0
i=0
J âˆ’1
A( J ) (z) Bâˆ’1) (z) = I;
(J
eâˆ’1) = cJ ;
(J
ci z i ;
= (3.50)
âˆ’1
i=0

â€˜j J ) := deg{A( J ) (z)} = J + [ j=2];
(
j

m( J ) := deg{Bj J ) (z)} = [(j + 1)=2]
(
(3.51)
j

and then (3.38) and (3.42) correspond to the case of J = 0.
For J Â¡ 0, we assume that c0 = 0, and deÃ¿ne
Ëœ Ëœ
g(z) = [f(z)]âˆ’1 = f(z)[f(z)f(z)]âˆ’1 (3.52)
corresponding to
g(z) = [ f (z)]âˆ’1 = f âˆ— (z)[ f (z) : f âˆ— (z)]âˆ’1 : (3.53)
(If c0 = 0, we would remove a maximal factor of z from f(z) and reformulate the problem.)
70 P.R. Graves-Morris et al. / Journal of Computational and Applied Mathematics 122 (2000) 51â€“80

Then, for J Â¡ 0,
âˆ’J
A( J ) (z) B0J ) (z)
(
e0J ) = [f(z)B0J ) (z)]1âˆ’J ;
( (
gi z i ;
= I; =
0
i=0
1âˆ’J
A( J ) (z) B1J ) (z)
(
e1J ) = [f(z)B1J ) (z)]2âˆ’J ;
( (
gi z i ;
= I; =
1
i=0

â€˜j J ) := deg{A( J ) (z)} = [ j=2];
(
j

m( J ) := deg{Bj J ) (z)} = [(j + 1)=2] âˆ’ J:
(
(3.54)
j

If an approximant of given type [â€˜=m] is required, there are usually two di erent staircase sequences
of the form
S (J ) = (A( J ) (z)[Bj J ) (z)]âˆ’1 ;
(
j = 0; 1; 2; : : :) (3.55)
j

which contain the approximant, corresponding to two values of J for which â€˜ = â€˜j J ) and m = m( J ) .
(
j
(J) (J)
[â€˜=m] [â€˜=m]
For ease of notation, we use p (z) â‰¡ Aj (z) and q (z) â‰¡ Bj (z). The construction based on
(3.41) is for right-handed OPAs, as in
f(z) = p[â€˜=m] (z)[q[â€˜=m] (z)]âˆ’1 + O(z â€˜+m+1 ); (3.56)
but the construction can easily be adapted to that for left-handed OPAs for which
f(z) = [q[â€˜=m] (z)]âˆ’1 p[â€˜=m] (z) + O(z â€˜+m+1 ): (3.57)
Although the left- and right-handed numerator and denominator polynomials usually are di erent,
the actual OPAs of given type are equal:

Theorem 3.8 (Uniqueness). Left-handed and right-handed OPAs; as speciÃ¿ed by (3.56) and (3.57)
are identical:
[â€˜=m](z) := p[â€˜=m] (z)[q[â€˜=m] (z)]âˆ’1 = [q[â€˜=m] (z)]âˆ’1 p[â€˜=m] (z) âˆˆ V (3.58)
C

and the OPA of type [â€˜=m] for f(z) is unique.

Proof. Cross-multiply (3.58), use (3.56), (3.57) and then (3.40) to establish the formula in (3.58).
Uniqueness of [â€˜=m](z) follows from this formula too, and its vector character follows from (3.43)
and (3.45).

The OPAs and the corresponding VPAs satisfy the compass (Ã¿ve-point star) identity amongst
approximants of the type shown in the same format as Fig. 3.

Theorem 3.9 (Wynnâ€™s compass identity [57,58]).
[N (z) âˆ’ C (z)]âˆ’1 + [S(z) âˆ’ C (z)]âˆ’1 = [E(z) âˆ’ C (z)]âˆ’1 + [W (z) âˆ’ C (z)]âˆ’1 : (3.59)
P.R. Graves-Morris et al. / Journal of Computational and Applied Mathematics 122 (2000) 51â€“80 71

Proof. We consider the accuracy-through-order equations for the operators:
Ë™Ë™
pN (z)qC (z) âˆ’ qN (z)pC (z) = z â€˜+m pN qC ;

Ë™Ë™
pC (z)qW (z) âˆ’ qC (z)pW (z) = z â€˜+m pC qW ;

Ë™Ë™
pN (z)qW (z) âˆ’ qN (z)pW (z) = z â€˜+m pN qW ;
where q ; p denote the leading coe cients of p (z); q (z), and care has been taken to respect
Ë™Ë™
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