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° q1 »
’1 ’1
cj+2 ’ cj+1 c0 c1 ··· c2j+2 ’ 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 Sylvester™s and Schweins™ identities.
We next construct the numerator and denominator polynomials of the OPAs of f(z) and prove
(3.31) using Berlekamp™s method [3], which leads on to the construction of vector Padà approxi-
e
mants.

Deÿnitions. Given the series expansion (3.22) of f(z), numerator and denominator polynomials
Aj (z); Bj (z) ∈ A[z] of degrees ˜j ; mj are deÿned sequentially for j = 0; 1; 2; : : : ; by
’1
Aj+1 (z) = Aj (z) ’ zAj’1 (z)ej’1 ej ; (3.34)

’1
Bj+1 (z) = Bj (z) ’ zBj’1 (z)ej’1 ej (3.35)
in terms of the error coe cients ej and auxiliary polynomials Dj (z) which are deÿned for j=0; 1; 2; : : :
by
˜
ej := [f(z)Bj (z)Bj (z)]j+1 ; (3.36)

˜ ’1
Dj (z) := Bj (z)Bj’1 (z)ej’1 : (3.37)

These deÿnitions are initialised with
A0 (z) = c0 ; B0 (z) = I; e 0 = c1 ;
A’1 (z) = 0; B’1 (z) = I; e’1 = c0 : (3.38)

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

D1 (z) = c1 ’ z c1 c’1 c1 :
’1
˜ ˜0 ’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) 51“80


Proof. See (3.35) and (3.38).

Theorem 3.7. With the deÿnitions above; for j = 0; 1; 2; : : : ;
f(z)Bj (z) ’ Aj (z) = O(z j+1 ):
(i) (3.41)
˜
(ii) ˜j := deg{Aj (z)} = [ j=2]; mj := deg{Bj (z)} = [(j + 1)=2]; deg{Aj (z)Bj (z)} = j: (3.42)
˜ ˜
(iii) Bj (z)Bj (z) = Bj (z)Bj (z) ∈ S[z]: (3.43)

(iv) ej ∈ V: (3.44)
C

˜
(v) Dj (z); Aj (z)Bj (z) ∈ V[z]: (3.45)
C

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

Proof. Cases j=0; 1 are veriÿed explicitly using (3.38) and (3.39). We make the inductive hypothesis
that (i) “ (vi) hold for index j as stated, and for index j ’ 1.
Part (i): Using (3.34), (3.35) and the inductive hypothesis (vi),
f(z)Bj+1 (z) ’ Aj+1 (z) = f(z)Bj (z) ’ Aj (z) ’ z( f(z)Bj’1 (z) ’ Aj’1 (z))ej’1 ej = O(z j+2 ):
’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) “ (iv) inductively,
˜ ˜ ˜ ˜˜
Bj+1 (z)Bj+1 (z) = Bj (z)Bj (z) + z 2 Bj’1 (z)Bj’1 (z)|ej |2 |ej’1 |’2 ’ z[Dj (z)ej + e j Dj (z)] ∈ S[z]
˜ ˜
and (iii) follows after postmultiplication by Bj+1 (z) and premultiplication by [Bj+1 (z)]’1 , see
[37, p. 45].
Part (iv): By deÿnition (3.36),
2mj+1
ej+1 = cj+2’i ÿi ;
i=0

˜
where each ÿi = [Bj+1 (z)Bj+1 (z)]i ∈ S is real. Hence
ej+1 ∈ V:
C

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

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

for some ∈ A. Hence,
j+1
˜ ˜ ˜
j+2
Bj+1 (z) + O(z j+3 ):
f(z)Bj+1 (z)Bj+1 (z) ’ 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) 51“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
˜
Qj (z) : I := Bj (z)Bj (z) (3.47)
deÿnes a real polynomial Qj (z). Part (iv) shows that the ej are images of vectors ej ∈ Cd ; part (vi)
(0)
justiÿes calling them error vectors but they are also closely related to the residuals b ’ A 2j of
˜
Example 3.1. Part (v) shows that Aj (z)Bj (z) is the image of some Pj (z) ∈ Cd [z], so that
d d
˜
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
ˆ
Pj (z) · Pj— (z) = Qj (z)Qj (z); (3.49)
ˆ ˆ ˜
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 Padà 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
speciÿcations (3.42) need to be changed.
For J ¿ 0, we use
J
A( J ) (z) B0J ) (z) = I;
(
e0J ) = cJ +1 ;
(
ci z i ;
=
0
i=0
J ’1
A( J ) (z) B’1) (z) = I;
(J
e’1) = cJ ;
(J
ci z i ;
= (3.50)
’1
i=0



˜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 ¡ 0, we assume that c0 = 0, and deÿne
˜ ˜
g(z) = [f(z)]’1 = f(z)[f(z)f(z)]’1 (3.52)
corresponding to
g(z) = [ f (z)]’1 = f — (z)[ f (z) : f — (z)]’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) 51“80


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


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


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

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

which contain the approximant, corresponding to two values of J for which ˜ = ˜j J ) and m = m( J ) .
(
j
(J) (J)
[˜=m] [˜=m]
For ease of notation, we use p (z) ≡ Aj (z) and q (z) ≡ Bj (z). The construction based on
(3.41) is for right-handed OPAs, as in
f(z) = p[˜=m] (z)[q[˜=m] (z)]’1 + O(z ˜+m+1 ); (3.56)
but the construction can easily be adapted to that for left-handed OPAs for which
f(z) = [q[˜=m] (z)]’1 p[˜=m] (z) + O(z ˜+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 speciÿed by (3.56) and (3.57)
are identical:
[˜=m](z) := p[˜=m] (z)[q[˜=m] (z)]’1 = [q[˜=m] (z)]’1 p[˜=m] (z) ∈ V (3.58)
C

and the OPA of type [˜=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 [˜=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 (ÿve-point star) identity amongst
approximants of the type shown in the same format as Fig. 3.

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


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

™™
pC (z)qW (z) ’ qC (z)pW (z) = z ˜+m pC qW ;

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

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