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(k)
Fn ({{sn }}; {{!n }}; {{x n }}) = =k ; (140)
(k) (k)
n ((x n )kâˆ’1 =!n ) x n =(x n )kâˆ’1 n ((x n )kâˆ’1 =!n )
where {{x n }} is an auxiliary sequence with limnâ†’âˆž 1=x n = 0 such that x n+â€˜ Â¿ x n for â€˜ âˆˆ N and
x0 Â¿ 1, i.e., a diverging sequence of monotonously increasing positive numbers. Using the deÃ¿nition
(k) (k)
(39) of the divided di erence operator n = n [{{x n }}], the coe cients may be taken as
k
k kâˆ’2 k
(x n+j )kâˆ’1 xn x n+j + m xn 1
(k)
= = : (141)
n; j
(x n )kâˆ’1 x n+j âˆ’ x n+i m=0 x n + m x n+j 1 âˆ’ x n+i =x n+j
i=0 i=0
i=j i=j

Assuming that the following limit exists such that
x n+1
lim = Â¿1 (142)
nâ†’âˆž x n

â—¦
holds, we see that one can deÃ¿ne a limiting transformation F(k) with coe cients
104 H.H.H. Homeier / Journal of Computational and Applied Mathematics 122 (2000) 81â€“147

k k
1 1 1
â—¦ (k) (k)
= (âˆ’1)k âˆ’k(k+1)=2
= lim = ; (143)
j n; j
1âˆ’ âˆ’j âˆ’
j â€˜âˆ’j âˆ’â€˜
nâ†’âˆž
â€˜=0 â€˜=0
â€˜=j â€˜=j

since
k
kâˆ’2 k k âˆ’l
x n+j + m xn 1 (kâˆ’1) j k(âˆ’j)
â†’ (144)
xn + m x n+j 1 âˆ’ x n+â€˜ =x n+j âˆ’
âˆ’â€˜ âˆ’j
m=0 â€˜=0 â€˜=0
â€˜=j â€˜=j

for n â†’ âˆž. Thus, the limiting transformation is given by
k k âˆ’j âˆ’â€˜
j=0 sn+j =!n+j 1=( âˆ’ )
â€˜=0
â€˜=j
â—¦ (k)
F ({{sn }}; {{!n }}; ) = : (145)
k k
1
1=( âˆ’ âˆ’â€˜ )
âˆ’j
â€˜=0
j=0 !n+j
â€˜=j

Comparison with deÃ¿nition (39) of the divided di erence operators reveals that the limiting trans-
formation can be rewritten as
(k) âˆ’n
n [{{ }}](sn =!n )
â—¦ (k)
F ({{sn }}; {{!n }}; ) = (k) : (146)
n [{{
âˆ’n }}](1=! )
n

Comparison to Eq. (133) shows that the limiting transformation is nothing but the W algorithm for
tn = âˆ’n . As characteristic polynomial we obtain
k k kâˆ’1
1âˆ’z j
1
â—¦
(k) j k(k+1)=2
(z) = z = : (147)
âˆ’j âˆ’ j+1 âˆ’ 1
âˆ’â€˜
j=0 j=0
â€˜=0
â€˜=j

â—¦
The last equality is easily proved by induction. Hence, the F transformation is convex since
â—¦
(k)
(1) = 0.
As shown in Appendix B, the F transformation may be computed using the recursive scheme
1 sn 1 1
Nn(0) = (0)
; Dn = ;
x n âˆ’ 1 !n x n âˆ’ 1 !n
(kâˆ’1)
(x n+k + k âˆ’ 2)Nn+1 âˆ’ (x n + k âˆ’ 2)Nn(kâˆ’1)
(k)
Nn = ;
x n+k âˆ’ x n
(kâˆ’1) (kâˆ’1)
(x n+k + k âˆ’ 2)Dn+1 âˆ’ (x n + k âˆ’ 2)Dn
(k)
Dn = ;
x n+k âˆ’ x n
Fn = Nn(k) =Dn :
(k) (k)
(148)
It follows directly from Eq. (146) and the recursion relation for divided di erences that the limiting
transformation can be computed via the recursive scheme
sn 1
â—¦ â—¦
N (0) = ; D (0) = ;
n n
!n !n
â—¦ â—¦
(kâˆ’1) (kâˆ’1)
N n+1 âˆ’ N n
â—¦ (k)
= âˆ’(n+k) ;
Nn
âˆ’ âˆ’n
H.H.H. Homeier / Journal of Computational and Applied Mathematics 122 (2000) 81â€“147 105

â—¦ â—¦
(kâˆ’1) (kâˆ’1)
D n+1 âˆ’ D n
â—¦ (k)
= âˆ’(n+k) ;
Dn
âˆ’ âˆ’n
â—¦ â—¦ â—¦
(k) (k) (k)
Fn = N n = D n : (149)

4.2.8. JD transformation
This transformation is newly introduced in this article. In Section 5.2.1, it is derived via (asymp-
totically) hierarchically consistent iteration of the D(2) transformation, i.e., of
2
(sn =!n )
sn = : (150)
2 (1=! )
n

The JD transformation may be deÃ¿ned via the recursive scheme
Nn(0) = sn =!n ; (0)
Dn = 1=!n ;
Ëœ (kâˆ’1) Ëœ (kâˆ’1) (kâˆ’1)
Nn(k) =  n Nn(kâˆ’1) ; (k)
Dn =  n Dn ;
(k) (k)
= Nn(k) =Dn ;
(k)
JDn ({{sn }}; {{!n }}; { n }) (151)
(k)
where the generalized di erence operator deÃ¿ned in Eq. (34) involves quantities n = 0 for k âˆˆ N0 .
(k)
Special cases of the JD transformation result from corresponding choices of the n . From Eq. (151)
one easily obtains the alternative representation
Ëœ (kâˆ’1) Ëœ (kâˆ’2) : : :  (0) [sn =!n ]
Ëœn
n n
(k) (k)
JDn ({{sn }}; {{!n }}; { n }) = (kâˆ’1) (kâˆ’2) : (152)
Ëœ (0) [1=!n ]
Ëœ n n Ëœ
 : : : n
Thus, the JD(k) is a Levin-type sequence transformation of order 2k.

4.2.9. H transformation and generalized H transformation
The H transformation was introduced by Homeier [34] and used or studied in a series of articles
[35,41â€“ 44,63]. Target of the H transformation are Fourier series
âˆž
s = A0 =2 + (Aj cos(j ) + Bj sin(j )) (153)
j=1

with partial sums sn = A0 =2 + n (Aj cos(j ) + Bj sin(j )) where the Fourier coe cients An and Bn
j=1
have asymptotic expansions of the form
âˆž
n
cj nâˆ’j
Cn âˆ¼ n (154)
j=0

for n â†’ âˆž with âˆˆ K; âˆˆ K and c0 = 0.
The H transformation was critized by Sidi [77] as very unstable and useless near singularities
of the Fourier series. However, Sidi failed to notice that â€“ as in the case of the d(1) transformation
with n = n â€“ one can apply also the H transformation (and also most other Levin-type sequence
transformations) to the subsequence {{s n }} of {{sn }}. The new sequence elements s n = s n can
be regarded as the partial sums of a Fourier series with â€“fold frequency. Using this -fold fre-
quency approach, one can obtain stable and accurate convergence acceleration even in the vicinity
of singularities [41â€“ 44].
106 H.H.H. Homeier / Journal of Computational and Applied Mathematics 122 (2000) 81â€“147

The H transformation may be deÃ¿ned as

Nn(0) = (n + Ã¿)âˆ’1 sn =!n ; Dn = (n + Ã¿)âˆ’1 =!n ;
(0)

(kâˆ’1) (kâˆ’1)
Nn(k) = (n + Ã¿)Nn(kâˆ’1) + (n + 2k + Ã¿)Nn+2 âˆ’ 2 cos( )(n + k + Ã¿)Nn+1 ;

(kâˆ’1) (kâˆ’1)
(k) (kâˆ’1)
Dn = (n + Ã¿)Dn + (n + 2k + Ã¿)Dn+2 âˆ’ 2 cos( )(n + k + Ã¿)Dn+1 ;

Hn ( ; Ã¿; {{sn }}; {{!n }}) = Nn(k) =Dn ;
(k) (k)
(155)

where cos = Â±1 and Ã¿ âˆˆ R+ .
It can also be represented in the explicit form [34]
P[P (2k) ( )][(n + Ã¿)kâˆ’1 sn =!n ]
(k)
Hn ( ; Ã¿; {{sn }}; {{!n }}) = ; (156)
P[P (2k) ( )][(n + Ã¿)kâˆ’1 =!n ]
where the pm ( ) and the polynomial P (2k) ( ) âˆˆ P(2k) are deÃ¿ned via
(2k)

2k
(2k) 2 k
pm ( )xm
(2k)
P ( )(x) = (x âˆ’ 2x cos + 1) = (157)
m=0

and P is the polynomial operator deÃ¿ned in Eq. (38). This shows that the H(k) transformation is a
Levin-type transformation of order 2k. It is not convex.
A subnormalized form is
kâˆ’1
2k (n+Ã¿+m) s
(2k)
pm ( ) (n+Ã¿+2k)kâˆ’1 !n+m
m=0
(k)
Hn ( ; Ã¿; {{sn }}; {{!n }}) = : (158)
n+m
(n+Ã¿+m)kâˆ’1 1
(2k)
2k
pm ( ) (n+Ã¿+2k)kâˆ’1 !n+m
m=0

This relation shows that the limiting transformation
P[P (2k) ( )][sn =!n ]
â—¦ (k)
= (159)
H
P[P (2k) ( )][1=!n ]
exists, and has characteristic polynomial P (2k) ( ).
A generalized H transformation was deÃ¿ned by Homeier [40,43]. It is given in terms of the
polynomial P (k; M ) (e) âˆˆ P(kM ) with
M kM
pâ€˜ M ) (e)xâ€˜ ;
(k;
(k; M ) k
P (e)(x) = (x âˆ’ em ) = (160)
m=1 â€˜=0

where e = (e1 ; : : : ; eM ) âˆˆ KM is a vector of constant parameters. Then, the generalized H transfor-
mation is deÃ¿ned as
P[P (k; M ) (e)][(n + Ã¿)kâˆ’1 sn =!n ]
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