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(k)
Fn ({{sn }}; {{!n }}; {{x n }}) = =k ; (140)
(k) (k)
n ((x n )k’1 =!n ) x n =(x n )k’1 n ((x n )k’1 =!n )
where {{x n }} is an auxiliary sequence with limn’∞ 1=x n = 0 such that x n+˜ ¿ x n for ˜ ∈ N and
x0 ¿ 1, i.e., a diverging sequence of monotonously increasing positive numbers. Using the deÿnition
(k) (k)
(39) of the divided di erence operator n = n [{{x n }}], the coe cients may be taken as
k
k k’2 k
(x n+j )k’1 xn x n+j + m xn 1
(k)
= = : (141)
n; j
(x n )k’1 x n+j ’ x n+i m=0 x n + m x n+j 1 ’ 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 = ¿1 (142)
n’∞ x n

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

k k
1 1 1
—¦ (k) (k)
= (’1)k ’k(k+1)=2
= lim = ; (143)
j n; j
1’ ’j ’
j ˜’j ’˜
n’∞
˜=0 ˜=0
˜=j ˜=j

since
k
k’2 k k ’l
x n+j + m xn 1 (k’1) j k(’j)
’ (144)
xn + m x n+j 1 ’ x n+˜ =x n+j ’
’˜ ’j
m=0 ˜=0 ˜=0
˜=j ˜=j

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

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

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

—¦
The last equality is easily proved by induction. Hence, the F transformation is convex since
—¦
(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 ’ 1 !n x n ’ 1 !n
(k’1)
(x n+k + k ’ 2)Nn+1 ’ (x n + k ’ 2)Nn(k’1)
(k)
Nn = ;
x n+k ’ x n
(k’1) (k’1)
(x n+k + k ’ 2)Dn+1 ’ (x n + k ’ 2)Dn
(k)
Dn = ;
x n+k ’ 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
—¦ —¦
N (0) = ; D (0) = ;
n n
!n !n
—¦ —¦
(k’1) (k’1)
N n+1 ’ N n
—¦ (k)
= ’(n+k) ;
Nn
’ ’n
H.H.H. Homeier / Journal of Computational and Applied Mathematics 122 (2000) 81“147 105

—¦ —¦
(k’1) (k’1)
D n+1 ’ D n
—¦ (k)
= ’(n+k) ;
Dn
’ ’n
—¦ —¦ —¦
(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 deÿned via the recursive scheme
Nn(0) = sn =!n ; (0)
Dn = 1=!n ;
˜ (k’1) ˜ (k’1) (k’1)
Nn(k) =  n Nn(k’1) ; (k)
Dn =  n Dn ;
(k) (k)
= Nn(k) =Dn ;
(k)
JDn ({{sn }}; {{!n }}; { n }) (151)
(k)
where the generalized di erence operator deÿned in Eq. (34) involves quantities n = 0 for k ∈ N0 .
(k)
Special cases of the JD transformation result from corresponding choices of the n . From Eq. (151)
one easily obtains the alternative representation
˜ (k’1) ˜ (k’2) : : :  (0) [sn =!n ]
˜n
n n
(k) (k)
JDn ({{sn }}; {{!n }}; { n }) = (k’1) (k’2) : (152)
˜ (0) [1=!n ]
˜ n n ˜
 : : : 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
[35,41“ 44,63]. Target of the H transformation are Fourier series

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

n
cj n’j
Cn ∼ n (154)
j=0

for n ’ ∞ with ∈ K; ∈ 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 “ as in the case of the d(1) transformation
with n = n “ 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 “fold frequency. Using this -fold fre-
quency approach, one can obtain stable and accurate convergence acceleration even in the vicinity
of singularities [41“ 44].
106 H.H.H. Homeier / Journal of Computational and Applied Mathematics 122 (2000) 81“147


The H transformation may be deÿned as

Nn(0) = (n + ÿ)’1 sn =!n ; Dn = (n + ÿ)’1 =!n ;
(0)



(k’1) (k’1)
Nn(k) = (n + ÿ)Nn(k’1) + (n + 2k + ÿ)Nn+2 ’ 2 cos( )(n + k + ÿ)Nn+1 ;

(k’1) (k’1)
(k) (k’1)
Dn = (n + ÿ)Dn + (n + 2k + ÿ)Dn+2 ’ 2 cos( )(n + k + ÿ)Dn+1 ;

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

where cos = ±1 and ÿ ∈ R+ .
It can also be represented in the explicit form [34]
P[P (2k) ( )][(n + ÿ)k’1 sn =!n ]
(k)
Hn ( ; ÿ; {{sn }}; {{!n }}) = ; (156)
P[P (2k) ( )][(n + ÿ)k’1 =!n ]
where the pm ( ) and the polynomial P (2k) ( ) ∈ P(2k) are deÿned via
(2k)


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

and P is the polynomial operator deÿned 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
k’1
2k (n+ÿ+m) s
(2k)
pm ( ) (n+ÿ+2k)k’1 !n+m
m=0
(k)
Hn ( ; ÿ; {{sn }}; {{!n }}) = : (158)
n+m
(n+ÿ+m)k’1 1
(2k)
2k
pm ( ) (n+ÿ+2k)k’1 !n+m
m=0

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

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

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