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[87]
(k) (n+1+(Ã¿+kâˆ’1)= )k
1
Cn ( ; Ã¿= ; {{sn }}; {{!n }}) ( n+Ã¿)j (n+(Ã¿+k)= )k+2
Weniger M transformation
(k) (n+1+ âˆ’(kâˆ’1))k
1
Mn ( ; {{sn }}; {{!n }}) (âˆ’nâˆ’ )j (n+ âˆ’k)k+2
Weniger S transformation
(k) 1
Sn (Ã¿; {{sn }}; {{!n }}) 1=(n + Ã¿)j (n+Ã¿+2k)2
Iterated Aitken process
[2,84]
(k)
An ({{sn }})
(k+1) 2 (k)
( An ({{sn }}))( An )({{sn }})
(k) (k)
=Jn ({{sn }}; {{ sn }}; { n }) Eq. (231) (k) (k)
( An ({{sn }}))( An+1 ({{sn }}))
Overholt process
[64]
Vn(k) ({{sn }})
( sn+k+1 ) [( sn+k )k+1 ]
(k) (k)
=Jn ({{sn }}; {{ sn }}; { n }) Eq. (231) ( sn+k )k+1

a
Refs. [36,38,40].
b
For the deÃ¿nition of the j; n see Eq. (5).
c
Factors independent of n are irrelevant.
H.H.H. Homeier / Journal of Computational and Applied Mathematics 122 (2000) 81â€“147 97

Ë† (k)
Nn
(k) (k)
Jn ({{sn }}; {{!n }}; { n }) = (k)
Ë†
Dn
with
(0) (1) (kâˆ’1)
n Â·Â·Â· n
n
(0) (k)
= 1; = ; kâˆˆN (97)
n n (0) (1) (kâˆ’1)
Â· Â· Â· n+1
n+1 n+1

and
Ëœ (0) Ëœ (0)
Dn = 1=!n ; N n = sn =!n ;
Ëœ (k) Ëœ (kâˆ’1) (kâˆ’1) Ëœ (kâˆ’1)
Dn = Dn+1 âˆ’ Dn ; k âˆˆN;
n

Ëœ (k) Ëœ (kâˆ’1) (kâˆ’1) Ëœ (kâˆ’1)
N n = N n+1 âˆ’ Nn ; k âˆˆ N; (98)
n

Ëœ (k)
Nn
(k) (k)
Jn ({{sn }}; {{!n }}; { n }) = (k)
Ëœ
Dn
with
(0) (1) (kâˆ’1)
n+k n+kâˆ’1 Â· Â· Â· n+1
(0) (k)
= 1; = ; k âˆˆ N: (99)
n n (0) (1) (kâˆ’1)
Â·Â·Â· n
n+kâˆ’1 n+kâˆ’2
(k)
The quantities n should not be mixed up with the k; n (u) as deÃ¿ned in Eq. (43).
Ë† (k) (k)
k
As shown in [46], the coe cients for the algorithm (96) that are deÃ¿ned via Dn = n; j =!n+j ,
j=0
satisfy the recursion
(k+1) (k) (k) (k)
= âˆ’ (100)
n; j n; j
n+1; jâˆ’1
n
(0) (k)
with starting values n; j = 1. This holds for all j if we deÃ¿ne n; j = 0 for j Â¡ 0 or j Â¿ k. Because
(k) (k)
(k)
n = 0, we have n; k = 0 such that { n; j } is a coe cient set for all k âˆˆ N0 .
(k)
Ëœ (k)
Similarly, the coe cients for algorithm (98) that are deÃ¿ned via Dn = k Ëœn; j =!n+j , satisfy the
j=0
recursion
Ëœ(k+1) = Ëœ(k) (k) Ëœ(k)
n+1; jâˆ’1 âˆ’ (101)
n; j n n; j
(0) (k)
with starting values Ëœn; j = 1. This holds for all j if we deÃ¿ne Ëœn; j = 0 for j Â¡ 0 or j Â¿ k. In this
(k) (k)
case, we have Ëœn; k = 1 such that { Ëœn; j } is a coe cient set for all k âˆˆ N0 .
Since the J transformation vanishes for {{sn }} = {{c!n }}, c âˆˆ K according to Eq. (95) for all
(1) (1)
k âˆˆ N, it is convex. This may also be shown by using induction in k using n; 1 = âˆ’ n; 0 = 1 and the
equation
k+1 k k
(k+1) (k) (k)
(k)
= âˆ’ (102)
n; j n; j
n+1; j
n
j=0 j=0 j=0

that follows from Eq. (100).
98 H.H.H. Homeier / Journal of Computational and Applied Mathematics 122 (2000) 81â€“147

(k)
Assuming that the limits = limnâ†’âˆž exist for all k âˆˆ N and noting that for k = 0 always
k n
â—¦
â—¦
= 1 holds, it follows that there exists a limiting transformation J [ ] that can be considered as
0
special variant of the J transformation and with coe cients given explicitly as [46, Eq. (16)]
kâˆ’1
â—¦ (k) kâˆ’j jm
= (âˆ’1) ( m) : (103)
j
m=0
j0 +j1 +:::+jkâˆ’1 =j;
j0 âˆˆ{0;1};:::; jkâˆ’1 âˆˆ{0;1}

As characteristic polynomial we obtain
k kâˆ’1
â—¦ â—¦ (k) j
(k)
(z) = jz = ( jz âˆ’ 1): (104)
j=0 j=0

â—¦ â—¦
Hence, the J transformation is convex since (k) (1) = 0 due to 0 = 1.
The p J Transformation: This is the special case of the J transformation corresponding to
1
(k)
= (105)
n
(n + Ã¿ + (p âˆ’ 1)k)2
or to [46, Eq. (18)] 2
ï£±
n+Ã¿+2 n+Ã¿
ï£´
ï£´ for p = 1;
ï£´
ï£´ pâˆ’1 pâˆ’1
ï£² k k
(k)
= (106)
n
ï£´ k
ï£´ n+Ã¿+2
ï£´
ï£´ for p = 1
ï£³
n+Ã¿
or to
ï£±
n+Ã¿+k âˆ’1 n+Ã¿+k +1
ï£´
ï£´ for p = 2;
ï£´
ï£´ pâˆ’2 pâˆ’2
ï£² k k
(k)
= (107)
n
ï£´ k
ï£´ n+Ã¿+k âˆ’1
ï£´
ï£´ for p = 2;
ï£³
n+Ã¿+k +1
that is,
(k) (k)
p Jn (Ã¿; {{sn }}; {{!n }}) = Jn ({{sn }}; {{!n }}; {1=(n + Ã¿ + (p âˆ’ 1)k)2 }): (108)
â—¦ â—¦
The limiting transformation p J of the p J transformation exists for all p and corresponds to the J
transformation with k = 1 for all k in N0 . This is exactly the Drummond transformation discussed
in Section 4.2.2, i.e., we have
â—¦
(k) (k)
p J n (Ã¿; {{sn }}; {{!n }}) = Dn ({{sn }}; {{!n }}): (109)

2
The equation in [46] contains an error.
H.H.H. Homeier / Journal of Computational and Applied Mathematics 122 (2000) 81â€“147 99

4.2.2. Drummond transformation
This transformation was given by Drummond [19]. It was also discussed by Weniger [84]. It may
be deÃ¿ned as
k
[sn =!n ]
(k)
Dn ({{sn }}; {{!n }}) = : (110)
k [1=! ]
n

Using the deÃ¿nition (32) of the forward di erence operator, the coe cients may be taken as
k
(k)
= (âˆ’1) j ; (111)
n; j
j
i.e., independent of n. As moduli, one has n = ( <k=2= ) = Ëœ (k) . Consequently, the Drummond trans-
k
(k)

formation is given in subnormalized form. As characteristic polynomial we obtain
k
k
(k)
(âˆ’1) j z j = (1 âˆ’ z)k :
n (z) = (112)
j
j=0

(k)
Hence, the Drummond transformation is convex since n (1) = 0. Interestingly, the Drummond
transformation is identical to its limiting transformation:
â—¦
D (k) ({{sn }}; {{!n }}) = Dn ({{sn }}; {{!n }}):
(k)
(113)
The Drummond transformation may be computed using the recursive scheme

Nn(0) = sn =!n ; (0)
Dn = 1=!n ;
Nn(k) = Nn(kâˆ’1) ; (k) (kâˆ’1)
Dn = Dn ;
Dn = Nn(k) =Dn :
(k) (k)
(114)

4.2.3. Levin transformation
This transformation was given by Levin [53]. It was also discussed by Weniger [84]. It may be
deÃ¿ned as 3
(n + Ã¿ + k)1âˆ’k k
[(n + Ã¿)kâˆ’1 sn =!n ]
(k)
Ln (Ã¿; {{sn }}; {{!n }}) = : (115)
(n + Ã¿ + k)1âˆ’k k [(n + Ã¿)kâˆ’1 =! ]
n

Using the deÃ¿nition (32) of the forward di erence operator, the coe cients may be taken as
k
(k)
= (âˆ’1) j (n + Ã¿ + j)kâˆ’1 =(n + Ã¿ + k)kâˆ’1 : (116)
n; j
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