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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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