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âˆ¼ : (279)
!n C( n ) n2m1

Proof. The proof proceeds as the proof of Theorem 12 but in the denominator we use
k
!n
(k)
âˆ¼ C( n ) (280)
n; j
!n+j
j=0

that follows from Lemma C.2 given in Appendix C.

Thus, the e ect of the sequence transformation in this case essentially depends on the question
whether ( n )âˆ’ nâˆ’2m1 goes to 0 for large n or not. In many important cases like the Levin transfor-
mation and the p J transformations, we have M = 1 and m1 = k. We note that Theorem 11 becomes
especially important in the case of logarithmic convergence since for instance for M =1 one observes
n
that (sn+1 âˆ’ s)=(sn âˆ’ s) âˆ¼ 1 and (sn âˆ’ s)=!n âˆ¼ 1 c1; 0 = 0 imply !n+1 =!n âˆ¼ 1= 1 for large n such that
the denominators vanish asymptotically. In this case, we have = m1 whence ( n )âˆ’ nâˆ’2m1 = O(nâˆ’m1 )
if n = O(1=n). This reduction of the speed of convergence of the acceleration process from O(nâˆ’2k )
to O(nâˆ’k ) in the case of logarithmic convergence is a generic behavior that is re ected in a number
of theorems regarding convergence acceleration properties of Levin-type sequence transformations.
Examples are Sidiâ€™s theorem for the Levin transformation given below (Theorem 15), and for the
p J transformation the Corollaries 18 and 19 given below, cf. also [84, Theorems 13:5, 13:9, 13:11,
13:12, 14:2].
The following theorem was given by Matos [59] where the proof may be found. To formulate it,
we deÃ¿ne that a sequence {{un }} has property M if it satisÃ¿es
un+1 â€˜
rn = o( â€˜ (1=n)) for n â†’ âˆž:
âˆ¼ 1 + + rn with rn = o(1=n); (281)
un n
H.H.H. Homeier / Journal of Computational and Applied Mathematics 122 (2000) 81â€“147 127

Theorem 14 (Matos [59, Theorem 13]). Let {{sn }} be a sequence such that
(1) (k)
sn âˆ’ s = !n (a1 g1 (n) + Â· Â· Â· + ak g1 (n) + n) (282)
with g1j+1) (n) = o(g1j) (n)); n = o(g1 (n)) for n â†’ âˆž. Let us consider an operator L of the form
( ( (k)

(193) for which we know a basis of solutions {{unj) }}; j = 1; : : : ; k; and each one can be written as
(

âˆž
( j)
unj)
( ( j) ( j)
gm+1 (n) = o(gmj) (n))
(
âˆ¼ m gm (n); (283)
m=1

as n â†’ âˆž for all m âˆˆ N and j = 1; : : : ; k. Suppose that
(a) g2j+1) (n) = o(g2j) (n)) for n â†’ âˆž; j = 1; : : : ; k âˆ’ 1;
( (

(1) (k) (1)
(b) g2 (n) = o(g1 (n)); and n âˆ¼ Kg2 (n) for n â†’ âˆž;
(284)
(c) {{gmj) (n)}} has property M for
(

m âˆˆ N; j = 1; : : : ; k:
Then
(k+1)
1. If {{!n }} satisÃ¿es limnâ†’âˆž !n =!n+1 = = 1; the sequence transformation Tn corresponding
to the operator L accelerates the convergence of {{sn }}. Moreover; the acceleration can be
measured by
(1)
(k+1)
Tn âˆ’s âˆ’k g2 (n)
âˆ¼ Cn ; n â†’ âˆž: (285)
(1)
sn âˆ’ s g1 (n)
(k+1)
2. If {{1=!n }} has property M; then the speed of convergence of Tn can be measured by
(1)
(k+1)
Tn âˆ’s g2 (n)
âˆ¼ C (1) ; n â†’ âˆž: (286)
sn âˆ’ s g1 (n)

7.2. Results for special cases

In the case that peculiar properties of a Levin-type sequence transformation are used, more stringent
theorems can often be proved as regards convergence acceleration using this particular transformation.
In the case of the Levin transformation, Sidi proved the following theorem:

Theorem 15 (Sidi [76] and Brezwski and Redivo Zaglia [14, Theorem 2:32]). If sn =s+!n fn where
fn âˆ¼ âˆž Ã¿j =nj with Ã¿0 = 0 and !n âˆ¼ âˆž j =nj+a with a Â¿ 0; 0 = 0 for n â†’ âˆž then; if Ã¿k = 0
j=0 j=0

0 Ã¿k
(k)
Â· nâˆ’aâˆ’k
Ln âˆ’ s âˆ¼ (n â†’ âˆž): (287)
âˆ’a
k

For the W algorithm and the d(1) transformation that may be regarded as direct generalizations
of the Levin transformation, Sidi has obtained a large number of results. The interested reader is
referred to the literature (see [77,78] and references therein).
Convergence results for the Levin transformation, the Drummond transformation and the Weniger
transformations may be found in Section 13 of Wenigerâ€™s report [84].
128 H.H.H. Homeier / Journal of Computational and Applied Mathematics 122 (2000) 81â€“147

Results for the J transformation and in particular, for the p J transformation are given in [39,40].
Here, we recall the following theorems:

Theorem 16. Assume that the following holds:
(A-0) The sequence {{sn }} has the (anti)limit s.
(A-1a) For every n; the elements of the sequence {{!n }} are strictly alternating in sign and do
not vanish.
(k) (k)
(A-1b) For all n and k; the elements of the sequence {{ n }} = {{ rn }} are of the same sign
and do not vanish.
(A-2) For all n âˆˆ N0 the ratio (sn âˆ’ s)=!n can be expressed as a series of the form
âˆž
sn âˆ’ s (0) (1) ( jâˆ’1)
= c0 + cj Â·Â·Â· (288)
n1 n2 nj
!n nÂ¿n1 Â¿n2 Â¿Â·Â·Â·Â¿nj
j=1

with c0 = 0.
(k) (k) (k)
Then the following holds for sn = Jn ({{sn }}; {{!n }}; {{ n }}) :
(k)
(a) The error sn âˆ’ s satisÃ¿es
(k)
bn
(k)
sn âˆ’ s = (kâˆ’1) (kâˆ’2) (289)
(0)
n n Â· Â· Â· n [1=!n ]
with
âˆž
(k) (k) (k+1) ( jâˆ’1)
bn = ck + cj Â·Â·Â· : (290)
nk+1 nk+2 nj
nÂ¿nk+1 Â¿nk+2 Â¿Â·Â·Â·Â¿nj
j=k+1
(k)
(b) The error sn âˆ’ s is bounded in magnitude according to
(k) (k) (0) (1) (kâˆ’1)
|sn âˆ’ s|6|!n bn Â·Â·Â· |: (291)
n n n

(c) For large n the estimate
(k)
sn âˆ’ s (0) (1) (kâˆ’1)
= O( Â·Â·Â· ) (292)
n n n
sn âˆ’ s
(k)
holds if bn = O(1) and (sn âˆ’ s)=!n = O(1) as n â†’ âˆž.

(k) (k) (k) (k) (k) (k)
Theorem 17. DeÃ¿ne sn =Jn ({{sn }}; {{!n }}; {{ n }}) and !n =1=Dn where the Dn are deÃ¿ned
(k)
(k) (k) (k) (k) (k)
as in Eq. (94). Put en = 1 âˆ’ !n+1 =!n and bn = (sn âˆ’ s)=!n . Assume that (A-0) of Theorem
16 holds and that the following conditions are satisÃ¿ed:
(B-1) Assume that
(k)
bn
lim = Bk (293)
nâ†’âˆž b(0)
n

exists and is Ã¿nite.
(B-2) Assume that
(k)
!n+1
= lim (k) = 0 (294)
k
nâ†’âˆž !
n
H.H.H. Homeier / Journal of Computational and Applied Mathematics 122 (2000) 81â€“147 129

and
(k)
n+1
Fk = lim =0 (295)
(k)
nâ†’âˆž
n
(k)
exist for all k âˆˆ N0 . Hence the limits = limnâ†’âˆž (cf. Eq. (97)) exist for all k âˆˆ N0 .
k n
Then; the following holds:
(a) If 0 âˆˆ { 0 = 1; 1 ; : : : ; kâˆ’1 }; then
âˆ’1
kâˆ’1
[ 0 ]k
(k)
sn âˆ’ s (l)
lim = Bk (296)
n kâˆ’1
nâ†’âˆž sn âˆ’ s
l=0 ( l âˆ’ 0)
l=0

and; hence;
(k)
sn âˆ’ s
= O( (0) (1) Â· Â· Â· n )
(kâˆ’1)
(297)
n n
sn âˆ’ s
holds in the limit n â†’ âˆž.
(b) If l = 1 for l âˆˆ {0; 1; 2; : : : ; k} then
âˆ’1
kâˆ’1
(k) (l)
sn âˆ’ s n
lim = Bk (298)
(l)
nâ†’âˆž sn âˆ’ s
l=0 en

and; hence;
kâˆ’1
(k) (l)
sn âˆ’ s n
=O (299)
(l)
sn âˆ’ s l=0 en

holds in the limit n â†’ âˆž.

This theorem has the following two corollaries for the p J transformation [39]:

Corollary 18. Assume that the following holds:
(C-1) Let Ã¿ Â¿ 0; pÂ¿1 and n = [(n+Ã¿+(pâˆ’1)k)âˆ’1 ]. Thus; we deal with the p J transformation
(k)
(k) (k)
and; hence; the equations Fk = limnâ†’âˆž n+1 = n = 1 and k = 1 hold for all k.
(C-2) Assumptions (A-2) of Theorem 16 and (B-1) of Theorem 17 are satisÃ¿ed for the particular
(k)
choice (C-1) for n .
(C-3) The limit 0 = limnâ†’âˆž !n+1 =!n exists; and it satisÃ¿es 0 âˆˆ {0; 1}. Hence; all the limits
(k) (k)
k = lim nâ†’âˆž !n+1 =!n exist for k âˆˆ N exist and satisfy k = 0 .
(k) (k)
Then the transformation sn =p Jn (Ã¿; {{sn }}; {{!n }}) satisÃ¿es
âˆ’1
kâˆ’1 k
(k)
sn âˆ’ s 0
(l)
lim = Bk (300)
n
nâ†’âˆž sn âˆ’ s 1âˆ’ 0
l=0

and; hence;
(k)
sn âˆ’ s
= O((n + Ã¿)âˆ’2k ) (301)
sn âˆ’ s
holds in the limit n â†’ âˆž.
130 H.H.H. Homeier / Journal of Computational and Applied Mathematics 122 (2000) 81â€“147

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