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m

Ã¿ki

(m; j) k kâˆ’1

s l =s + ( + )[ a l] ; j6l6j + N (49)

l

( l + )i

i=0

k=1

with Â¿ 0; N = m nk and the N +1 unknowns s(m; j) and Ã¿k i . The [ k aj ] are deÃ¿ned via [ 0 aj ]=aj

k=1

k kâˆ’1

aj+1 ] âˆ’ [ kâˆ’1 aj ]; k = 1; 2; : : : . In most cases, all nk are chosen equal and one

and [ aj ] = [

puts = (n; n; : : : ; n). Apart from the value of , only the input of m and of â€˜ is required from the

(m; 0)

user. As transformed sequence, often one chooses the elements s(n; :::; n) for n = 0; 1; : : : . The u variant

of the Levin transformation is obtained for m = 1; = Ã¿ and l = l. The deÃ¿nition above di ers

slightly from the original one [54] and was given in Ref. [22] with = 1.

Ford and Sidi have shown, how these transformations can be calculated recursively with the

W(m) algorithms [22]. The d(m) transformations are the best known special cases of the generalised

Richardson Extrapolation process (GREP) as deÃ¿ned by Sidi [72,73,78].

The d(m) transformations are derived by asymptotic analysis of the remainders sr âˆ’ s for r â†’ âˆž

Ëœ (m)

for the family B of sequences {{ar }} as deÃ¿ned in Ref. [54]. For such sequences, the ar satisfy

a di erence equation of order m of the form

m

k

ar = pk (r) ar : (50)

k=1

90 H.H.H. Homeier / Journal of Computational and Applied Mathematics 122 (2000) 81â€“147

The pk (r) satisfy the asymptotic relation

âˆž

pkâ€˜

ik

pk (r) âˆ¼ r for r â†’ âˆž: (51)

râ€˜

â€˜=0

The ik are integers satisfying ik 6k for k = 1; : : : ; m. This family of sequences is very large. But still,

Levin and Sidi could prove [54, Theorem 2] that under mild additional assumptions, the remainders

for such sequences satisfy

m âˆž

Ã¿kâ€˜

jk kâˆ’1

sr âˆ’ s âˆ¼ r( ar ) for r â†’ âˆž: (52)

râ€˜

k=1 â€˜=0

The jk are integers satisfying jk 6k for k = 1; : : : ; m. A corresponding result for m = 1 was proven

by Sidi [71, Theorem 6:1].

System (49) now is obtained by truncation of the expansions at â€˜ = nn , evaluation at r = l , and

some further obvious substitutions.

The introduction of suitable l was shown to improve the accuracy and stability in di cult situ-

ations considerably [77].

3.3. Shanks transformation and epsilon algorithm

An important special case of the E algorithm is the choice gj (n) = sn+jâˆ’1 leading to the Shanks

transformation [70]

(k)

En [{{sn }}; {{ sn+jâˆ’1 }}]

ek (sn ) = (k) : (53)

En [{{1}}; {{ sn+jâˆ’1 }}]

Instead of using one of the recursive schemes for the E algorithms, the Shanks transformation may

be implemented using the epsilon algorithm [104] that is deÃ¿ned by the recursive scheme

(n) (n)

= 0; = sn ;

âˆ’1 0

(n) (n+1) (n+1) (n)

= + 1=[ âˆ’ k ]: (54)

k+1 kâˆ’1 k

The relations

(n) (n)

= ek (sn ); = 1=ek ( sn ) (55)

2k 2k+1

(n)

hold and show that the elements 2k+1 are only auxiliary quantities.

The kernel of the Shanks transformation ek is given by sequences of the form

kâˆ’1

sn = s + cj sn+j : (56)

j=0

See also [14, Theorem 2:18].

Additionally, one can use the Shanks transformation â€“ and hence the epsilon algorithm â€“ to

compute the upper-half of the PadÃƒ table according to [70,104]

e

ek (fn (z)) = [n + k=k]f (z) (kÂ¿0; nÂ¿0); (57)

H.H.H. Homeier / Journal of Computational and Applied Mathematics 122 (2000) 81â€“147 91

where

n

cj z j

fn (z) = (58)

j=0

are the partial sums of a power series of a function f(z). PadÃƒ approximants of f(z) are rational

e

functions in z given as ratio of two polynomials pâ€˜ âˆˆ P and qm âˆˆ P(m) according to

(â€˜)

[â€˜=m]f (z) = pâ€˜ (z)=qm (z); (59)

where the Taylor series of f and [â€˜=m]f are identical to the highest possible power of z, i.e.,

f(z) âˆ’ pâ€˜ (z)=qm (z) = O(z â€˜+m+1 ): (60)

Methods for the extrapolation of power series will be treated later.

3.4. Aitken process

(n) 2

The special case = e1 (sn ) is identical to the famous method of Aitken [2]

2

(sn+1 âˆ’ sn )2

(1)

sn = sn âˆ’ (61)

sn+2 âˆ’ 2sn+1 + sn

with kernel

sn = s + c (sn+1 âˆ’ sn ); n âˆˆ N0 : (62)

2

Iteration of the method yields the iterated Aitken process [14,84,102]

(0)

An = sn ;

(k)

(An+1 âˆ’ An )2

(k)

(k+1) (k)

An = An

âˆ’ (k) : (63)

(k) (k)

An+2 âˆ’ 2An+1 + An

The iterated Aitken process and the epsilon algorithm accelerate linear convergence and can some-

times be applied successfully for the summation of alternating divergent series.

3.5. Overholt process

The Overholt process is deÃ¿ned by the recursive scheme [64]

Vn(0) ({{sn }}) = sn ;

(kâˆ’1)

( sn+kâˆ’1 )k Vn+1 ({{sn }}) âˆ’ ( sn+k )k Vn(kâˆ’1) ({{sn }})

Vn(k) ({{sn }})

= (64)

( sn+kâˆ’1 )k âˆ’ ( sn+k )k

for k âˆˆ N and n âˆˆ N0 . It is important for the convergence acceleration of Ã¿xed point iterations.

4. Levin-type sequence transformations

4.1. DeÃ¿nitions for Levin-type transformations

(k)

A set (k) = { n; j âˆˆ K | n âˆˆ N0 ; 06j6k} is called a coe cient set of order k with k âˆˆ N if

(k)

= { (k) | k âˆˆ N} is called coe cient set. Two coe cient sets

n; k = 0 for all n âˆˆ N0 . Also,

92 H.H.H. Homeier / Journal of Computational and Applied Mathematics 122 (2000) 81â€“147

(k)

and Ë† = {{ Ë†n; j }} are called equivalent, if for all n and k, there is a constant cn = 0

(k) (k)

= {{ n; j }}

(k)

such that Ë†n; j = cn n; j for all j with 06j6k.

(k) (k)

(k)

For each coe cient set (k) = { n; j |n âˆˆ N0 ; 06j6k} of order k, one may deÃ¿ne a Levin-type

sequence transformation of order k by

(k)

] : SK Ã— Y(k) â†’ SK

T[

(k)

: ({{sn }}; {{!n }}) â†’ {{sn }} = T[ ]({{sn }}; {{!n }}) (65)

with

(k)

k

j=0 n; j sn+j =!n+j

(k)

sn = Tn ({{sn }}; {{!n }}) = (66)

(k)

k

j=0 n; j =!n+j

and

ï£± ï£¼

ï£² ï£½

k

(k)

Y(k) = {{!n }} âˆˆ OK : n; j =!n+j = 0 for all n âˆˆ N0 : (67)

ï£³ ï£¾

j=0

We call T[ ] = {T[ (k) ]| k âˆˆ N} the Levin-type sequence transformation corresponding to the

coe cient set = { (k) | k âˆˆ N}. We write T(k) and T instead of T[ (k) ] and T[ ], respectively,

and Ë† are

(k)

whenever the coe cients n; j are clear from the context. Also, if two coe cient sets

equivalent, they give rise to the same sequence transformation, i.e., T[ ] = T[ Ë† ], since

Ë†(k) sn+j =!n+j (k)

k k

j=0 n; j sn+j =!n+j (k)

for Ë†n; j = cn

n; j

j=0 (k) (k)

= (68)

n

(k) (k)

k

Ë† =!n+j j=0 n; j =!n+j

k

n; j

j=0

(k)

with arbitrary cn = 0.

(k)

The number Tn are often arranged in a two-dimensional table

(0) (1) (2)

T0 T0 T0 Â·Â·Â·

(0) (1) (2)

T1 T1 T1 Â·Â·Â·

(69)

(0) (1) (2)

T2 T2 T2 Â·Â·Â·

. . . ..

. . . .

. . .

that is called the T table. The transformations T(k) thus correspond to columns, i.e., to following

vertical paths in the table. The numerators and denominators such that Tn = Nn(k) =Dn also are

(k) (k)

often arranged in analogous N and D tables.

Note that for Ã¿xed N , one may also deÃ¿ne a transformation

(k)

TN : {{sn+N }} â†’ {{TN }}âˆž : (70)

k=0

This corresponds to horizontal paths in the T table. These are sometimes called diagonals, because

rearranging the table in such a way that elements with constant values of n + k are members of the

(k)

same row, TN for Ã¿xed N correspond to diagonals of the rearranged table.

For a given coe cient set deÃ¿ne the moduli by

(k)

(k)

= max {| n; j |} (71)

n

06j6k

H.H.H. Homeier / Journal of Computational and Applied Mathematics 122 (2000) 81â€“147 93

and the characteristic polynomials by

k

(k) j

(k) (k) (k)

âˆˆP : n (z) = n; j z (72)

n

j=0

for n âˆˆ N0 and k âˆˆ N.

(k)

Then, T[ ] is said to be in normalized form if n = 1 for all k âˆˆ N and n âˆˆ N0 . Is is said to be

in subnormalized form if for all k âˆˆ N there is a constant Ëœ (k) such that n 6 Ëœ (k) for all n âˆˆ N0 .

(k)

Any Levin-type sequence transformation T[ ] can rewritten in normalized form. To see this, use

(k) (k)

cn = 1= (73)

n

in Eq. (68). Similarly, each Levin-type sequence transformation can be rewritten in (many di erent)

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