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1 (z) =âˆ’ ; n âˆˆ N0 ; (4.19b)

[ n+2 z âˆ’ n+1 ] âˆ’ n+1 [ n+3 z âˆ’ n+2 ]

n+3

(n)

Nk+1 (z)

(n) (n+3)

k+1 (z) = (z) âˆ’ (n) ; k; n âˆˆ N0 ; (4.19c)

k

Dk+1 (z)

(n) (n+2) (n+2) (n)

(z)][ n+3k+2 z âˆ’ n+3k+1 + 2 k (z)]

Nk+1 (z) = [ + (z)]{[ n+3k+3 +

n+3k+3 k k

(n+1) (n) (n+2)

(z)]2 âˆ’ [ n+3k+1 +

+[ n+3k+2 + k (z)][ n+3k+3 + (z)]}; (4.19d)

k k

(n) (n+2) (n)

(z)][ n+3k+2 z âˆ’ n+3k+1 + 2

Dk+1 (z) = [ + k (z)]

n+3k+3 k

(n) (n+1)

2

âˆ’[ + k (z)][ n+3k+3 z âˆ’ n+3k+2 + (z)]: (4.19e)

n+3k+1 k

(n+2) (n+2)

(z) and 2 k (z) are deÃ¿ned by (2:15).

Here, k

A comparison of (4.6) and (4.18) yields

(n) (n)

k (z) = Gk + O(z); z â†’ 0: (4.20)

(n) (n)

Consequently, the z-independent part Gk of k (z) is the prediction for the Ã¿rst coe cient n+3k+1

(n)

not used for the computation of Jk .

If we set z = 0 in the recursive scheme (4:19) and use (4.20), we obtain the following recursive

(n)

scheme for the predictions Gk :

(n)

G0 = 0; n âˆˆ N0 ; (4.21a)

2

n+3 {[ n+2 ] âˆ’2 n+1 n+3 }

(n)

G1 =âˆ’ ; n âˆˆ N0 ; (4.21b)

n+1 n+2

(n)

Fk+1

(n) (n+3)

Gk+1 = Gk âˆ’ (n) ; k; n âˆˆ N0 ; (4.21c)

Hk+1

346 E.J. Weniger / Journal of Computational and Applied Mathematics 122 (2000) 329â€“356

(n) (n+2) (n+1) (n) (n+2)

âˆ’ Gk ]2 âˆ’ 2[

Fk+1 = [ âˆ’ Gk ]{[ âˆ’ Gk ][ âˆ’ Gk ]}; (4.21d)

n+3k+3 n+3k+2 n+3k+1 n+3k+3

(n) (n) (n+1)

Hk+1 = [ âˆ’ Gk ][ âˆ’ Gk ]: (4.21e)

n+3k+1 n+3k+2

(n) (n) (n) (n)

The z-independent parts Ck of Rk (z) and Gk of k (z), respectively, are connected. A com-

parison of (4.13), (4.15), (4.18), and (4.20) yields

(n) (n)

Gk = Ck + n+3k+1 : (4.22)

As in the case of Aitkenâ€™s iterated 2 process or Wynnâ€™s epsilon algorithm, a new approximation

to the limit will be computed after the computation of each new partial sum. Thus, if the index m

of the last partial sum fm (z) is a multiple of 3, m = 3 , we use as approximation to the limit f(z)

the transformation

{f0 (z); f1 (z); : : : ; f3 (z)} â†’ J (0) ; (4.23)

if we have m = 3 + 1, we use the transformation

â†’ J(1) ;

{f1 (z); f2 (z); : : : ; f3 +1 (z)} (4.24)

and if we have m = 3 + 2, we use the transformation

â†’ J(2) ;

{f2 (z); f3 (z); : : : ; f3 +2 (z)} (4.25)

These three relationships can be combined into a single equation, yielding [95, Eq. (10:4-7)]

(mâˆ’3<m=3=)

{fmâˆ’3<m=3= (z); fmâˆ’3<m=3=+1 (z); : : : ; fm (z)} â†’ J<m=3= ; m âˆˆ N0 : (4.26)

5. Applications

In this article, two principally di erent kinds of results were derived. The Ã¿rst group of

results â€” the accuracy-through-order relationships (2.13), (3.9), and (4.13) and the corresponding

(n)

recursive schemes (2:14), (3.9), and (4:14) â€” deÃ¿nes the transformation error terms z n+2k+1 Rk (z),

z n+2k+1 r2k (z), and z n+3k+1 Rk (z). These quantities describe how the rational approximants A(n) , 2k ,

(n) (n) (n)

k

(n)

and Jk di er from the function f(z) which is to be approximated. Obviously, the transformation

error terms must vanish if the transformation process converges.

The second group of results â€” (2.19), (3.14), and (4.18) and the corresponding recursive schemes

(n) (n) (n)

(2:20), (3:15), and (4:19) â€” deÃ¿nes the terms z n+2k+1 k (z), z n+2k+1 â€™2k (z), and z n+3k+1 k (z). These

(n) (n) (n)

quantities describe how the rational approximants Ak , 2k , and Jk di er from the partial sums

fn+2k (z) and fn+3k (z), respectively, from which they were constructed. Hence, the Ã¿rst group of

results essentially describes what is still missing in the transformation process, whereas the second

group describes what was gained by constructing rational expressions from the partial sums.

The recursive schemes (2:14), (3.9), and (4:14) of the Ã¿rst group use as input data the remainder

terms

âˆž

fn (z) âˆ’ f(z)

=âˆ’ n+ +1 z : (5.1)

z n+1 =0

In most practically relevant convergence acceleration and summation problems, only a Ã¿nite number

of series coe cients are known. Consequently, the remainder terms (5.1) are usually not known

E.J. Weniger / Journal of Computational and Applied Mathematics 122 (2000) 329â€“356 347

explicitly, which means that the immediate practical usefulness of the Ã¿rst group of results is quite

limited. Nevertheless, these results are of interest because they can be used to study the convergence

of the sequence transformations of this article for model problems.

As an example, let us consider the following series expansion for the logarithm

âˆž

(âˆ’z)m

ln(1 + z)

= 2 F1 (1; 1; 2; âˆ’z) = ; (5.2)

z m+1

m=0

which converges for all z âˆˆ C with |z| Â¡ 1. The logarithm possesses the integral representation

1

ln(1 + z) dt

= ; (5.3)

z 1 + zt

0

which shows that ln(1+z)=z is a Stieltjes function and that the hypergeometric series on the right-hand

side of (5.2) is the corresponding Stieltjes series (a detailed treatment of Stieltjes functions and

Stieltjes series can for example be found in Section 5 of Baker and Graves-Morris [8]). Consequently,

ln(1 + z)=z possesses the following representation as a partial sum plus an explicit remainder which

is given by a Stieltjes integral (compare for example Eq. (13:1-5) of Weniger [95]):

n

(âˆ’z)m 1

t n+1 dt

ln(1 + z)

+ (âˆ’z)n+1

= ; n âˆˆ N0 : (5.4)

z m+1 1 + zt

0

m=0

For |z| Â¡ 1, the denominator of the remainder integral on the right-hand side can be expanded.

Interchanging summation and integration then yields

âˆž

1

t n+1 dt (âˆ’1)n+m+1 z m

n+1

(âˆ’1) = : (5.5)

1 + zt m=0 n + m + 2

0

Next, we use for 06n66 the negative of these remainder integrals as input data in the recursive

schemes (2:14), (3.9), and (4:14), and do a Taylor expansion of the resulting expressions. Thus, we

obtain according to (2.13), (3.9), and (4.13)

421z 7 796321z 8 810757427z 9

ln(1 + z)

A(0) + O(z 10 );

= + âˆ’ + (5.6a)

3

z 16537500 8682187500 4051687500000

z7 31z 8 113z 9

ln(1 + z)

(0)

+ O(z 10 );

= + âˆ’ + (5.6b)

6

z 9800 77175 120050

z7 19z 8 z9

ln(1 + z)

(0)

+ O(z 10 ):

J2 = + âˆ’ + (5.6c)

z 37800 198450 4725

All calculations were done symbolically, using the exact rational arithmetics of Maple. Consequently,

the results in (5.6) are exact and free of rounding errors.

The leading coe cients of the Taylor expansions of the transformation error terms for A(0) and

3

(0) (0)

J2 are evidently smaller than the corresponding coe cients for 6 . This observation provides

considerable evidence that Aitkenâ€™s iterated 2 process and Brezinskiâ€™s iterated theta algorithm are

in the case of the series (5.2) for ln(1 + z)=z more e ective than Wynnâ€™s epsilon algorithm which

according to (3.2) produces PadÃƒ approximants.

e

This conclusion is also conÃ¿rmed by the following numerical example in Table 1, in which the

convergence of the series (5.2) for ln(1+z)=z is accelerated for z =0:95. The numerical values of the

348 E.J. Weniger / Journal of Computational and Applied Mathematics 122 (2000) 329â€“356

Table 1

âˆž

(âˆ’z)m =

Convergence of the transformation error terms. Transformation of ln(1 + z)=z = m=0

(m + 1) for z = 0:95

âˆž (âˆ’1)n+m z m (nâˆ’2<n=2=) (nâˆ’2<n=2=) (nâˆ’3<n=3=)

z n+1 R<n=2= z n+1 r2<n=2= z n+1 R<n=3=

n (z) (z) (z)

m=0 n+m+2

Eq. (2.13) Eq. (3.9) Eq. (4.13)

0:312654 Â· 100

0 0 0 0

âˆ’0:197206 Â· 100

1 0 0 0

0:143292 Â· 100 0:620539 Â· 10âˆ’2 0:620539 Â· 10âˆ’2

2 0

âˆ’0:112324 Â· 100 âˆ’0:230919 Â· 10âˆ’2 âˆ’0:230919 Â· 10âˆ’2 0:113587 Â· 10âˆ’2

3

0:922904 Â· 10âˆ’1 0:109322 Â· 10âˆ’3 0:156975 Â· 10âˆ’3 âˆ’0:367230 Â· 10âˆ’3

4

âˆ’0:782908 Â· 10âˆ’1 âˆ’0:333267 Â· 10âˆ’4 âˆ’0:466090 Â· 10âˆ’4 0:148577 Â· 10âˆ’3

5

0:679646 Â· 10âˆ’1 0:131240 Â· 10âˆ’5 0:413753 Â· 10âˆ’5 0:137543 Â· 10âˆ’5

6

âˆ’0:600373 Â· 10âˆ’1 âˆ’0:371684 Â· 10âˆ’6 âˆ’0:108095 Â· 10âˆ’5 âˆ’0:392983 Â· 10âˆ’6

7

0:537619 Â· 10âˆ’1 0:111500 Â· 10âˆ’7 0:110743 Â· 10âˆ’6 0:131377 Â· 10âˆ’6

8

âˆ’0:486717 Â· 10âˆ’1 âˆ’0:311899 Â· 10âˆ’8 âˆ’0:266535 Â· 10âˆ’7 0:412451 Â· 10âˆ’9

9

0:444604 Â· 10âˆ’1 0:689220 Â· 10âˆ’10 0:298638 Â· 10âˆ’8 âˆ’0:139178 Â· 10âˆ’9

10

âˆ’0:409189 Â· 10âˆ’1 âˆ’0:199134 Â· 10âˆ’10 âˆ’0:678908 Â· 10âˆ’9 0:475476 Â· 10âˆ’10

11

0:378992 Â· 10âˆ’1 0:282138 Â· 10âˆ’12 0:808737 Â· 10âˆ’10 âˆ’0:316716 Â· 10âˆ’12

12

remainder terms (5.5) were used as input data in the recursive schemes (2:14), (3.9), and (4.13) to

compute numerically the transformation error terms in (2.13), (3.9), and (4.13). The transformation

error terms, which are listed in columns 3â€“5, were chosen in agreement with (2.27), (3.21), and

(4.26), respectively.

The zeros, which are found in columns 3â€“5 of Table 1, occur because Aitkenâ€™s iterated 2 process

and Wynnâ€™s epsilon algorithm can only compute a rational approximant if at least three consecutive

partial sums are available, and because the iteration of Brezinskiâ€™s theta algorithm requires at least

four partial sums.

The result in Table 1 show once more that Aitkenâ€™s iterated 2 process and Brezinskiâ€™s iterated

theta algorithm are in the case of series (5.2) for ln(1 + z)=z apparently more e ective than Wynnâ€™s

epsilon algorithm.

The second group of results of this article â€” (2.19), (3.14), and (4.18) and the corresponding

recursive schemes (2:20), (3:15), and (4:19) â€” can for example be used to demonstrate how rational

approximants work if a divergent power series is to be summed.

Let us therefore assume that the partial sums, which occur in (2.19), (3.14), and (4.18), diverge

if the index becomes large. Then, a summation to a Ã¿nite generalized limit f(z) can only be

(n) (n)

accomplished if z n+2k+1 k (z) and z n+2k+1 â€™2k (z) in (2.19) and (3.14), respectively, converge to the

(n)

negative of fn+2k (z), and if z n+3k+1 k (z) in (4.18) converges to the negative of fn+3k (z).

Table 2 shows that this is indeed the case. We again consider the inÃ¿nite series (5.2) for ln(1+z)=z,

but this time we choose z = 5:0, which is clearly outside the circle of convergence. We use the

numerical values of the partial sums n (âˆ’z)m =(m+1) with 06n610 as input data in the recursive

m=0

schemes (2:20); (3:15), and (4:19) to compute the transformation terms in (2.19), (3.14), and (4.18).

The transformation terms, which are listed in columns 3â€“5 of Table 2, were chosen in agreement

with (2.27), (3.21), and (4.26), respectively. All calculations were done using the oating point

arithmetics of Maple.

E.J. Weniger / Journal of Computational and Applied Mathematics 122 (2000) 329â€“356 349

Table 2

Convergence of transformation terms to the partial sums. Transformation of ln(1 + z)=z =

âˆž

(âˆ’z)m =(m + 1) for z = 5:0

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