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Plotted is the negative decadic logarithm of the relative error.
An : F(n) ({{Sn (z; a)}}; {{(2 + ln(n + a)) Sn (z; a)}}; {{1 + ln(n + a)}});
0
Bn : W0(n) ({{Sn (z; a)}}; {{(2 + ln(n + a)) Sn (z; a)}}; {{1=(1 + ln(n + a))}});
Cn : F(n) ({{Sn (z; a)}}; {{(n + 1) Sn (z; a)}}; {{1 + n + a}});
0
Dn : W0(n) ({{Sn (z; a)}}; {{(n + 1) Sn (z; a)}}; {{1=(1 + n + a)}});
En : F(n) ({{Sn (z; a)}}; {{ Sn (z; a)}}; {{1 + ln(n + a)}});
0
Fn : W0(n) ({{Sn (z; a)}}; {{ Sn (z; a)}}; {{1=(1 + ln(n + a))}});
Gn : F(n) ({{Sn (z; a)}}; {{ Sn (z; a)}}; {{1 + n + a}});
0
(n)
Hn : W0 ({{Sn (z; a)}}; {{ Sn (z; a)}}; {{1=(1 + n + a)}}).


with partial sums
j’1
n
1
j
Sn (z; a) = 1 + z (335)
ln(a + ˜)
j=1 ˜=0

for z = 1:2 and a = 1:01. Since the terms aj satisfy aj+1 =aj = z=ln(a + j), the ratio test reveals
that S(z; a) converges for all z and, hence, represents an analytic function. Nevertheless, only for
j¿ ’ a + exp(|z|), the ratio of the terms becomes less than unity in absolute value. Hence, for larger
z the series converges rather slowly.
It should be noted that for cases Cn and Gn , the F transformation is identical to the Weniger
transformation S, i.e., to the 3 J transformation, and for cases Cn and Hn the W algorithm is identical
to the Levin transformation. In the upper part of the table, we use u-type remainder estimates while
138 H.H.H. Homeier / Journal of Computational and Applied Mathematics 122 (2000) 81“147

Table 4
Acceleration of (’1=10 + 10i; 1; 95=100) with the J transformationa

n An Bn Cn Dn En Fn Gn Hn

10 2:59e ’ 05 3:46e + 01 2:11e ’ 05 4:67e + 01 1:84e ’ 05 4:14e + 01 2:63e ’ 05 3:90e + 01
20 1:72e ’ 05 6:45e + 05 2:53e ’ 05 5:53e + 07 1:38e ’ 04 3:40e + 09 1:94e ’ 05 2:47e + 06
30 2:88e ’ 05 3:52e + 10 8:70e ’ 06 2:31e + 14 8:85e ’ 05 1:22e + 17 2:02e ’ 05 6:03e + 11
40 4:68e ’ 06 1:85e + 15 8:43e ’ 08 1:27e + 20 4:06e ’ 06 2:78e + 23 1:50e ’ 06 9:27e + 16
42 2:59e ’ 06 1:46e + 16 2:61e ’ 08 1:51e + 21 2:01e ’ 06 4:70e + 24 6:64e ’ 07 8:37e + 17
44 1:33e ’ 06 1:10e + 17 7:62e ’ 09 1:76e + 22 1:73e ’ 06 7:85e + 25 2:76e ’ 07 7:24e + 18
46 6:46e ’ 07 8:00e + 17 1:80e ’ 09 2:02e + 23 1:31e ’ 05 1:30e + 27 1:09e ’ 07 6:08e + 19
48 2:97e ’ 07 5:62e + 18 1:07e ’ 08 2:29e + 24 1:52e ’ 04 2:12e + 28 4:16e ’ 08 5:00e + 20
50 1:31e ’ 07 3:86e + 19 1:51e ’ 07 2:56e + 25 1:66e ’ 03 3:43e + 29 1:54e ’ 08 4:05e + 21
(n) (n)
a
An : relative error of 1 J0 (1; {{sn }}; {{(n+1)(sn ’sn’1 )}}), Bn : stability index of 1 J0 (1; {{sn }}; {{(n+1)(sn ’sn’1 )}}),
(n) (n)
Cn : relative error of 2 J0 (1; {{sn }}; {{(n+1)(sn ’sn’1 )}}), Dn : stability index of 2 J0 (1; {{sn }}; {{(n+1)(sn ’sn’1 )}}),
(n) (n)
En : relative error of 3 J0 (1; {{sn }}; {{(n + 1)(sn ’ sn’1 )}}), Fn : stability index of 3 J0 (1; {{sn }}; {{(n + 1)(sn ’
(n)
sn’1 )}}), Gn : relative error of J0 ({{sn }}; {{(n + 1)(sn ’ sn’1 )}}; {1=(n + 1) ’ 1=(n + k + 2)}), Hn : Stability index
(n)
of J0 ({{sn }}; {{(n + 1)(sn ’ sn’1 )}}; {1=(n + 1) ’ 1=(n + k + 2)}).

in the lower part, we use t˜ variants. It is seen that the choices x n = 1 + ln(a + n) for the F
transformation and tn =1=(1+ln(a+n)) for the W algorithm perform for both variants nearly identical
(columns An ; Bn ; En and Fn ) and are superior to the choices x n = 1 + n + a and tn = 1=(1 + n + a),
respectively, that correspond to the Weniger and the Levin transformation as noted above. For the
latter two transformations, the Weniger t˜S transformation is slightly superior the t˜L transformation
for this particular example (columns Gn vs. Hn ) while the situation is reversed for the u-type variants
displayed in colums Cn and Dn .
The next example is taken from [46], namely the “in ated Riemann function”, i.e., the series

qj
( ; 1; q) = ; (336)
(j + 1)
j=0

that is a special case of the Lerch zeta function (s; b; z) (cf. [30, p. 142, Eq. (6:9:7); 20, Section
1:11]). The partial sums are deÿned as
n
qj
sn = : (337)
(j + 1)
j=0

The series converges linearly for 0 ¡ |q| ¡ 1 for any complex . In fact, we have in this case
= limn’∞ (sn+1 ’ s)=(sn ’ s) = q. We choose q = 0:95 and = ’0:1 + 10i. Note that for this value
of , there is a singularity of ( ; 1; q) at q = 1 where the deÿning series diverges since R ( ) ¡ 1.
The results of applying u variants of the p J transformation with p = 1; 2; 3 and of the Levin
transformation to the sequence of partial sums is displayed in Table 4. For each of these four
variants of the J transformation, we give the relative error and the stability index. The true value of
the series (that is used to compute the errors) was computed using a more accurate method described
below. It is seen that the 2 J transformation achieves the best results. The attainable accuracy for this
transformation is limited to about 9 decimal digits by the fact that the stability index displayed in the
column Dn of Table 4 grows relatively fast. Note that for n = 46, the number of digits (as given by
H.H.H. Homeier / Journal of Computational and Applied Mathematics 122 (2000) 81“147 139

Table 5
Acceleration of (’1=10 + 10i; 1; 95=100) with the J transformation ( = 10)a

n An Bn Cn Dn En Fn Gn Hn

10 2:10e ’ 05 2:08e + 01 8:17e ’ 06 3:89e + 01 1:85e ’ 05 5:10e + 01 1:39e ’ 05 2:52e + 01
12 2:49e ’ 06 8:69e + 01 1:43e ’ 07 3:03e + 02 9:47e ’ 06 8:98e + 02 1:29e ’ 06 1:26e + 02
14 1:93e ’ 07 3:11e + 02 5:98e ’ 09 1:46e + 03 8:24e ’ 07 4:24e + 03 6:86e ’ 08 5:08e + 02
16 1:11e ’ 08 9:82e + 02 2:02e ’ 11 6:02e + 03 6:34e ’ 08 2:09e + 04 2:57e ’ 09 1:77e + 03
18 5:33e ’ 10 2:87e + 03 1:57e ’ 12 2:29e + 04 4:08e ’ 09 9:52e + 04 7:81e ’ 11 5:66e + 03
20 2:24e ’ 11 7:96e + 03 4:15e ’ 14 8:26e + 04 2:31e ’ 10 4:12e + 05 2:07e ’ 12 1:73e + 04
22 8:60e ’ 13 2:14e + 04 8:13e ’ 16 2:89e + 05 1:16e ’ 11 1:73e + 06 4:95e ’ 14 5:08e + 04
24 3:07e ’ 14 5:61e + 04 1:67e ’ 17 9:87e + 05 5:17e ’ 13 7:07e + 06 1:10e ’ 15 1:46e + 05
26 1:04e ’ 15 1:45e + 05 3:38e ’ 19 3:31e + 06 1:87e ’ 14 2:84e + 07 2:33e ’ 17 4:14e + 05
28 3:36e ’ 17 3:69e + 05 6:40e ’ 21 1:10e + 07 3:81e ’ 16 1:13e + 08 4:71e ’ 19 1:16e + 06
30 1:05e ’ 18 9:30e + 05 1:15e ’ 22 3:59e + 07 1:91e ’ 17 4:43e + 08 9:19e ’ 21 3:19e + 06
(n) (n)
a
An : relative error of 1 J0 (1; {{s10 n }}; {{(10 n + 1)(s10 n ’ s10 n’1 )}}), Bn : stability index of 1 J0 (1; {{s10 n }}; {{(10 n +
(n)
1)(s10 n ’ s10 n’1 )}}), Cn : relative error of 2 J0 (1; {{s10 n }}; {{(10 n + 1)(s10 n ’ s10 n’1 )}}), Dn : stability index of
(n) (n)
2 J0 (1; {{s10 n }}; {{(10 n + 1)(s10 n ’ s10 n’1 )}}), En : relative error of 3 J0 (1; {{s10 n }}; {{(10 n + 1)(s10 n ’ s10 n’1 )}}),
(n) (n)
Fn : stability index of 3 J0 (1; {{s10 n }}; {{(10 n + 1)(s10 n ’ s10 n’1 )}}), Gn : relative error of J0 ({{s10 n }}; {{(10 n +
(n)
1)(s10 n ’ s10 n’1 )}}; {1=(10 n + 10) ’ 1=(10 n + 10 k + 10)}) Hn : Stability index of Hn : J0 ({{s10 n }}; {{(10 n + 1)(s10 n ’
s10 n’1 )}}; {1=(n + 1) ’ 1=(n + k + 2)}).




the negative decadic logarithm of the relative error) and the decadic logarithm of the stability index
sum up to approximately 32 which corresponds to the maximal number of decimal digits that could
be achieved in the run. Since the stability index increases with n, indicating decreasing stability, it
is clear that for higher values of n the accuracy will be lower.
The magnitude of the stability index is largely controlled by the value of , compare Corollary
24. If one can treat a related sequence with a smaller value of , the stability index will be smaller
and thus, the stability of the extrapolation will be greater.
Such a related sequence is given by putting s˜ = s ˜ for ˜ ∈ N0 , where the sequence ˜ is

a monotonously increasing sequence of nonnegative integers. In the case of linear convergent se-
quences, the choice ˜ = ˜ with ∈ N can be used as in the case of the d(1) transformation. It is
easily seen that the new sequence also converges linearly with = limn’∞ („n+1 ’ s)=(„n ’ s) = q . For
s s
¿ 1, both the e ectiveness and the stability of the various transformations are increased as shown
in Table 5 for the case = 10. Note that this value was chosen to display basic features relevant to
the stability analysis, and is not necessarily the optimal value. As in Table 4, the relative errors and
the stability indices of some variants of the J transformation are displayed. These are nothing but
the p J transformation for p = 1; 2; 3 and the Levin transformation as applied to the sequence {{„n }}
s
with remainder estimates !n = (n + ÿ)(sn ’ sn ’1 ) for ÿ = 1. Since constant factors in the remainder
estimates are irrelevant since the J transformation is invariant under any scaling !n ’ !n for
= 0, the same results would have been obtained for !n = (n + ÿ= )(sn ’ sn ’1 ).
If the Levin transformation is applied to the series with partial sums sn = s n , and if the remainder

estimates !n = (n + ÿ= )(s n ’ s( n)’1 ) are used, then one obtains nothing but the d(1) transformation
with ˜ = ˜ for ∈ N [46,77].
140 H.H.H. Homeier / Journal of Computational and Applied Mathematics 122 (2000) 81“147

Table 6
Stability indices for the 2 J transformation ( = 10)
(1) (2) (3) (4) (5) (6) (7)
n n n n n n n n


2.70 101 7.55 101 2.02 102 5.20 102 1.29 103
20 3.07 9.26
1.19 101 3.81 101 1.16 102 3.36 102 9.36 102 2.51 103
30 3.54
1.33 101 4.49 101 1.44 102 4.42 102 1.30 103 3.71 103
40 3.75
1.34 101 4.54 101 1.46 102 4.51 102 1.34 103 3.82 103
41 3.77
1.35 101 4.59 101 1.49 102 4.60 102 1.37 103 3.93 103
42 3.78
1.36 101 4.64 101 1.51 102 4.69 102 1.40 103 4.05 103
43 3.79
1.37 101 4.68 101 1.53 102 4.77 102 1.43 103
44 3.80
1.38 101 4.73 101 1.55 102 4.85 102
45 3.81
1.39 101 4.77 101 1.57 102
46 3.82
1.39 101 4.81 101
47 3.83
1.40 101
48 3.84
49 3.85

1.59 101 6.32 101 2.52 102 1.00 103 4.00 103 1.59 104
Extr. 4.01
1.59 101 6.32 101 2.52 102 1.00 103 4.00 103 1.59 104
Corollary 24 3.98



It is seen from Table 5 that again the best accuracy is obtained for the 2 J transformation. The
(1)
d transformation is worse, but better than the p J transformations for p = 1 and 3. Note that
the stability indices are now much smaller and do not limit the achievable accuracy for any of the
transformations up to n = 30. The true value of the series was computed numerically by applying the
2 J transformation to the further sequence {{s40n }} and using 64 decimal digits in the calculation. In
this way, a su ciently accurate approximation was obtained that was used to compute the relative
errors in Tables 4 and 5. A comparison value was computed using the representation [20, p. 29,
Eq. (8)]

(log q) j
(1 ’ s)
(log 1=q)s’1 + z ’1
(s; 1; q) = (s ’ j) (338)
z j!
j=0

that holds for |log q| ¡ 2 and s ∈ N. Here, (z) denotes the Riemann zeta function. Both values
agreed to all relevant decimal digits.
In Table 6, we display stability indices corresponding to the acceleration of sn with the 2 J trans-

formation columnwise, as obtainable by using the sequence elements up to s50 = s500 . In the row

(k)
labelled Corollary 24, we display the limits of the n for large n, i.e., the quantities
k
1+q
(k)
lim = ; (339)
n
1’q
n’∞

that are the limits according to Corollary 24. It is seen that the values for ÿnite n are still relatively
far o the limits. In order to check numerically the validity of the corollary, we extrapolated the
(k)
values of all n for ÿxed k with n up to the maximal n for which there is an entry in the
corresponding column of Table 6 using the u variant of the 1 J transformation. The results of the
extrapolation are displayed in the row labelled Extr in Table 6 and coincide nearly perfectly with
the values expected according to Corollary 24.
H.H.H. Homeier / Journal of Computational and Applied Mathematics 122 (2000) 81“147 141

Table 7
Extrapolation of series representation (340) of the Fm (z) function using the 2 J transformation
(z = 8; m = 0)
u t K
n sn !n !n !n

5 ’13:3 0.3120747 0.3143352 0.3132981
6 ’14:7 0.3132882 0.3131147 0.3133070
7 ’13:1 0.3132779 0.3133356 0.3133087
8 11:4 0.3133089 0.3133054 0.3133087
9 ’8:0 0.3133083 0.3133090 0.3133087


As a ÿnal example, we consider the evaluation of the Fm (z) functions that are used in quantum
chemistry calculations via the series representation

(’z) j =j!(2m + 2j + 1)
Fm (z) = (340)
j=0

with partial sums
n
(’z) j =j!(2m + 2j + 1):
sn = (341)
j=0

In this case, for larger z, the convergence is rather slow although the convergence ÿnally is hyper-
linear. As a K variant, one may use
« 
n
!n =  (’z) j =(j + 1)! ’ (1 ’ e’z )=z  :
k
(342)
j=0

since (1 ’ e’z )=z is a Kummer related series. The results for several variants in Table 7 show that
the K variant is superior to u and t variants in this case.

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