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( j)
than {B2m }j=0 . Again, we have veriÿed the superiority of our new approach to the direct approach,
at least with respect to column sequences.
We would like to add that the theory of [11] applies to the more general class of functions A(y)
that have asymptotic expansions of the form A(y) ∼ A + ∞ k (y)( qk ki (log y)i ) as y ’ 0+,
k=1 i=0
where qk are arbitrary nonnegative integers.


4. Application to inÿnite series via the d (1) -transformation: the (d=d )d (1) -transformation

4.1. General usage

Let {Sm } be the sequence of partial sums of the inÿnite series vk , namely,
k=1
m
Sm = vk ; m = 1; 2; : : : : (4.1)
k=1

Assume that

’i
vm ∼ ‚i m as m ’ ∞; ‚0 = 0; + 1 = 0; 1; 2; : : : : (4.2)
i=0

As is known, limm’∞ Sm exists and is ÿnite if and only if R + 1 ¡ 0. When R + 1¿0 but
+ 1 = 0; 1; 2; : : : ; {Sm } diverges but has a well deÿned and useful antilimit as has been shown in
Theorem 4.1 of [10]. For all in (4.2) this theorem reads as follows:

Theorem 4.1. With Sm as in (4:1) and (4:2); we have

ÿi m’i
Sm ∼ S + mvm as m ’ ∞; ÿ0 = 0: (4.3)
i=0

Here S = limm’∞ Sm when R + 1 ¡ 0; and S is the antilimit of {Sm } otherwise.

The part of Theorem 4.1 concerning convergent sequences {Sm } is already contained in
Theorem 2 of [6].
268 A. Sidi / Journal of Computational and Applied Mathematics 122 (2000) 251“273


From Theorem 4.1 it is clear that GREP(1) can be applied to the sequence {Sm } by drawing the
analogy a(t) ” Sm ; t ” m’1 ; ™(t) ” mvm , and A ” S, and by picking tl = 1=Rl for some positive
integers Rl ; 16R0 ¡ R1 ¡ R2 ¡ · · · ; and the W-algorithm can be used to implement it. This GREP(1)
is simply the Levin“Sidi d(1) -transformation, and we denote its A( j) by Sn j) .
(
n
As already explained in [10,4], for the type of sequences considered here we should pick the Rl
such that {Rl } increases exponentially to ensure the best stability and convergence properties in the
Sn j) . Exponential increase in the Rl can be achieved by picking them, for example, as in
(


R0 = 1 and Rl+1 = Rl + 1; l = 0; 1; : : : ; for some ¿ 1: (4.4)
(With = 1 we have Rl = l + 1; l = 0; 1; : : : ; for which the d(1) -transformation becomes the Levin
[5] u-transformation.) This gives Rl = O( l ) as l ’ ∞: Needless to say, should not be picked
too far from 1 to avoid too quick a growth in the Rl . We have found that between 1:1 and 1:5 is
su cient for most purposes. Since tl = 1=Rl , (4.4) implies that
tl tl
6tl+1 ¡ ; l = 0; 1; : : : (4.5)
+ tl
as a result of which {tl } satisÿes (3.1) with ! = 1= ∈ (0; 1). Therefore, Theorem 3.1 applies to the
approximations Smj) to S obtained via the d(1) -transformation, as has been shown in [10]. Clearly,
(

= ’ ’ 1 in (3.2) and (3.6) for this case.

If, in addition, vm and S are di erentiable functions of a parameter , S is the limit or antilimit

of {S m }, and
vm = vm [K log m + L + o(1)]
™ as m ’ ∞; for some constants K = 0 and L (4.6)
and the asymptotic expansion in (4.3) can be di erentiated with respect to term by term, then
™( j) ∞
Theorems 3.2“3.4 apply to {S n }j=0 without any modiÿcations. We shall denote this method that
™( j) ™( j) ™
produces the S n the (d=d )d(1) -transformation for short. The rate of convergence of the S n to S is
almost identical to the rate of convergence of the Sn j) to S as we have observed in many numerical
(

examples, and as we have proved in Theorem 3.4 for the column sequences.
To summarize the relevant convergence results for the d(1) - and (d=d )d(1) -transformations as these

are applied to {Sm } and {S m } above, we have from Theorems 3.1 and 3.4
Sn j) ’ S = O(vRj Rj
( ’n+1 ( +1’n) j
) = O( ) as j ’ ∞;

™( j) ™ ’n+1 ( +1’n) j
S n ’ S = O(vRj Rj log Rj ) = O(j ) as j ’ ∞: (4.7)
Of course, these results are not optimal. Optimal results follow from (3.3) and (3.19), and we leave
them to the reader. The results for nj) and nj) that pertain to stability can be obtained from
( (

Theorems 3.1“3.3.
Rj
For the sake of completeness we note that the (d=d )W-algorithm takes tj = 1=Rj ; a(tj ) = k=1 vk ;
Rj
a(tj ) = k=1 vk ; ™(tj ) = Rj vRj , and ™(tj ) = Rj vRj as input for this problem.
™ ™ ™ ™

It is worth mentioning that we can also compute S by applying the d(2) -transformation directly

to {S m }. The d(2) -transformation is a GREP(2) . As we mentioned earlier, this is less e ective than
the application of the (d=d )d(1) -transformation to {Sm }. We shall see this also through numerical
examples in the next section.
A. Sidi / Journal of Computational and Applied Mathematics 122 (2000) 251“273 269


4.2. A special application

We next turn to an interesting application of the (d=d )d(1) -transformation to the summation of a
class of inÿnite series ∞ vk , where vm has the form
k=1 ˜ ˜

’i
vm = [log (m)]vm ;
˜ (m) ∼ im as m ’ ∞; = 0 and = 0; (4.8)
0
i=0

with vm as in (4.2). (When = 0 the d(1) -transformation is very e ective on the series vk .) To
˜
k=1
this end ÿrst let us consider the inÿnite series ∞ uk ( ), where
k=1

um ( ) = vm [ (m)] ; m = 1; 2; : : : : (4.9)
(Here vm and (m) do not depend on ). Now it can be shown that [ (m)] ∼ ∞ i m ’i as i=0
m ’ ∞, where 0 = 0 = 0 and = . Consequently, um ( ) ∼ ∞ ‚i m ’i as m ’ ∞, where i=0
‚0 =‚0 = 0 and = + , so that um ( ) is of the form described in (4.1) for all . That is to say,
the d(1) -transformation can be applied to sum ∞ uk ( ) for any . Next, u m ( ) = um ( ) log (m) ∼

k=1
um ( )[ log m + log 0 + o(1)] as m ’ ∞; cf. (4.6). Therefore, the (d=d )d(1) -transformation can
be used for summing ∞ u k ( ) for any . Finally, um (0) = vm and u m (0) = vm , and hence the
k=1 ™ ™ ˜

(1)
(d=d )d -transformation can be used for summing k=1 vk in particular. This can be done by setting
˜
Rj Rj
tj = 1=Rj ; a(tj ) = k=1 vk ; a(tj ) = k=1 vk ; ™(tj ) = Rj vRj , and ™(tj ) = Rj vRj in the (d=d )W-algorithm.
™ ˜ ™ ˜


5. Numerical examples

In this section we wish to demonstrate numerically the e ectiveness of (d=d )GREP(1) via the
(d=d )d(1) -transformation on some inÿnite series, convergent or divergent. We will do this with two
examples. The ÿrst one of these examples has already been treated in [14] within the framework of
the Richardson extrapolation process.

Example 5.1. Consider the series ∞ k ’ ’1 that converges for R ¿ 0 and deÿnes the Riemann
k=1
zeta function ( + 1). As is known, (z) can be continued analytically to the entire complex plane
except z = 1, where it has a simple pole. As the term vm = m’ ’1 is of the form described in the
previous section, Theorem 4.1 applies to Sm = m k ’ ’1 with S = ( + 1) and = , whether
k=1

limm’∞ Sm exists or not. Furthermore, the asymptotic expansion of S m = m (’log k)k ’ ’1 can be
k=1
obtained by term-by-term di erentiation of the expansion in (4.3), as has already been mentioned in

[14]. This implies that the (d=d )d(1) -transformation can be applied to the computation of S= ( +1),
and Theorems 3.2“3.4 are valid with = . In particular, (4.7) is valid with = ’ ’ 1 there.

We applied the (d=d )d(1) -transformation to this problem to compute S = ( + 1). We picked
the integers Rl as in (4.4) with = 1:2 there. We considered the two cases (i) = 1 and (ii)
™ ™
= ’0:5. Note that in case (i) both limm’∞ Sm and limm’∞ S m exist and are S = (2) and S = (2),

respectively, while in case (ii) these limits do not exist and S = (0:5) and S = (0:5) are the

corresponding antilimits. We also applied the d(2) -transformation directly to {S m } with the same
Rl ™s, the resulting approximations being denoted Bnj) , as in (2.19). The numerical results are shown
(

in Tables 1“3.
270 A. Sidi / Journal of Computational and Applied Mathematics 122 (2000) 251“273

Table 1
Numerical results on Process I for (z) in Example 5.1, where (z) is the Riemann zeta function,

with z=2. The d(1) - and (d=d )d(1) -transformations on {Sm } and {S m } and the d(2) -transformation
™( j+1) ’

on {S m } are implemented with =1:2 in (4.4). Here Pn j) =|Sn j+1) ’S|=|Sn j) ’S|; Qn j) =|S n
( ( ( (

™ ™( j) ™ ™( j)
™ ™
S|=|S n ’ S|, and Zn j) = |Bn j+1) ’ S|=|Bn j) ’ S|, where Sn j) , S n , and Bn j) are the approximations
( ( ( ( (

obtained from the d(1) -, (d=d )d(1) -, and d(2) -transformations, respectively. All six columns are
tending to ’7 = 0:279 : : :

P5 j)
(
Q5 j)
( ( j)
P6 j)
(
Q6 j)
( ( j)
j Z10 Z12

0 1:53D ’ 01 1:62D ’ 01 3:18D ’ 01 1:09D ’ 03 2:25D ’ 02 3:08D ’ 02
2 1:94D ’ 01 2:09D ’ 01 2:01D ’ 01 3:23D ’ 01 3:19D ’ 01 1:06D ’ 01
4 1:97D ’ 01 2:10D ’ 01 1:58D ’ 01 2:30D ’ 01 2:41D ’ 01 1:58D ’ 02
6 2:33D ’ 01 2:46D ’ 01 2:02D ’ 01 2:44D ’ 01 2:56D ’ 01 4:27D ’ 01
8 2:45D ’ 01 2:57D ’ 01 4:95D ’ 01 2:51D ’ 01 2:63D ’ 01 2:93D ’ 01
10 2:50D ’ 01 2:61D ’ 01 3:56D ’ 01 2:56D ’ 01 2:66D ’ 01 2:65D ’ 01
12 2:65D ’ 01 2:75D ’ 01 3:22D ’ 01 2:66D ’ 01 2:75D ’ 01 2:58D ’ 01
14 2:67D ’ 01 2:76D ’ 01 3:07D ’ 01 2:68D ’ 01 2:77D ’ 01 2:60D ’ 01
16 2:70D ’ 01 2:79D ’ 01 3:03D ’ 01 2:71D ’ 01 2:79D ’ 01 2:69D ’ 01
18 2:70D ’ 01 2:79D ’ 01 2:99D ’ 01 2:71D ’ 01 2:79D ’ 01 2:78D ’ 01
20 2:74D ’ 01 2:82D ’ 01 2:97D ’ 01 2:74D ’ 01 2:82D ’ 01 2:85D ’ 01




Table 2
Numerical results on Process II for (z) in Example 5.1, where (z) is the Riemann zeta function,

with z=2. The d(1) - and (d=d )d(1) -transformations on {Sm } and {S m } and the d(2) -transformation
( j)
™ ™
on {S m } are implemented with = 1:2 in (4.4). Here Sn j) , S n , and Bn j) are the approximations
( (

obtained from the d(1) -, (d=d )d(1) -, and d(2) -transformations, respectively. (The inÿnite series
converge.)
(0)
™ ™ ™ ™ ™
(0) (0)
n Rn |SRn ’ S| |Sn ’ S| |S Rn ’ S| |S n ’ S| |Bn ’ S|

0 1 6:45D ’ 01 1:64D + 00 9:38D ’ 01 9:38D ’ 01 9:38D ’ 01
2 3 2:84D ’ 01 1:99D ’ 02 6:42D ’ 01 3:67D ’ 02 4:56D ’ 01
4 5 1:81D ’ 01 3:12D ’ 05 4:91D ’ 01 1:07D ’ 04 3:28D ’ 03
6 9 1:05D ’ 01 7:08D ’ 07 3:42D ’ 01 1:56D ’ 06 6:19D ’ 04
8 14 6:89D ’ 02 8:18D ’ 09 2:53D ’ 01 2:35D ’ 08 3:26D ’ 05
10 21 4:65D ’ 02 3:71D ’ 11 1:89D ’ 01 1:25D ’ 10 8:26D ’ 07
12 32 3:08D ’ 02 6:95D ’ 14 1:38D ’ 01 2:70D ’ 13 5:11D ’ 07
14 47 2:11D ’ 02 2:55D ’ 17 1:02D ’ 01 1:44D ’ 16 4:17D ’ 09
16 69 1:44D ’ 02 8:28D ’ 20 7:54D ’ 02 3:03D ’ 19 1:60D ’ 11
18 100 9:95D ’ 03 1:14D ’ 22 5:58D ’ 02 4:90D ’ 22 4:32D ’ 13
20 146 6:83D ’ 03 5:75D ’ 26 4:09D ’ 02 2:72D ’ 25 2:14D ’ 16
22 212 4:71D ’ 03 1:52D ’ 29 2:99D ’ 02 4:53D ’ 29 4:53D ’ 17
24 307 3:25D ’ 03 2:44D ’ 30 2:19D ’ 02 3:52D ’ 29 1:97D ’ 19




Table 1 shows the validity of the theory for Process I given in Sections 2“ 4 very clearly. The
results of this table that have been computed with = 1 can be understood as follows:
A. Sidi / Journal of Computational and Applied Mathematics 122 (2000) 251“273 271

Table 3
Numerical results on Process II for (z) in Example 5.1, where (z) is the Riemann zeta

function, with z = 0:5. The d(1) - and (d=d )d(1) -transformations on {Sm } and {S m } and the
( j)
™ ™
d(2) -transformation on {S m } are implemented with = 1:2 in (4.4). Here Sn j) , S n , and Bn j) are
( (

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