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Dnj) {(d=d )g(t)}. Consequently, from (2.16) we have
(

d a(t) a(t)
Ë™ a(t)â€™(t)
Ë™
Ë™ ( j)
M n = Dnj)
(
= Dnj)
(
âˆ’ : (2.23)
[â€™(t)]2
d â€™(t) â€™(t)

Next, substituting (2.23) in (2.22), and using the fact that Dnj) is a linear operator, we obtain
(

Ë™( j)
An = Y1 + Y2 + Y3 ; (2.24)

where
n
Dnj) {a(t)=â€™(t)}
(
Ë™ ( j)
Y1 = = ni a(tj+i );
Ë™
Nn( j) i=0

Ë™ ( j) ( Ë™ ( j) n
N n Dnj) {a(t)=â€™(t)} Nn ( j)
Y2 = âˆ’ = âˆ’ ( j) ni a(tj+i );
( j) 2
[Nn ] Nn i=0

n
Dnj) {a(t)â€™(t)=[â€™(t)]2 }
(
Ë™ ( j)
Y3 = âˆ’ =âˆ’ ni a(tj+i ) (2.25)
Nn( j) i=0

( j) ( j)
with = ni â€™(tj+i )=â€™(tj+i ).
Ë™ Here we have used the fact that
ni

n
Dnj) {h(t)=â€™(t)}
(
( j)
= ni h(tj+i ) for any h(t): (2.26)
Dnj) {1=â€™(t)}
(
i=0

Recalling (2.20), we identify

Ë™ ( j)
Nn ( j) ( j)
Ë™( j) = âˆ’ ( j) âˆ’ ni ; i = 0; 1; : : : ; n: (2.27)
ni
ni
Nn
Therefore,

Ë™ ( j) Ë™ ( j) â€™(tj+i )
n n n n
Nn Nn Ë™
( j) ( j) ( j) ( j) ( j)
( j)
= | ni | + + = | ni | + | ni | + : (2.28)
ni ni
n
Nn( j) ( j)
â€™(tj+i )
Nn
i=0 i=0 i=0 i=0
262 A. Sidi / Journal of Computational and Applied Mathematics 122 (2000) 251â€“273

Now even though the Ã¿rst summation is simply nj) , and hence can be computed very inexpensively,
(

Ë™ ( j)
the second sum cannot, as its general term depends also on N n =Nn( j) ; hence on j and n. We can,
( j)
however, compute, again very inexpensively, an upper bound Ëœ n on nj) , deÃ¿ned by
(

( j)
Ë™ n
|N n |
Ëœ ( j) ( j)
( j) ( j) ( j) ( j)
= + ( j) + where â‰¡ | ni | (2.29)
n n n n n
|Nn | i=0

which is obtained by manipulating the second summation in (2.28) appropriately. This can be
achieved by Ã¿rst realizing that
|Dnj) {v(t)}|
(
( j)
= ; (2.30)
n
|Nn( j) |
where v(t) is arbitrarily deÃ¿ned for all t except for t0 ; t1 ; : : : ; for which it is deÃ¿ned by
v(tl ) = (âˆ’1)l |â€™(tl )|=|â€™(tl )|2 ;
Ë™ l = 0; 1; : : : (2.31)
and then by applying Lemma 2.1.

Ë™( j)
2.2.3. The (d=d )W-algorithm for An
Ë™( j)
Combining all of the developments above, we can now extend the W-algorithm to compute An
( j)
and Ëœ n . We shall denote the resulting algorithm the (d=d )W-algorithm. Here are the steps of this
algorithm.

1. For j = 0; 1; : : : ; set
a(tj ) 1
M0( j) = N0( j) = H0( j) = (âˆ’1)j |N0( j) |;
; ; and
â€™(tj ) â€™(tj )

Ë™ ( j) a(tj ) âˆ’ a(tj )â€™(tj ) ;
Ë™ Ë™ â€™(tj )
Ë™ Ëœ ( j)
Ë™ ( j) Ë™ ( j)
H 0 = (âˆ’1)j |N 0 |:
M0 = N0 = âˆ’ ; (2.32)
[â€™(tj )]2 [â€™(tj )]2
â€™(tj )
Ëœ ( j)
Ë™ ( j) Ë™ ( j)
2. For j =0; 1; : : : ; and n=1; 2; : : : ; compute Mn( j) ; Nn( j) ; Hn( j) ; M n ; N n , and H n recursively from
( j+1) ( j)
Qnâˆ’1 âˆ’ Qnâˆ’1
Qnj)
(
= : (2.33)
tj+n âˆ’ tj
3. For all j and n set
Mn( j) |Hn( j) |
A( j) ( j)
= ( j) ; = ( j) ; and
n n
Nn |Nn |
ï£« ï£¶
Ëœ ( j)
Ë™ ( j) Ë™ ( j) Ë™ ( j)
Mn ( j) N n |H n | ï£­ |N n | ï£¸
( j)
( j)
Ëœn =
Ë™ ( j)
An = ( j) âˆ’ An ( j) ; + 1 + ( j) n: (2.34)
|Nn( j) |
Nn Nn |Nn |
It is interesting to note that we need six tables of the form (1.11) in order to carry out the
(d=d )W-algorithm. This is twice the number of tables needed to carry out the W-algorithm. Note
( j)
Ë™( j)
also that no tables need to be saved for A( j) ; nj) ; An ; and Ëœ n . This seems to be the situation for
(
n
all extrapolation methods.
A. Sidi / Journal of Computational and Applied Mathematics 122 (2000) 251â€“273 263

3. Column convergence for (d=d )GREP (1)
( j) âˆž
Ë™
In this section we shall give a detailed analysis of the column sequences {An }j=0 with n Ã¿xed
for the case in which the tl are picked such that
tm+1
t0 Â¿ t1 Â¿ Â· Â· Â· Â¿ 0 and lim =! for some ! âˆˆ (0; 1): (3.1)
mâ†’âˆž tm

We also assume that
â€™(tm+1 )
lim =! for some (complex) = 0; âˆ’1; âˆ’2; : : : : (3.2)
mâ†’âˆž â€™(tm )

Recalling from DeÃ¿nition 2.1 that Ã¿(y) â‰¡ B(t) âˆ¼ âˆž Ã¿i t i as t â†’ 0+; we already have the following
i=0
optimal convergence and stability results for An and nj) , see Theorems 2:1 and 2:2 in [10].
( j) (

Theorem 3.1. Under the conditions given in (3:1) and (3:2); we have
n
cn+ +1 âˆ’ ci
Ã¿n+ â€™(tj )tjn+
A( j) âˆ’Aâˆ¼ as j â†’ âˆž; (3.3)
n
1 âˆ’ ci
i=1

where Ã¿n+ is the Ã¿rst nonzero Ã¿i with iÂ¿n; and
n n n
z âˆ’ ci
( j) i
Ëœni z i ;
lim ni z = â‰¡ Un (z) â‰¡ (3.4)
1 âˆ’ ci
jâ†’âˆž
i=0 i=1 i=0

so that for each Ã¿xed n
n
1 + |ci |
( j) ( j)
lim = hence sup Â¡ âˆž: (3.5)
n n
|1 âˆ’ ci |
jâ†’âˆž j
i=1

Here
+kâˆ’1
ck = ! ; k = 1; 2; : : : : (3.6)

We shall see below that what we need for the analysis of (d=d )GREP(1) are the asymptotic
behaviors of ( j) and Ë™( j) . Now that we know the behavior of ( j) as j â†’ âˆž from (3.4), we turn to
ni ni
ni
( j)
n n
cnij) i
(
Tn( j) (z)
( j) i
Tn( j) (z)
ni z = ( j) with = z; (3.7)
â€™(tj+i )
Tn (1)
i=0 i=0

which follows from the fact that ( j) = [cnij) =â€™(tj+i )]=Dnj) {1=â€™(t)}. Of course, Tn( j) (1) = Dnj) {1=â€™(t)}.
( ( (
ni
Ë™ ( j)
Di erentiating (3.7) with respect to , and denoting T n (z) = (d=d )Tn( j) (z), we obtain
Ë™ ( j) Ë™ ( j)
n
T n (z)Tn( j) (1) âˆ’ Tn( j) (z)T n (1)
Ë™( j) z i = : (3.8)
ni
[Tn( j) (1)]2
i=0
264 A. Sidi / Journal of Computational and Applied Mathematics 122 (2000) 251â€“273

Obviously,

n
â€™(tj+i ) i
Ë™
Ë™ ( j) cnij)
(
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