<s>
In	O
mathematics	O
,	O
series	B-Algorithm
acceleration	I-Algorithm
is	O
one	O
of	O
a	O
collection	O
of	O
sequence	O
transformations	O
for	O
improving	O
the	O
rate	B-Architecture
of	I-Architecture
convergence	I-Architecture
of	O
a	O
series	O
.	O
</s>
<s>
Techniques	O
for	O
series	B-Algorithm
acceleration	I-Algorithm
are	O
often	O
applied	O
in	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
where	O
they	O
are	O
used	O
to	O
improve	O
the	O
speed	O
of	O
numerical	B-Algorithm
integration	I-Algorithm
.	O
</s>
<s>
Series	B-Algorithm
acceleration	I-Algorithm
techniques	O
may	O
also	O
be	O
used	O
,	O
for	O
example	O
,	O
to	O
obtain	O
a	O
variety	O
of	O
identities	O
on	O
special	O
functions	O
.	O
</s>
<s>
If	O
the	O
original	O
sequence	O
is	O
divergent	B-Algorithm
,	O
the	O
sequence	O
transformation	O
acts	O
as	O
an	O
extrapolation	O
method	O
to	O
the	O
antilimit	B-Algorithm
.	O
</s>
<s>
The	O
mappings	O
from	O
the	O
original	O
to	O
the	O
transformed	O
series	O
may	O
be	O
linear	B-Architecture
(	O
as	O
defined	O
in	O
the	O
article	O
sequence	O
transformations	O
)	O
,	O
or	O
non-linear	O
.	O
</s>
<s>
In	O
general	O
,	O
the	O
non-linear	O
sequence	O
transformations	O
tend	O
to	O
be	O
more	O
powerful	O
.	O
</s>
<s>
Two	O
classical	O
techniques	O
for	O
series	B-Algorithm
acceleration	I-Algorithm
are	O
Euler	O
's	O
transformation	O
of	O
series	O
and	O
Kummer	B-Algorithm
's	I-Algorithm
transformation	I-Algorithm
of	I-Algorithm
series	I-Algorithm
.	O
</s>
<s>
A	O
variety	O
of	O
much	O
more	O
rapidly	O
convergent	O
and	O
special-case	O
tools	O
have	O
been	O
developed	O
in	O
the	O
20th	O
century	O
,	O
including	O
Richardson	B-Algorithm
extrapolation	I-Algorithm
,	O
introduced	O
by	O
Lewis	O
Fry	O
Richardson	O
in	O
the	O
early	O
20th	O
century	O
but	O
also	O
known	O
and	O
used	O
by	O
Katahiro	O
Takebe	O
in	O
1722	O
;	O
the	O
Aitken	B-Algorithm
delta-squared	I-Algorithm
process	I-Algorithm
,	O
introduced	O
by	O
Alexander	O
Aitken	O
in	O
1926	O
but	O
also	O
known	O
and	O
used	O
by	O
Takakazu	O
Seki	O
in	O
the	O
18th	O
century	O
;	O
the	O
given	O
by	O
Peter	O
Wynn	O
in	O
1956	O
;	O
the	O
Levin	O
u-transform	O
;	O
and	O
the	O
Wilf-Zeilberger-Ekhad	O
method	O
or	O
WZ	O
method	O
.	O
</s>
<s>
For	O
alternating	O
series	O
,	O
several	O
powerful	O
techniques	O
,	O
offering	O
convergence	B-Architecture
rates	I-Architecture
from	O
all	O
the	O
way	O
to	O
for	O
a	O
summation	O
of	O
terms	O
,	O
are	O
described	O
by	O
Cohen	O
et	O
al	O
.	O
</s>
<s>
A	O
basic	O
example	O
of	O
a	O
linear	B-Architecture
sequence	O
transformation	O
,	O
offering	O
improved	O
convergence	O
,	O
is	O
Euler	O
's	O
transform	O
.	O
</s>
<s>
If	O
the	O
original	O
series	O
,	O
on	O
the	O
left	O
hand	O
side	O
,	O
is	O
only	O
slowly	O
converging	O
,	O
the	O
forward	B-Algorithm
differences	I-Algorithm
will	O
tend	O
to	O
become	O
small	O
quite	O
rapidly	O
;	O
the	O
additional	O
power	O
of	O
two	O
further	O
improves	O
the	O
rate	O
at	O
which	O
the	O
right	O
hand	O
side	O
converges	O
.	O
</s>
<s>
A	O
particularly	O
efficient	O
numerical	O
implementation	O
of	O
the	O
Euler	O
transform	O
is	O
the	O
van	B-Algorithm
Wijngaarden	I-Algorithm
transformation	I-Algorithm
.	O
</s>
<s>
The	O
function	O
f(z )	O
can	O
have	O
singularities	O
in	O
the	O
complex	O
plane	O
(	O
branch	O
point	O
singularities	O
,	O
poles	O
or	O
essential	O
singularities	O
)	O
,	O
which	O
limit	B-Algorithm
the	O
radius	O
of	O
convergence	O
of	O
the	O
series	O
.	O
</s>
<s>
The	O
conformal	O
transform	O
needs	O
to	O
be	O
chosen	O
such	O
that	O
,	O
and	O
one	O
usually	O
chooses	O
a	O
function	O
that	O
has	O
a	O
finite	O
derivative	B-Algorithm
at	O
w	O
=	O
0	O
.	O
</s>
<s>
Examples	O
of	O
such	O
nonlinear	O
sequence	O
transformations	O
are	O
Padé	O
approximants	O
,	O
the	O
Shanks	B-Algorithm
transformation	I-Algorithm
,	O
and	O
Levin-type	O
sequence	O
transformations	O
.	O
</s>
<s>
Especially	O
nonlinear	O
sequence	O
transformations	O
often	O
provide	O
powerful	O
numerical	B-General_Concept
methods	I-General_Concept
for	O
the	O
summation	O
of	O
divergent	B-Algorithm
series	I-Algorithm
or	O
asymptotic	O
series	O
that	O
arise	O
for	O
instance	O
in	O
perturbation	O
theory	O
,	O
and	O
may	O
be	O
used	O
as	O
highly	O
effective	O
extrapolation	O
methods	O
.	O
</s>
<s>
This	O
transformation	O
is	O
commonly	O
used	O
to	O
improve	O
the	O
rate	B-Architecture
of	I-Architecture
convergence	I-Architecture
of	O
a	O
slowly	O
converging	O
sequence	O
;	O
heuristically	O
,	O
it	O
eliminates	O
the	O
largest	O
part	O
of	O
the	O
absolute	B-Algorithm
error	I-Algorithm
.	O
</s>
