<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
the	O
Shanks	B-Algorithm
transformation	I-Algorithm
is	O
a	O
non-linear	O
series	B-Algorithm
acceleration	I-Algorithm
method	O
to	O
increase	O
the	O
rate	B-Architecture
of	I-Architecture
convergence	I-Architecture
of	O
a	O
sequence	O
.	O
</s>
<s>
Further	O
speed-up	O
may	O
be	O
obtained	O
by	O
repeated	O
use	O
of	O
the	O
Shanks	B-Algorithm
transformation	I-Algorithm
,	O
by	O
computing	O
etc	O
.	O
</s>
<s>
is	O
more	O
numerically	B-General_Concept
stable	O
than	O
the	O
expression	O
to	O
its	O
left	O
(	O
i.e.	O
</s>
<s>
Both	O
Aitken	B-Algorithm
's	I-Algorithm
method	I-Algorithm
and	O
the	O
Shanks	B-Algorithm
transformation	I-Algorithm
operate	O
on	O
a	O
sequence	O
,	O
but	O
the	O
sequence	O
the	O
Shanks	B-Algorithm
transformation	I-Algorithm
operates	O
on	O
is	O
usually	O
thought	O
of	O
as	O
being	O
a	O
sequence	O
of	O
partial	O
sums	O
,	O
although	O
any	O
sequence	O
may	O
be	O
viewed	O
as	O
a	O
sequence	O
of	O
partial	O
sums	O
.	O
</s>
<s>
In	O
the	O
table	O
below	O
,	O
the	O
partial	O
sums	O
,	O
the	O
Shanks	B-Algorithm
transformation	I-Algorithm
on	O
them	O
,	O
as	O
well	O
as	O
the	O
repeated	O
Shanks	B-Algorithm
transformations	I-Algorithm
and	O
are	O
given	O
for	O
up	O
to	O
12	O
.	O
</s>
<s>
The	O
figure	O
to	O
the	O
right	O
shows	O
the	O
absolute	O
error	O
for	O
the	O
partial	O
sums	O
and	O
Shanks	B-Algorithm
transformation	I-Algorithm
results	O
,	O
clearly	O
showing	O
the	O
improved	O
accuracy	O
and	O
convergence	B-Architecture
rate	I-Architecture
.	O
</s>
<s>
The	O
Shanks	B-Algorithm
transformation	I-Algorithm
already	O
has	O
two-digit	O
accuracy	O
,	O
while	O
the	O
original	O
partial	O
sums	O
only	O
establish	O
the	O
same	O
accuracy	O
at	O
Remarkably	O
,	O
has	O
six	O
digits	O
accuracy	O
,	O
obtained	O
from	O
repeated	O
Shank	O
transformations	O
applied	O
to	O
the	O
first	O
seven	O
terms	O
As	O
said	O
before	O
,	O
only	O
obtains	O
6-digit	O
accuracy	O
after	O
about	O
summing	O
400,000	O
terms	O
.	O
</s>
<s>
The	O
generalized	O
kth-order	O
Shanks	B-Algorithm
transformation	I-Algorithm
is	O
given	O
as	O
the	O
ratio	O
of	O
the	O
determinants	O
:	O
</s>
<s>
By	O
evaluating	O
the	O
above	O
equation	O
at	O
the	O
elements	O
and	O
solving	O
for	O
the	O
above	O
expression	O
for	O
the	O
kth-order	O
Shanks	B-Algorithm
transformation	I-Algorithm
is	O
obtained	O
.	O
</s>
<s>
The	O
first-order	O
generalized	O
Shanks	B-Algorithm
transformation	I-Algorithm
is	O
equal	O
to	O
the	O
ordinary	O
Shanks	B-Algorithm
transformation	I-Algorithm
:	O
</s>
<s>
The	O
generalized	O
Shanks	B-Algorithm
transformation	I-Algorithm
is	O
closely	O
related	O
to	O
Padé	O
approximants	O
and	O
Padé	B-Algorithm
tables	I-Algorithm
.	O
</s>
