<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
Aitken	B-Algorithm
's	I-Algorithm
delta-squared	I-Algorithm
process	I-Algorithm
or	O
Aitken	O
extrapolation	O
is	O
a	O
series	B-Algorithm
acceleration	I-Algorithm
method	O
,	O
used	O
for	O
accelerating	O
the	O
rate	B-Architecture
of	I-Architecture
convergence	I-Architecture
of	O
a	O
sequence	O
.	O
</s>
<s>
From	O
a	O
practical	O
point	O
of	O
view	O
,	O
one	O
does	O
in	O
general	O
rather	O
consider	O
only	O
the	O
first	O
few	O
terms	O
of	O
the	O
sequence	O
,	O
which	O
usually	O
provide	O
the	O
needed	O
precision	B-Architecture
.	O
</s>
<s>
Moreover	O
,	O
when	O
numerically	B-General_Concept
computing	O
the	O
sequence	O
,	O
one	O
has	O
to	O
take	O
care	O
to	O
stop	O
the	O
computation	O
when	O
rounding	B-Algorithm
errors	I-Algorithm
in	O
the	O
denominator	O
become	O
too	O
large	O
,	O
where	O
the	O
Δ²	O
operation	O
may	O
cancel	O
too	O
many	O
significant	B-Architecture
digits	I-Architecture
.	O
</s>
<s>
(	O
It	O
would	O
be	O
better	O
for	O
numerical	B-General_Concept
calculation	I-General_Concept
to	O
use	O
rather	O
than	O
.	O
)	O
</s>
<s>
Aitken	B-Algorithm
's	I-Algorithm
delta-squared	I-Algorithm
process	I-Algorithm
is	O
a	O
method	O
of	O
acceleration	B-Algorithm
of	I-Algorithm
convergence	I-Algorithm
,	O
and	O
a	O
particular	O
case	O
of	O
a	O
nonlinear	O
sequence	O
transformation	O
.	O
</s>
<s>
Although	O
the	O
new	O
process	O
does	O
not	O
in	O
general	O
converge	O
quadratically	O
,	O
it	O
can	O
be	O
shown	O
that	O
for	O
a	O
fixed	O
point	O
process	O
,	O
that	O
is	O
,	O
for	O
an	O
iterated	B-Algorithm
function	I-Algorithm
sequence	O
for	O
some	O
function	O
converging	O
to	O
a	O
fixed	O
point	O
,	O
the	O
convergence	O
is	O
quadratic	O
.	O
</s>
<s>
One	O
can	O
also	O
show	O
that	O
if	O
goes	O
to	O
its	O
limit	O
at	O
a	O
rate	O
strictly	O
greater	O
than	O
1	O
,	O
does	O
not	O
have	O
a	O
better	O
rate	B-Architecture
of	I-Architecture
convergence	I-Architecture
.	O
</s>
<s>
100	O
correct	O
decimal	B-Architecture
places	I-Architecture
after	O
5	O
resp	O
.	O
</s>
<s>
Care	O
must	O
be	O
taken	O
,	O
however	O
,	O
to	O
avoid	O
introducing	O
errors	O
due	O
to	O
insufficient	O
precision	B-Architecture
when	O
calculating	O
the	O
differences	O
in	O
the	O
numerator	O
and	O
denominator	O
of	O
the	O
expression	O
.	O
</s>
<s>
It	O
is	O
worth	O
noting	O
here	O
that	O
Aitken	B-Algorithm
's	I-Algorithm
method	I-Algorithm
does	O
not	O
save	O
two	O
iteration	O
steps	O
;	O
computation	O
of	O
the	O
first	O
three	O
Ax	O
values	O
required	O
the	O
first	O
five	O
x	O
values	O
.	O
</s>
<s>
In	O
this	O
example	O
,	O
Aitken	B-Algorithm
's	I-Algorithm
method	I-Algorithm
is	O
applied	O
to	O
a	O
sublinearly	O
converging	O
series	O
,	O
accelerating	O
convergence	O
considerably	O
.	O
</s>
<s>
For	O
instance	O
,	O
if	O
the	O
sequence	O
is	O
given	O
by	O
with	O
starting	O
point	O
then	O
the	O
function	O
will	O
be	O
which	O
has	O
as	O
a	O
fixed	O
point	O
(	O
see	O
Methods	B-Library
of	I-Library
computing	I-Library
square	I-Library
roots	I-Library
)	O
;	O
it	O
is	O
this	O
fixed	O
point	O
whose	O
value	O
will	O
be	O
approximated	O
.	O
</s>
