Aitken's delta-squared process
numerical analysis series acceleration method
In numerical analysis, Aitken's delta-squared process or Aitken extrapolation is a series acceleration method used for accelerating the rate of convergence of a sequence. It is named after Alexander Aitken, who introduced this method in 1926 as part of an extension to Bernoulli's method. It is most useful for accelerating the convergence of a sequence that is converging linearly. A precursor form was known to Seki Kōwa (1642 – 1708) and applied to the rectification of the circle, i.e., to the calculation of π.
Definition
Given a sequence
X
=
(
x
n
)
{\displaystyle X={(x_{n})}}
with
n
=
0
,
1
,
2
,
3
,
…
,
{\displaystyle n=0,1,2,3,\ldots ,}
Aitken's delta-squared process associates to this sequence the new sequence
A
[
X
]
=
(
a
n
)
=
(
x
n
x
n
+
2
−
x
n
+
1
2
x
n
+
x
n
+
2
−
2
x
n
+
1
)
,
{\displaystyle A[X]=(a_{n})={\left({\frac {x_{n}\,x_{n+2}-x_{n+1}^{2}}{x_{n}+x_{n+2}-2\,x_{n+1}}}\right)},}
which can also be written as
A
[
X
]
=
(
x
n
−
(
Δ
x
n
)
2
Δ
2
x
n
)
,
{\displaystyle A[X]=\left(x_{n}-{\frac {(\Delta x_{n})^{2}}{\Delta ^{2}x_{n}}}\right),}
with
Δ
x
n
=
x
n
+
1
−
x
n
{\textstyle \Delta x_{n}=x_{n+1}-x_{n}}
and
Δ
2
x
n
=
x
n
−
2
x
n
+
1
+
x
n
+
2
=
Δ
x
n
+
1
−
Δ
x
n
.
{\textstyle \Delta ^{2}x_{n}=x_{n}-2x_{n+1}+x_{n+2}=\Delta x_{n+1}-\Delta x_{n}.}
Both are the same sequence algebraically but the latter has improved numerical stability in computational implementation.
A
[
X
]
{\textstyle A[X]}
is ill-defined if the sequence
Δ
2
[
X
]
=
(
Δ
2
x
n
)
{\textstyle \Delta ^{2}[X]=(\Delta ^{2}x_{n})}
contains a zero element, which occurs if the sequence of forward differences,
Δ
[
X
]
=
(
Δ
x
n
)
,
{\textstyle \Delta [X]=(\Delta x_{n}),}
has any repeated term. From a theoretical point of view, if that occurs only for a finite number of indices, one could apply the Aitken process to only the part of the sequence
X
{\displaystyle X}
with indices
n
>
n
0
{\displaystyle n>n_{0}}
such that
n
0
{\displaystyle n_{0}}
is the last index for which the sequence
Δ
[
X
]
{\textstyle \Delta [X]}
repeats. In practice, the first few terms of the sequence usually provide desired precision; also, when numerically computing the sequence, one has to take care to stop the computation before rounding errors in the denominator become too large, as the
Δ
2
{\textstyle \Delta ^{2}}
sequence transformation may cancel significant digits.
Properties
Aitken's delta-squared process is an acceleration of convergence method and a particular case of a nonlinear sequence transformation.
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