# Aitken's delta-squared process

# Aitken's delta-squared process

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.^{[1]} Its early form was known to Seki Kōwa (end of 17th century) and was found for rectification of the circle, i.e. the calculation of π. It is most useful for accelerating the convergence of a sequence that is converging linearly.

Definition

`Given a sequence, one associates with this sequence the new sequence`

which can, with improved numerical stability, also be written as

or equivalently as

where

and

`for`

`Obviously,is ill-defined ifcontains a zero element, or equivalently, if the sequence offirst differenceshas a repeating term.`

`From a theoretical point of view, if that occurs only for a finite number of indices, one could easily agree to consider the sequencerestricted to indiceswith a sufficiently large. From a practical point of view, one does in general rather consider only the first few terms of the sequence, which usually provide the needed precision. Moreover, when numerically computing the sequence, one has to take care to stop the computation whenrounding errorsin the denominator become too large, where the Δ² operation may cancel too manysignificant digits. (It would be better for numerical calculation to userather than.)`

Properties

Aitken's delta-squared process is a method of acceleration of convergence, and a particular case of a nonlinear sequence transformation.

`willconverge linearlytoif there exists a number μ ∈ (0, 1) such that`

`Aitken's method will accelerate the sequenceif`

`is not a linear operator, but a constant term drops out, viz:, ifis a constant. This is clear from the expression ofin terms of thefinite differenceoperator.`

`Although the new process does not in general converge quadratically, it can be shown that for afixed pointprocess, that is, for aniterated functionsequencefor some function, converging to a fixed point, the convergence is quadratic. In this case, the technique is known asSteffensen's method.`

`Empirically, the`

*A*-operation eliminates the "most important error term". One can check this by considering a sequence of the form, where: The sequencewill then go to the limit likegoes to zero.`One can also show that ifgoes to its limitat a rate strictly greater than 1,does not have a better rate of convergence. (In practice, one rarely has e.g. quadratic convergence which would mean over 30 resp. 100 correct decimal places after 5 resp. 7 iterations (starting with 1 correct digit); usually no acceleration is needed in that case.)`

`In practice,converges much faster to the limit thandoes, as demonstrated by the example calculations below. Usually, it is much cheaper to calculate(involving only calculation of differences, one multiplication and one division) than to calculate many more terms of the sequence. Care must be taken, however, to avoid introducing errors due to insufficient precision when calculating the`

*differences*in the numerator and denominator of the expression.Example calculations

**Example 1**: The value ofcan be approximated by assuming an initial value forand iterating the following:`Starting with`

n | x = iterated value | Ax |

0 | 1 | 1.4285714 |

1 | 1.5 | 1.4141414 |

2 | 1.4166667 | 1.4142136 |

3 | 1.4142157 | -- |

4 | 1.4142136 | -- |

It is worth noting here that Aitken's method does not save two iteration steps; computation of the first three *Ax* values required the first five *x* values. Also, the second Ax value is decidedly inferior to the 4th x value, mostly due to the fact that Aitken's process assumes linear, rather than quadratic, convergence.

**Example 2**: The value ofmay be calculated as an infinite sum:n | term | x = partial sum | Ax |

0 | 1 | 1 | 0.79166667 |

1 | −0.33333333 | 0.66666667 | 0.78333333 |

2 | 0.2 | 0.86666667 | 0.78630952 |

3 | −0.14285714 | 0.72380952 | 0.78492063 |

4 | 0.11111111 | 0.83492063 | 0.78567821 |

5 | −9.0909091×10^{−2} | 0.74401154 | 0.78522034 |

6 | 7.6923077×10^{−2} | 0.82093462 | 0.78551795 |

7 | -6.6666667×10^{−2} | 0.75426795 | -- |

8 | 5.8823529×10^{−2} | 0.81309148 | -- |

In this example, Aitken's method is applied to a sublinearly converging series, accelerating convergence considerably. It is still sublinear, but much faster than the original convergence: the first Ax value, whose computation required the first three x values, is closer to the limit than the eighth x value.

Example pseudocode for Aitken extrapolation

`The following is an example of using the Aitken extrapolation to help find the limit of the sequencewhen given, which we assume to be the fixed point. For instance, we could havewithwhich has the fixed pointso that(seeMethods of computing square roots).`

`This pseudo code also computes the Aitken approximation to. The Aitken extrapolates will be denoted byaitkenX. We must check if during the computation of the extrapolate the denominator becomes too small, which could happen if we already have a large amount of accuracy, since otherwise a large amount of error could be introduced. We denote this small number byepsilon.`

See also

Rate of convergence

Limit of a sequence

Fixed point iteration

Richardson extrapolation

Sequence transformation

Shanks transformation

Steffensen's method

## References

*Proceedings of the Royal Society of Edinburgh*(1926)

**46**pp. 289–305.