Aitken's delta-squared process

In numerical analysis, Aitken's delta-squared process 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 [Alexander Aitken, "On Bernoulli's numerical solution of algebraic equations", "Proceedings of the Royal Society of Edinburgh" (1926) 46 pp. 289-305.] . It is most useful for accelerating the convergence of a sequence that is converging linearly.

Definition

Given a sequence $x=\{(x\_n)\}\_\{ninN\}$, one associates to this sequence the new sequence

:$Ax=\{left(frac\{x\_\{n+2\},x\_n-(x\_\{n+1\})^2\}\{x\_\{n+2\}-2,x\_\{n+1\}+x\_n\}\; ight)\}\_\{ninN\}$

which can also be written as

:$Ax=x-frac\{(Delta\; x)^2\}\{Delta^2\; x\}$ with $Delta\; x=\{(x\_\{n\}-x\_\{n-1\})\}\_\{ninN^*\}.$

(To use this nice operator notation, one has to allow for the indices to start from "n" = 2 on, or apply a translation operator which first shifts the sequence indices by two, or to adopt the convention that "x_n" = 0 for all "n" < 0.)

Obviously, "Ax" is ill-defined if Δ²x contains a zero element, or equivalently, if the sequence of first differences has a repeating term. From a theoretical point of view, assuming that this occurs only for a finite number of indices, one could easily agree to consider the sequence "Ax" restricted to indices "n>n₀" with a sufficiently large "n₀". 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 when rounding errors become too important in the denominator, where the Δ² operation may cancel too many significant digits.

Properties

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

When $x$ converges linearly to $ell$, then $limfrac\{A\_n-ell\}\{x\_n-ell\}=0.$

$A$ is not a linear operator, but a constant term drops out, viz: $A\; [x-ell]\; =\; Ax\; -\; ell$, if $ell$ is a constant. This is clear from the expression of $Ax$ in terms of the finite difference operator $Delta$.

Although the new process does not in general converge quadratically, it can be shown that for a fixed point process, that is, for an iterated function sequence $x\_\{n+1\}=f(x\_n)$ for some function $f$, converging to a fixed point, the convergence is quadratic. In this case, the technique is known as Steffensen's method.

Empirically, the A-operation eliminates the "most important error term". One can check this by considering a sequence of the form $x\_n=ell+a^n+b^n$, where

One can also show that if $x$ goes to its limit $ell$ at a rate strictly greater than 1, $Ax$ has not 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, $Ax$ converges much faster to the limit than $x$ does [work out an example for standard sequences converging to &radic;2 or &pi;] .Usually, it is much cheaper to calculate $Ax$ (involving only calculation of differences, one multiplication and one division) than to calculate much more terms of the sequence $x$. 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.

ee also

* rate of convergence
* limit of a sequence
* fixed point iteration
* Sequence transformations

References

* William H. Press, "et al", "Numerical Recipes in C", (1987) Cambridge University Press, ISBN 0-521-43108-5 "(See section 5.1)"
* Abramowitz and Stegun, "Handbook of Mathematical Functions", section 3.9.7