Shanks transformation

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In numerical analysis, the Shanks transformation is a non-linear series acceleration method to increase the rate of convergence of a sequence. This method is named after Daniel Shanks, who rediscovered this sequence transformation in 1955. It was first derived and published by R. Schmidt in 1941.^[1]

One can calculate only a few terms of a perturbation expansion, usually no more than two or three, and almost never more than seven. The resulting series is often slowly convergent, or even divergent. Yet those few terms contain a remarkable amount of information, which the investigator should do his best to extract.
This viewpoint has been persuasively set forth in a delightful paper by Shanks (1955), who displays a number of amazing examples, including several from fluid mechanics.

Milton D. Van Dyke (1975) Perturbation methods in fluid mechanics, p. 202.

Formulation

For a sequence $\left\{a_{m}\right\}_{{m\in {\mathbb {N}}}}$ the series

$A=\sum _{{m=0}}^{\infty }a_{m}\,$

is to be determined. First, the partial sum $A_{n}$ is defined as:

$A_{n}=\sum _{{m=0}}^{n}a_{m}\,$

and forms a new sequence $\left\{A_{n}\right\}_{{n\in {\mathbb {N}}}}$ . Provided the series converges, $A_{n}$ will approach in the limit to $A$ as $n\to \infty .$ The Shanks transformation $S(A_{n})$ of the sequence $A_{n}$ is defined as^[2]^[3]

$S(A_{n})={\frac {A_{{n+1}}\,A_{{n-1}}\,-\,A_{n}^{2}}{A_{{n+1}}-2A_{n}+A_{{n-1}}}}$

and forms a new sequence. The sequence $S(A_{n})$ often converges more rapidly than the sequence $A_{n}.$ Further speed-up may be obtained by repeated use of the Shanks transformation, by computing $S^{2}(A_{n})=S(S(A_{n})),$ $S^{3}(A_{n})=S(S(S(A_{n}))),$ etc.

Note that the non-linear transformation as used in the Shanks transformation is of similar form as used in Aitken's delta-squared process. But while Aitken's method operates on the coefficients $\left\{a_{m}\right\}$ of the original sequence, the Shanks transformation operates on the partial sums $A_{n}.$

Example

Absolute error as a function of $n$ in the partial sums $A_{n}$ and after applying the Shanks transformation once or several times: $S(A_{n}),$ $S^{2}(A_{n})$ and $S^{3}(A_{n}).$ The series used is $\scriptstyle 4\left(1-{\frac 13}+{\frac 15}-{\frac 17}+{\frac 19}-\cdots \right),$ which has the exact sum $\pi .$

As an example, consider the slowly convergent series^[3]

$4\sum _{{k=0}}^{\infty }(-1)^{k}{\frac {1}{2k+1}}=4\left(1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots \right)$

which has the exact sum π ≈ 3.14159265. The partial sum $A_{6}$ has only one digit accuracy, while six-figure accuracy requires summing about 400,000 terms.

In the table below, the partial sums $A_{n}$ , the Shanks transformation $S(A_{n})$ on them, as well as the repeated Shanks transformations $S^{2}(A_{n})$ and $S^{3}(A_{n})$ are given for $n$ up to 12. The figure to the right shows the absolute error for the partial sums and Shanks transformation results, clearly showing the improved accuracy and convergence rate.

$n$	$A_{n}$	$S(A_{n})$	$S^{2}(A_{n})$	$S^{3}(A_{n})$
0	4.00000000	—	—	—
1	2.66666667	3.16666667	—	—
2	3.46666667	3.13333333	3.14210526	—
3	2.89523810	3.14523810	3.14145022	3.14159936
4	3.33968254	3.13968254	3.14164332	3.14159086
5	2.97604618	3.14271284	3.14157129	3.14159323
6	3.28373848	3.14088134	3.14160284	3.14159244
7	3.01707182	3.14207182	3.14158732	3.14159274
8	3.25236593	3.14125482	3.14159566	3.14159261
9	3.04183962	3.14183962	3.14159086	3.14159267
10	3.23231581	3.14140672	3.14159377	3.14159264
11	3.05840277	3.14173610	3.14159192	3.14159266
12	3.21840277	3.14147969	3.14159314	3.14159265

The Shanks transformation $S(A_{1})$ already has two-digit accuracy, while the original partial sums only establish the same accuracy at $A_{{24}}.$ Remarkably, $S^{3}(A_{3})$ has six digits accuracy, obtained from repeated Shank transformations applied to the first seven terms $A_{0}$ , ... , $A_{6}.$ As said before, $A_{n}$ only obtains 6-digit accuracy after about summing 400,000 terms.

Motivation

The Shanks transformation is motivated by the observation that — for larger $n$ — the partial sum $A_{n}$ quite often behaves approximately as^[2]

$A_{n}=A+\alpha q^{n},\,$

with $|q|<1$ so that the sequence converges transiently to the series result $A$ for $n\to \infty .$ So for $n-1,$ $n$ and $n+1$ the respective partial sums are:

$A_{{n-1}}=A+\alpha q^{{n-1}}\quad ,\qquad A_{n}=A+\alpha q^{n}\qquad {\text{and}}\qquad A_{{n+1}}=A+\alpha q^{{n+1}}.$

These three equations contain three unknowns: $A,$ $\alpha$ and $q.$ Solving for $A$ gives^[2]

$A={\frac {A_{{n+1}}\,A_{{n-1}}\,-\,A_{n}^{2}}{A_{{n+1}}-2A_{n}+A_{{n-1}}}}.$

In the (exceptional) case that the denominator is equal to zero: then $A_{n}=A$ for all $n.$

Generalized Shanks transformation

The generalized kth-order Shanks transformation is given as the ratio of the determinants:^[4]

$S_{k}(A_{n})={\frac {{\begin{vmatrix}A_{{n-k}}&\cdots &A_{{n-1}}&A_{n}\\\Delta A_{{n-k}}&\cdots &\Delta A_{{n-1}}&\Delta A_{{n}}\\\Delta A_{{n-k+1}}&\cdots &\Delta A_{{n}}&\Delta A_{{n+1}}\\\vdots &&\vdots &\vdots \\\Delta A_{{n-1}}&\cdots &\Delta A_{{n+k-2}}&\Delta A_{{n+k-1}}\\\end{vmatrix}}}{{\begin{vmatrix}1&\cdots &1&1\\\Delta A_{{n-k}}&\cdots &\Delta A_{{n-1}}&\Delta A_{{n}}\\\Delta A_{{n-k+1}}&\cdots &\Delta A_{{n}}&\Delta A_{{n+1}}\\\vdots &&\vdots &\vdots \\\Delta A_{{n-1}}&\cdots &\Delta A_{{n+k-2}}&\Delta A_{{n+k-1}}\\\end{vmatrix}}}},$

with $\Delta A_{p}=A_{{p+1}}-A_{p}.$ It is the solution of a model for the convergence behaviour of the partial sums $A_{n}$ with $k$ distinct transients:

$A_{n}=A+\sum _{{p=1}}^{k}\alpha _{p}q_{p}^{n}.$

This model for the convergence behaviour contains $2k+1$ unknowns. By evaluating the above equation at the elements $A_{{n-k}},A_{{n-k+1}},\ldots ,A_{{n+k}}$ and solving for $A,$ the above expression for the kth-order Shanks transformation is obtained. The first-order generalized Shanks transformation is equal to the ordinary Shanks transformation: $S_{1}(A_{n})=S(A_{n}).$

The generalized Shanks transformation is closely related to Padé approximants and Padé tables.^[4]

Notes

↑ Weniger (2003).
↑ 2.0 2.1 2.2 Bender & Orszag (1999), pp. 368–375.
↑ 3.0 3.1 Van Dyke (1975), pp. 202–205.
↑ 4.0 4.1 Bender & Orszag (1999), pp. 389–392.

References

Shanks, D. (1955), "Non-linear transformation of divergent and slowly convergent sequences", Journal of Mathematics and Physics 34: 1–42
Schmidt, R. (1941), "On the numerical solution of linear simultaneous equations by an iterative method", Philosophical Magazine 32: 369–383
Van Dyke, M.D. (1975), Perturbation methods in fluid mechanics (annotated ed.), Parabolic Press, ISBN 0-915760-01-0
Bender, C.M.; Orszag, S.A. (1999), Advanced mathematical methods for scientists and engineers, Springer, ISBN 0-387-98931-5
Weniger, E.J. (2003). "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series". arXiv:math.NA/0306302v1.

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