Shanks transformation

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.

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 [math]\displaystyle{ \left\{a_m\right\}_{m\in\mathbb{N}} }[/math] the series

[math]\displaystyle{ A = \sum_{m=0}^\infty a_m\, }[/math]

is to be determined. First, the partial sum [math]\displaystyle{ A_n }[/math] is defined as:

[math]\displaystyle{ A_n = \sum_{m=0}^n a_m\, }[/math]

and forms a new sequence [math]\displaystyle{ \left\{A_n\right\}_{n\in\mathbb{N}} }[/math]. Provided the series converges, [math]\displaystyle{ A_n }[/math] will also approach the limit [math]\displaystyle{ A }[/math] as [math]\displaystyle{ n\to\infty. }[/math] The Shanks transformation [math]\displaystyle{ S(A_n) }[/math] of the sequence [math]\displaystyle{ A_n }[/math] is the new sequence defined by

[math]\displaystyle{ S(A_n) = \frac{A_{n+1}\, A_{n-1}\, -\, A_n^2}{A_{n+1}-2A_n+A_{n-1}} = A_{n+1} - \frac{(A_{n+1}-A_{n})^2}{(A_{n+1}-A_{n})-(A_{n}-A_{n-1})} }[/math]

where this sequence [math]\displaystyle{ S(A_n) }[/math] often converges more rapidly than the sequence [math]\displaystyle{ A_n. }[/math] Further speed-up may be obtained by repeated use of the Shanks transformation, by computing [math]\displaystyle{ S^2(A_n)=S(S(A_n)), }[/math] [math]\displaystyle{ S^3(A_n)=S(S(S(A_n))), }[/math] etc.

Note that the non-linear transformation as used in the Shanks transformation is essentially the same as used in Aitken's delta-squared process so that as with Aitken's method, the right-most expression in [math]\displaystyle{ S(A_n) }[/math]'s definition (i.e. [math]\displaystyle{ S(A_n) = A_{n+1} - \frac{(A_{n+1}-A_{n})^2}{(A_{n+1}-A_{n})-(A_{n}-A_{n-1})} }[/math]) is more numerically stable than the expression to its left (i.e. [math]\displaystyle{ S(A_n) = \frac{A_{n+1}\, A_{n-1}\, -\, A_n^2}{A_{n+1}-2A_n+A_{n-1}} }[/math]). Both Aitken's method and the Shanks transformation operate on a sequence, but the sequence the Shanks transformation operates on is usually thought of as being a sequence of partial sums, although any sequence may be viewed as a sequence of partial sums.

Example

Absolute error as a function of [math]\displaystyle{ n }[/math] in the partial sums [math]\displaystyle{ A_n }[/math] and after applying the Shanks transformation once or several times: [math]\displaystyle{ S(A_n), }[/math] [math]\displaystyle{ S^2(A_n) }[/math] and [math]\displaystyle{ S^3(A_n). }[/math] The series used is [math]\displaystyle{ \scriptstyle 4\left(1-\frac13+\frac15-\frac17+\frac19-\cdots\right), }[/math] which has the exact sum [math]\displaystyle{ \pi. }[/math]

As an example, consider the slowly convergent series

[math]\displaystyle{ 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) }[/math]

which has the exact sum π ≈ 3.14159265. The partial sum [math]\displaystyle{ A_6 }[/math] has only one digit accuracy, while six-figure accuracy requires summing about 400,000 terms.

In the table below, the partial sums [math]\displaystyle{ A_n }[/math], the Shanks transformation [math]\displaystyle{ S(A_n) }[/math] on them, as well as the repeated Shanks transformations [math]\displaystyle{ S^2(A_n) }[/math] and [math]\displaystyle{ S^3(A_n) }[/math] are given for [math]\displaystyle{ n }[/math] 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.

[math]\displaystyle{ n }[/math]	[math]\displaystyle{ A_n }[/math]	[math]\displaystyle{ S(A_n) }[/math]	[math]\displaystyle{ S^2(A_n) }[/math]	[math]\displaystyle{ S^3(A_n) }[/math]
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 [math]\displaystyle{ S(A_1) }[/math] already has two-digit accuracy, while the original partial sums only establish the same accuracy at [math]\displaystyle{ A_{24}. }[/math] Remarkably, [math]\displaystyle{ S^3(A_3) }[/math] has six digits accuracy, obtained from repeated Shank transformations applied to the first seven terms [math]\displaystyle{ A_0, \ldots, A_6. }[/math] As mentioned before, [math]\displaystyle{ A_n }[/math] only obtains 6-digit accuracy after summing about 400,000 terms.

Motivation

The Shanks transformation is motivated by the observation that — for larger [math]\displaystyle{ n }[/math] — the partial sum [math]\displaystyle{ A_n }[/math] quite often behaves approximately as

[math]\displaystyle{ A_n = A + \alpha q^n, \, }[/math]

with [math]\displaystyle{ |q|\lt 1 }[/math] so that the sequence converges transiently to the series result [math]\displaystyle{ A }[/math] for [math]\displaystyle{ n\to\infty. }[/math] So for [math]\displaystyle{ n-1, }[/math] [math]\displaystyle{ n }[/math] and [math]\displaystyle{ n+1 }[/math] the respective partial sums are:

[math]\displaystyle{ 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}. }[/math]

These three equations contain three unknowns: [math]\displaystyle{ A, }[/math] [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ q. }[/math] Solving for [math]\displaystyle{ A }[/math] gives

[math]\displaystyle{ A = \frac{A_{n+1}\, A_{n-1}\, -\, A_n^2}{A_{n+1}-2A_n+A_{n-1}}. }[/math]

In the (exceptional) case that the denominator is equal to zero: then [math]\displaystyle{ A_n=A }[/math] for all [math]\displaystyle{ n. }[/math]

Generalized Shanks transformation

The generalized kth-order Shanks transformation is given as the ratio of the determinants:

[math]\displaystyle{ 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} }, }[/math]

with [math]\displaystyle{ \Delta A_p = A_{p+1} - A_p. }[/math] It is the solution of a model for the convergence behaviour of the partial sums [math]\displaystyle{ A_n }[/math] with [math]\displaystyle{ k }[/math] distinct transients:

[math]\displaystyle{ A_n = A + \sum_{p=1}^k \alpha_p q_p^n. }[/math]

This model for the convergence behaviour contains [math]\displaystyle{ 2k+1 }[/math] unknowns. By evaluating the above equation at the elements [math]\displaystyle{ A_{n-k}, A_{n-k+1}, \ldots, A_{n+k} }[/math] and solving for [math]\displaystyle{ A, }[/math] the above expression for the kth-order Shanks transformation is obtained. The first-order generalized Shanks transformation is equal to the ordinary Shanks transformation: [math]\displaystyle{ S_1(A_n)=S(A_n). }[/math]

The generalized Shanks transformation is closely related to Padé approximants and Padé tables.

Note: The calculation of determinants requires many arithmetic operations to make, however Peter Wynn discovered a recursive evaluation procedure called epsilon-algorithm which avoids calculating the determinants.

See also

• Aitken's delta-squared process
• Anderson acceleration
• Rate of convergence
• Richardson extrapolation
• Sequence transformation

References

• Shanks, D. (1955), "Non-linear transformation of divergent and slowly convergent sequences", Journal of Mathematics and Physics 34 (1–4): 1–42, doi:10.1002/sapm19553411 
• Schmidt, R.J. (1941), "On the numerical solution of linear simultaneous equations by an iterative method", Philosophical Magazine 32 (214): 369–383, doi:10.1080/14786444108520797 
• 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. (1989). "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series". Computer Physics Reports 10 (5–6): 189–371. doi:10.1016/0167-7977(89)90011-7. Bibcode: 1989CoPhR..10..189W. 
• Brezinski, C.; Redivo-Zaglia, M.; Saad, Y. (2018), "Shanks sequence transformations and Anderson acceleration", SIAM Review 60 (3): 646–669, doi:10.1137/17M1120725 
• Senhadji, M.N. (2001), "On condition numbers of the Shanks transformation", J. Comput. Appl. Math. 135 (1): 41–61, doi:10.1016/S0377-0427(00)00561-6, Bibcode: 2001JCoAM.135...41S 
• Wynn, P. (1956), "On a device for computing the e_m(S_n) transformation", Mathematical Tables and Other Aids to Computation 10 (54): 91–96, doi:10.2307/2002183 
• Wynn, P. (1962), "Acceleration techniques for iterated vector and matrix problems", Math. Comp. 16 (79): 301–322, doi:10.1090/S0025-5718-1962-0145647-X