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.^[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 also approach the limit A as n\to\infty. The Shanks transformation S(A_n) of the sequence A_n is the new sequence defined by^[2][3]

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})}

where this 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 essentially the same as used in Aitken's delta-squared process so that as with Aitken's method, the right-most expression in S(A_n)'s definition (i.e. S(A_n) = A_{n+1} - \frac{(A_{n+1}-A_{n})^2}{(A_{n+1}-A_{n})-(A_{n}-A_{n-1})}) is more numerically stable than the expression to its left (i.e. S(A_n) = \frac{A_{n+1}\, A_{n-1}\, -\, A_n^2}{A_{n+1}-2A_n+A_{n-1}}). 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

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.

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 Shanks transformations applied to the first seven terms A_0, \ldots, A_6. As mentioned before, A_n only obtains 6-digit accuracy after summing about 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]

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.^[5][6]

See also

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

Notes

1. ^ Weniger (2003).
2. ^ Bender & Orszag (1999), pp. 368–375.
3. ^ Van Dyke (1975), pp. 202–205.
4. ^ Bender & Orszag (1999), pp. 389–392.
5. ^ Wynn (1956)
6. ^ Wynn (1962)

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. arXiv:math.NA/0306302. Bibcode:1989CoPhR..10..189W. doi:10.1016/0167-7977(89)90011-7
• Brezinski, C.; Redivo-Zaglia, M.; Saad, Y. (2018), "Shanks sequence transformations and Anderson acceleration", SIAM Review. 60 (3): 646–669, doi:10.1137/17M1120725. hdl:11577/3270110
• Senhadji, M.N. (2001), "On condition numbers of the Shanks transformation", J. Comput. Appl. Math.. 135 (1): 41–61, Bibcode:2001JCoAM.135...41S. doi:10.1016/S0377-0427(00)00561-6
• 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. JSTOR 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