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.
For a sequence the series
is to be determined. First, the partial sum is defined as:
and forms a new sequence. The sequence often converges more rapidly than the sequence Further speed-up may be obtained by repeated use of the Shanks transformation, by computing 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. Both 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.
As an example, consider the slowly convergent series
which has the exact sum π ≈ 3.14159265. The partial sum has only one digit accuracy, while six-figure accuracy requires summing about 400,000 terms.
In the table below, the partial sums , the Shanks transformation on them, as well as the repeated Shanks transformations and are given for 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 already has two-digit accuracy, while the original partial sums only establish the same accuracy at Remarkably, has six digits accuracy, obtained from repeated Shank transformations applied to the first seven terms , ... , As said before, only obtains 6-digit accuracy after about summing 400,000 terms.
The Shanks transformation is motivated by the observation that — for larger — the partial sum quite often behaves approximately as
with so that the sequence converges transiently to the series result for So for and the respective partial sums are:
These three equations contain three unknowns: and Solving for gives
In the (exceptional) case that the denominator is equal to zero: then for all
Generalized Shanks transformation
with It is the solution of a model for the convergence behaviour of the partial sums with distinct transients:
This model for the convergence behaviour contains unknowns. By evaluating the above equation at the elements and solving for the above expression for the kth-order Shanks transformation is obtained. The first-order generalized Shanks transformation is equal to the ordinary Shanks transformation:
- Weniger (2003).
- Bender & Orszag (1999), pp. 368–375.
- Van Dyke (1975), pp. 202–205.
- Bender & Orszag (1999), pp. 389–392.
- 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.