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.
is defined as: and forms a new sequence
Further speed-up may be obtained by repeated use of the Shanks transformation, by computing
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
) is more numerically stable than the expression to its left (i.e.
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.
As an example, consider the slowly convergent series[3] which has the exact sum π ≈ 3.14159265.
In the table below, the partial sums
on them, as well as the repeated Shanks transformations
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.
has six digits accuracy, obtained from repeated Shank transformations applied to the first seven terms
only obtains 6-digit accuracy after summing about 400,000 terms.
The Shanks transformation is motivated by the observation that — for larger
so that the sequence converges transiently to the series result
the respective partial sums are: These three equations contain three unknowns:
gives[2] In the (exceptional) case that the denominator is equal to zero: then
The generalized kth-order Shanks transformation is given as the ratio of the determinants:[4] with
It is the solution of a model for the convergence behaviour of the partial sums
distinct transients: This model for the convergence behaviour contains
By evaluating the above equation at the elements
the above expression for the kth-order Shanks transformation is obtained.
The first-order generalized Shanks transformation is equal to the ordinary Shanks transformation:
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.
