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 to be determined. First, the partial sum is defined as:
and forms a new sequence . Provided the series converges, will also approach the limit as
The Shanks transformation of the sequence is the new sequence defined by
where this 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 so that as with Aitken's method, the right-most expression in 's definition (i.e. ) 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.
Absolute error as a function of in the partial sums and after applying the Shanks transformation once or several times: and The series used is which has the exact sum
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 generalized kth-order Shanks transformation is given as the ratio of the determinants:
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: