Shanks Transformation - Example

Example

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.

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 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 till As said before, only obtains 6-digit accuracy after about summing 400,000 terms.

Read more about this topic:  Shanks Transformation

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