kigochen / WynnEpsilon

Implementation of Wynn's Epsilon Method For Convergence Acceleration

Contact

Patrick Lohr: [e-mail]

Implementation of Wynn's Epsilon Method For Convergence Acceleration

For most convergent series, the rate of convergence can be accelerated by applying the Shanks Transformation, which approximates the limit of a sequence in a non-linear fashion. That is, for a series of the form:

The Shanks Transformation is defined as:

Where A_p are the pth partial sums of the sequence, and ΔA_p = A_p+1 - A_p. This gives a method to solve for the k^th-order Shank's Transformation. Peter Wynn introduced an efficient algorithm for computing the k^th Shanks Transformation of a partial sum and is given by the following:

Here, each ε_2k is the equivalent of applying the k^th Shanks Transformation to the n^th partial sum S_n. ε₀ is defined as S_n, and the primary diagonal gives the next approximation. For this example, take the Gregory-Leibniz series for calculating π:

This series will eventually converge to π, but it will take billions of terms to get even a decent a approximation! Just take a look at http://www.eveandersson.com/pi/gregory-leibniz. After 1000 terms, the series is only accurate to 2 decimal places. Even after 10⁹ iterations, the approximation only reaches 8 decimals of precision.

wynnpi.py takes an integer argument N and implements Wynn's epsilon method to find a highly precise approximation for π after N terms of the Gregory-Leibniz series. Make sure to include NumPy.

Expected Behavior

(biosystems-analytics) Patricks-MacBook-Pro:14-open patricklohr$ ./wynnpi.py -h
usage: wynnpi.py [-h] [-n INT]

Wynn Epsilon for Gregory-Leibniz series

optional arguments:
  -h, --help            show this help message and exit
  -n INT, --numterms INT
                        Number of series terms (1<=n<=22) (default: 10)
(biosystems-analytics) Patricks-MacBook-Pro:14-open patricklohr$ 

(biosystems-analytics) dhcp-10-142-223-217:14-open patricklohr$ ./wynnpi.py -n 22

Series approximation for π (22 terms):      3.096161526463642
Wynn Epsilon accelerated result (22 terms): 3.141592653589795
Actual Value:                               3.141592653589793
Relative Error:                              7.067899292141149e-16

(biosystems-analytics) dhcp-10-142-223-217:14-open patricklohr$ 

Test Suite

(biosystems-analytics) Patricks-MacBook-Pro:14-open patricklohr$ make test
pytest -v test.py
============================ test session starts ============================
platform darwin -- Python 3.7.2, pytest-4.3.1, py-1.8.0, pluggy-0.9.0 -- /Users/patricklohr/
anaconda3/envs/biosystems-analytics/bin/python
cachedir: .pytest_cache
rootdir: /Users/patricklohr/Documents/Repo/biosys-analytics/assignments/14-open, inifile:
collected 3 items                                                           

test.py::test_usage PASSED                                            [ 33%]
test.py::test_bad_input PASSED                                        [ 66%]
test.py::test_good_input PASSED                                       [100%]

========================= 3 passed in 0.59 seconds ==========================
(biosystems-analytics) Patricks-MacBook-Pro:14-open patricklohr$