Patrick Lohr: [email protected]
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 Ap are the pth partial sums of the sequence, and ΔAp = Ap+1 - Ap. This gives a method to solve for the kth-order Shank's Transformation. Peter Wynn introduced an efficient algorithm for computing the kth Shanks Transformation of a partial sum and is given by the following:
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.
(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$
(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$
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