Computes Pi using Newton's method as described here
Computes Pi using Binomial Theorem with n=1/2 by computing Int[Sqrt(1-x^2)dx,{x,0,1/2}] == Pi/12+Sqrt(3)/8 Wolfram Alpha
Int[Sqrt(1-x^2)dx,{x,0,1/2}]=[x-1/2x^3/3-1/8x^5/5-1/16x^7/7-5/128x^9/9-... ] {x,0,1/2} = [a_0 x - a_1 x^3/3 - a_2 x^5/5 - a_3 x^7/7 - a_4 x^9/9 - ...] at x = 1/2, where
a_0 = 1 a_n = 1/n (3-2n)/2 a_{n-1}
First 50 iterations:
$./newton_pi.py
Iter Term Error Pi (only correct digits)
0 0.500000 -0.260331 3.
1 -0.0208333 -0.0103311 3.1
2 -0.000781250 -0.000956135 3.14
3 -6.97545e-5 -0.000119081 3.141
4 -8.47711e-6 -1.73562e-5 3.141
5 -1.21377e-6 -2.79102e-6 3.14159
6 -1.92569e-7 -4.80191e-7 3.14159
7 -3.27826e-8 -8.68004e-8 3.141592
8 -5.87555e-9 -1.62938e-8 3.1415926
9 -1.09522e-9 -3.15115e-9 3.14159265
10 -2.10570e-10 -6.24310e-10 3.14159265
11 -4.15105e-11 -1.26184e-10 3.141592653
12 -8.35399e-12 -2.59356e-11 3.141592653
13 -1.71066e-12 -5.40766e-12 3.1415926535
14 -3.55511e-13 -1.14153e-12 3.1415926535
15 -7.48293e-14 -2.43584e-13 3.1415926535
16 -1.59260e-14 -5.24717e-14 3.141592653589
17 -3.42276e-15 -1.13986e-14 3.141592653589
18 -7.41982e-16 -2.49481e-15 3.14159265358979
19 -1.62090e-16 -5.49730e-16 3.141592653589793
20 -3.56548e-17 -1.21873e-16 3.141592653589793
21 -7.89202e-18 -2.71684e-17 3.1415926535897932
22 -1.75677e-18 -6.08708e-18 3.1415926535897932
23 -3.93080e-19 -1.37012e-18 3.14159265358979323
24 -8.83678e-20 -3.09706e-19 3.141592653589793238
25 -1.99521e-20 -7.02811e-20 3.141592653589793238
26 -4.52288e-21 -1.60066e-20 3.1415926535897932384
27 -1.02907e-21 -3.65778e-21 3.14159265358979323846
28 -2.34942e-22 -8.38478e-22 3.14159265358979323846
29 -5.38093e-23 -1.92766e-22 3.141592653589793238462
30 -1.23607e-23 -4.44379e-23 3.1415926535897932384626
31 -2.84730e-24 -1.02703e-23 3.1415926535897932384626
32 -6.57582e-25 -2.37933e-24 3.14159265358979323846264
33 -1.52239e-25 -5.52464e-25 3.141592653589793238462643
34 -3.53261e-26 -1.28551e-25 3.141592653589793238462643
35 -8.21491e-27 -2.99722e-26 3.141592653589793238462643
36 -1.91423e-27 -7.00140e-27 3.1415926535897932384626433
37 -4.46913e-28 -1.63844e-27 3.14159265358979323846264338
38 -1.04531e-28 -3.84075e-28 3.141592653589793238462643383
39 -2.44914e-29 -9.01788e-29 3.141592653589793238462643383
40 -5.74773e-30 -2.12061e-29 3.141592653589793238462643383
41 -1.35100e-30 -4.99404e-30 3.1415926535897932384626433832
42 -3.18025e-31 -1.17774e-30 3.1415926535897932384626433832
43 -7.49688e-32 -2.78116e-31 3.141592653589793238462643383279
44 -1.76964e-32 -6.57591e-32 3.1415926535897932384626433832795
45 -4.18265e-33 -1.55673e-32 3.1415926535897932384626433832795
46 -9.89811e-34 -3.68958e-33 3.14159265358979323846264338327950
47 -2.34512e-34 -8.75438e-34 3.14159265358979323846264338327950
48 -5.56248e-35 -2.07940e-34 3.14159265358979323846264338327950
49 -1.32082e-35 -4.94422e-35 3.141592653589793238462643383279502