Two Simple Equations to Compute PI

Computing PI can be dated back long long time ago, where without the help of computer, mathematicans can only compute several digts i.e. 15 in months. Later, in 20 century, the digits of PI have been hunt with the help of computers. Some equations of computing PI converge quickly but they are not easy to solve/program. There are two most simple equations to compute PI, One is Gregory-Leibniz Series.

However, this equation converges slower than the Nilakantha method.

The below python code solves the above two equations.

from math import *

EPSILON = 0.001

# Nilakantha
step = 0
ans = 3
j = 2
while 1:
    step += 1
    if step % 2 == 1:
        ans += 4.0 / (j * (j + 1) * (j + 2))
    else:
        ans -= 4.0 / (j * (j + 1) * (j + 2))
    j += 2
    if abs(pi - ans) <= EPSILON:
        break
    print step, ans, abs(pi - ans)

# Gregory–Leibniz
step = 0
ans = 0
j = 1
while 1:
    step += 1
    if step % 2 == 1:
        ans += 4.0 / j
    else:
        ans -= 4.0 / j
    j += 2
    if abs(pi - ans) <= EPSILON:
        break
    print step, ans, abs(pi - ans)

The output is:

1 3.16666666667 0.0250740130769
2 3.13333333333 0.00825932025646
3 3.14523809524 0.0036454416483
4 3.13968253968 0.00191011390725
5 3.14271284271 0.00112018912305
6 3.14088134088 0.000711312708452
7 3.14207181707 0.000479163482024
8 3.14125482361 0.000337829982028
9 3.14183961893 0.000246965339609
...
...
23 3.14161069904 1.804545068e-05
24 3.14157668544 1.5968154762e-05
25 3.14160685135 1.4197757757e-05
26 3.14157997396 1.2679627611e-05
27 3.14160402399 1.13703964391e-05
28 3.14158241825 1.0235342045e-05

1 4.0 0.85840734641
2 2.66666666667 0.474925986923
3 3.46666666667 0.325074013077
4 2.89523809524 0.246354558352
5 3.33968253968 0.198089886093
6 2.97604617605 0.165546477544
7 3.28373848374 0.142145830149
8 3.01707181707 0.124520836518
9 3.25236593472 0.110773281129
10 3.04183961893 0.0997530346604
....
...
...
992 3.14058458933 0.00100806426003
993 3.14259970268 0.0010070490901
994 3.14058661763 0.00100603596275
995 3.14259767846 0.00100502487184
996 3.14058863778 0.00100401581123
997 3.14259566236 0.00100300877482
998 3.14059064983 0.00100200375651
999 3.14259365434 0.00100100075025

It is obvious that the first method is less efficient than the second one. For Gregory-Leibniz, it takes around 999 iterations to converge to PI with error smaller than 0.001 but the Nilakantha takes just 28 iterations to approach the value.

–EOF (The Ultimate Computing & Technology Blog) —