Gauss-Legendre Algorithm in python
Question
I need some help calculating Pi. I am trying to write a python program that will calculate Pi to X digits. I have tried several from the python mailing list, and it is to slow for my use. I have read about the Gauss-Legendre Algorithm, and I have tried porting it to Python with no success.
I am reading from Here, and I would appreciate any input as to where I am going wrong!
It outputs: 0.163991276262
from __future__ import division
import math
def square(x):return x*x
a = 1
b = 1/math.sqrt(2)
t = 1/4
x = 1
for i in range(1000):
y = a
a = (a+b)/2
b = math.sqrt(b*y)
t = t - x * square((y-a))
x = 2* x
pi = (square((a+b)))/4*t
print pi
raw_input()
Solution
You forgot parentheses around
4*t
:pi = (a+b)**2 / (4*t)
You can use
decimal
to perform calculation with higher precision.#!/usr/bin/env python from __future__ import with_statement import decimal def pi_gauss_legendre(): D = decimal.Decimal with decimal.localcontext() as ctx: ctx.prec += 2 a, b, t, p = 1, 1/D(2).sqrt(), 1/D(4), 1 pi = None while 1: an = (a + b) / 2 b = (a * b).sqrt() t -= p * (a - an) * (a - an) a, p = an, 2*p piold = pi pi = (a + b) * (a + b) / (4 * t) if pi == piold: # equal within given precision break return +pi decimal.getcontext().prec = 100 print pi_gauss_legendre()
Output:
3.141592653589793238462643383279502884197169399375105820974944592307816406286208\
998628034825342117068
OTHER TIPS
- If you want to calculate PI to 1000 digits you need to use a data type that supports 1000 digits of precision (e.g., mxNumber)
- You need to calculate a,b,t, and x until |a-b| < 10**-digits, not iterate digits times.
- Calculate square and pi as @J.F. suggests.
pi = (square((a+b)))/4*t
should be
pi = (square((a+b)))/(4*t)
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