Question

Define a well-commented main function in Python that determines the number of terms (N1, N2, and N3) required for each method of calculating pi to a given level of precision E = 10^-3. Store the number of terms in a dictionary with key/pair defined as {n1: ’myPiOne’, N2: ‘myPiTwo’, N3:’myPiThree’}. Sort the keys from low to high and print out the associated Method Number pair.

*Leibnitz, Adamchik, and Nilakantha to be used

Answer 1

Problem: Find the number to terms it took for different approximations for PI and print sorted methods.

Attached images at the end for clarity and indentation errors.

Python Code Begin:

import math
def main():
terms_dict = {}
#Lebinitz approximation of pi
#pi = 4*(1 - 1/3 + 1/5 - 1/7 + 1/9....)
#Initializing pi and other values for Lebinitz method
denom = 1.0
multiplier = 1.0
pi = (4.0 / denom)
i=0
#Looping until error reaches threshod of 0.001
while(abs(math.pi-pi) > 1e-3):
denom += 2.0
multiplier *= -1.0
pi += ( (4.0 / denom) * multiplier )
i += 1
#print("It took {} terms to error be 0.001 and pi value is {}".format(i,pi))
terms_dict[i] = "Leibnitz"
  
#Adamchik approximation of pi
#pi = sum from i=0 to i = infinity((1/16)^i*(4/(8*i+1)-2/(8*i+4)-1/(8*i+5)-1/(8*i+6)))
#Initializing pi and other values for Adamchik method
pi=0
i=0
#Looping until error reaches threshod of 0.001
while(abs(math.pi-pi) > 1e-3):
term1 = 4.0/(8*i+1)
term2 = 2.0/(8*i+4)
term3 = 1.0/(8*i+5)
term4 = 1.0/(8*i+6)
pi += (1.0/16**i)*(term1-term2-term3-term4)
i += 1
#print("It took {} terms to error be 0.001 and pi value is {}".format(i,pi))
terms_dict[i] = "Adamchik"
  
#Nilakantha approximation of pi
#pi = 3 + (4/(2*3*4)-4/(4*5*6)+4/(6*7*8).....)
#Initializing pi and other values for Nilakantha method
multiplier = 1.0
denom = 2.0
pi = 3.0
i=0
#Looping until error reaches threshod of 0.001
while(abs(math.pi-pi) > 1e-3):
pi += ( (4.0 / (denom * (denom + 1.0) * (denom + 2.0)) ) * multiplier)
denom += 2.0
multiplier *= -1.0
i+=1
#print("It took {} terms to error be 0.001 and pi value is {}".format(i,pi))
terms_dict[i] = "Nilakantha"
for key in sorted(terms_dict):
print("{} method took {} terms to approximate pi with 0.001 error.".format(terms_dict[key],key))

Python Code End.