3) Stirling\'s approximation NNeNV2TN is commonly used to approx- imate the fact

Question

3) Stirling's approximation NNeNV2TN is commonly used to approx- imate the factorial of large numbers. a) What is the behavior of the relative error for this approximation as a function of N E [1; 100]?. Hint1: For this sort of question I highly recommend using a plot to support your answer (don't forget the la- bels). Hint2: In Matlab/Python the exact value of N! is given by gamma(N+1) (in Python import the gamma function from the mod ule "math" first) b) What is the smallest integer value of N that could be used in Stirling's approximation with less than 0.5% relative error?

Explanation / Answer

a.

from math import gamma

from math import factorial as fact

import matplotlib.pyplot as plt

from math import log

import numpy as np

import math as math

def stirling(n):

return math.sqrt(2*math.pi*n)*(n/math.e)**n

t = np.arange(1, 100, 1);

error = lambda t : (gamma(t+1) - stirling(t))/stirling(t)

errorFunction = np.vectorize(error)

plt.plot(t, errorFunction(t+1) )

plt.show()

b.

from math import gamma

from math import factorial as fact

import matplotlib.pyplot as plt

from math import log

import numpy as np

import math as math

def stirling(n):

return math.sqrt(2*math.pi*n)*(n/math.e)**n

for x in range (1,100):

if(gamma(x+1) - stirling(x))/stirling(x) < 0.005:

print(x)

break;

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