Euler's method in python

Question

I'm trying to implement euler's method to approximate the value of e in python. This is what I have so far:

def Euler(f, t0, y0, h, N):
    t = t0 + arange(N+1)*h
    y = zeros(N+1)
    y[0] = y0
    for n in range(N):
        y[n+1] = y[n] + h*f(t[n], y[n])
        f = (1+(1/N))^N
    return y

However, when I try to call the function, I get the error "ValueError: shape <= 0". I suspect this has something to do with how I defined f? I tried inputting f directly when euler is called, but gave me errors related to variables not being defined. I also tried defining f as its own function, which gave me a division by 0 error.

def f(N):
    for n in range(N): 
        return (1+(1/n))^n

(not sure if N was the appropriate variable to use here...)

Answer 1

The formula you are trying to use is not Euler's method, but rather the exact value of e as n approaches infinity wiki,

$n = \lim_{n\to\infty} (1 + \frac{1}{n})^n$

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Euler's method is used to solve first order differential equations.

Here are two guides that show how to implement Euler's method to solve a simple test function: beginner's guide and numerical ODE guide.

To answer the title of this post, rather than the question you are asking, I've used Euler's method to solve usual exponential decay:

$\frac{dN}{dt} = -\lambda N$

Which has the solution,

$N(t) = N_0 e^{-\lambda t}$

Code:

import numpy as np
import matplotlib.pyplot as plt
from __future__ import division

# Concentration over time
N = lambda t: N0 * np.exp(-k * t)
# dN/dt
def dx_dt(x):
    return -k * x

k = .5
h = 0.001
N0 = 100.

t = np.arange(0, 10, h)
y = np.zeros(len(t))

y[0] = N0
for i in range(1, len(t)):
    # Euler's method
    y[i] = y[i-1] + dx_dt(y[i-1]) * h

max_error = abs(y-N(t)).max()
print 'Max difference between the exact solution and Euler's approximation with step size h=0.001:'

print '{0:.15}'.format(max_error)

Output:

Max difference between the exact solution and Euler's approximation with step size h=0.001:
0.00919890254720457

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Note: I'm not sure how to get LaTeX displaying properly.