#!/usr/bin/python3
# Euler's method, numerical stability

import numpy as np
import matplotlib.pyplot as plt

lam = 10.

def f(x, t):
    return -lam * x
    
a, b, = 0.0, 2.0        # interval


for h in [0.03, 0.15, 0.3]:
    x = 1.0                 
    tpoints = np.arange(a, b, h)
    t2 = np.linspace(a, tpoints[-1], 100)
    xpoints = []

    for t in tpoints:
        xpoints.append(x)
        x += h * f(x, t)
    plt.figure()
    plt.xlabel("t")
    plt.ylabel("x(t)")    
    plt.plot(tpoints, xpoints, "ro")
    plt.plot(t2, np.exp(-lam * t2))
plt.show()