finite_differences.md

import math
import scipy.integrate as integrate
import numpy as np
import matplotlib.pyplot as plt

def f(x):
    return math.sqrt(2)*math.sin(math.pi * x)

h = 0.1
def f_approx(x):
    return ((-1/2)*f(x-2*h) + f(x-1*h) - f(x+1*h) + (1/2)*f(x+2*h))/(h**3)

def f_approx_4(x):
    return ((1/8)*f(x-3*h) - f(x-2*h) + (13/8)*f(x-h) - (13/8)*f(x+h) + f(x + 2*h) - (1/8)*f(x + 3*h))/(h**3)

def f_exact(x):
    return -math.sqrt(2)*(math.pi**3)*math.cos(math.pi*x)

def integrand(x):
    return ((f_approx_4(x) - f_exact(x)) / f_exact(x)) ** 2

integrand(1)

3.113442238266441e-07

integrate.quad(integrand, 0, 1)

(3.1134422382661596e-07, 5.551115123125783e-17)

X = np.linspace(0, 1, 100)
vfunc = np.vectorize(integrand)
plt.plot(X, vfunc(X))
plt.show()