Read and understand the code provided to you here . Modify the code to compute the forward differences, and compare the errors obtained with the central differences. a. Implement the forward differences and modify the code to compute: i. Dc derivative in central differences ii. Df derivative in central differences iii. Ec Truncation Error in central differences iv. Ef Truncation Error in forward differences b. What do you observe in the log-log plot? c. Can you assess the order of magnitude of the truncation error just looking at the graph?

import numpy as np

import matplotlib.pyplot as plt

def diffex(func, dfunc, x, n):

dftrue = dfunc(x)

h = 1

H = np.zeros(n)

D = np.zeros(n)

E = np.zeros(n)

H[0] = h

D[0] = (func(x+h) - func(x-h))/2/h

E[0] = abs(dftrue - D[0])

for i in range(1,n):

h = h/10

H[i] = h

D[i] = (func(x+h) - func(x-h))/2/h

E[i] = abs(dftrue - D[i])

return H, D, E

ff = lambda x : -0.1 * x ** 4 - 0.15 * x ** 3 - 0.5 * x **2 - 0.25 * x + 1.2

df = lambda x : - 0.4 * x ** 3 - 0.45 * x **2 - x - 0.25

# Example 4.5 --> n = 11, but can change

n = 11

H, D, E = diffex(ff, df, 0.5, n)

print(' step size finite differences true error')

for i in range(n):

print('{0:14.10f} {1:16.14f} {2:16.13f}'.format(H[i], D[i], E[i]))

plt.loglog(H, E, "k",label = "Total Error Central")

plt.grid()

plt.title("Plot of the Error(E) versus Step Size(H))")

plt.xlabel("step size")

plt.ylabel("Error")

plt.legend()

plt.show()