First/Second/Third order Taylor Approximation of f(x) = x ** 3

main.py

import numpy as np
from autograd import grad
import matplotlib.pyplot as plt


def f(x):
    return x ** 3

def f_first(x):
    return 3 * x ** 2

def f_second(x):
    return 6 * x

def _test():
    """ test phase """
    for x in np.arange(1, 6, dtype=np.float32):
        # y = f(x)
        # print("y: {}".format(y))

        grad_f = grad(f)
        y1 = grad_f(x)
        y2 = f_first(x)
        print("First Order Gradient| autograd: {}, actual: {}".format(y1, y2))

        grad_f = grad(grad(f))
        y1 = grad_f(x)
        y2 = f_second(x)
        print("Second Order Gradient| autograd: {}, actual: {}".format(y1, y2))

def taylor_approx():
    num_data = 1000

    # specify the point at which we approximate the graph f
    point = 2.0

    # get the correct answer
    x = np.linspace(-10, 10, num_data, dtype=np.float32)
    ans = f(x)

    # first order taylor approximation
    grad_first = grad(f)
    approx_first = f(point) + grad_first(point)*(x - point)

    # second order taylor approximation
    grad_second = grad(grad_first)
    approx_second = f(point) + grad_first(point)*(x - point) + 0.5*grad_second(point)*(x - point)**2

    # third order taylor approximation
    grad_third = grad(grad_second)
    approx_third = f(point) + grad_first(point)*(x - point) + 0.5*grad_second(point)*(x - point)**2 + (1/6)*grad_third(point)*(x - point)**3

    # visualise the result
    plt.plot(x, ans, label="f(x)")
    plt.plot(x, approx_first, label="approx-first")
    plt.plot(x, approx_second, label="approx-second")
    plt.plot(x, approx_third, label="approx-third")
    plt.grid()
    plt.legend()
    plt.show()


if __name__ == '__main__':
    # _test()
    taylor_approx()

Just to play with autograd