Newton's Method

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

import seaborn as sns
sns.set(font_scale=2)
sns.set_style("whitegrid")

Here's a function:

def f(x):
    return x**3 - x +1

def df(x):
    return 3*x**2 - 1

xmesh = np.linspace(-4, 4, 100)
plt.ylim([-5, 10])
plt.plot(xmesh, f(xmesh))

[<matplotlib.lines.Line2D at 0x1a16ba1f60>]

guesses = [-.9]
guesses = [1.5]

Evaluate this cell many times in-place (using Ctrl-Enter)

x = guesses[-1] # grab last guess

slope = df(x)

# plot approximate function
plt.plot(xmesh, f(xmesh))
plt.plot(xmesh, f(x) + slope*(xmesh-x))
plt.plot(x, f(x), "o")
plt.xlim([-4, 4])
plt.ylim([-5, 10])
plt.axhline(0, color="black")

# Compute approximate root
xnew = x - f(x) / slope
guesses.append(xnew)
print(xnew)

-1.3247187886152572

f(xnew)

0.6960453045686934

print(guesses)