How to use the Newton's method in python ?

In numerical analysis, Newton's method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function. wikipedia. Example of implementation using python:

Solution 1

from scipy import misc

def NewtonsMethod(f, x, tolerance=0.000001):
    while True:
        x1 = x - f(x) / misc.derivative(f, x) 
        t = abs(x1 - x)
        if t < tolerance:
            break
        x = x1
    return x

def f(x):
    return (1.0/4.0)*x**3+(3.0/4.0)*x**2-(3.0/2.0)*x-2

x = 4

x0 = NewtonsMethod(f, x)

print('x: ', x)
print('x0: ', x0)
print("f(x0) = ", ((1.0/4.0)*x0**3+(3.0/4.0)*x0**2-(3.0/2.0)*x0-2 ))

returns

x:  4
x0:  2.0000002745869883
f(x0) =  1.2356416165815176e-06

Solution 2 (scipy)

from scipy.optimize import newton

def f(x):
    return (1.0/4.0)*x**3+(3.0/4.0)*x**2-(3.0/2.0)*x-2

x = 4

x0 = newton(f, x, fprime=None, args=(), tol=1.48e-08, maxiter=50, fprime2=None)

print('x: ', x)
print('x0: ', x0)
print("f(x0) = ", ((1.0/4.0)*x0**3+(3.0/4.0)*x0**2-(3.0/2.0)*x0-2 ))

returns

x:  4
x0:  2.000000000000008
f(x0) =  3.597122599785507e-14

Plot the above figure using matplotlib

#!/usr/bin/env python

from pylab import *

t = arange(-6.0, 4.0, 0.01)
s= t*t*t/4.0+3.0*t*t/4.0-3*t/2.0-2.0

ax = subplot(111)

ax.plot(t, s)

ax.scatter([-4,-1,2],[0,0,0])

ax.grid(True)

ax.spines['left'].set_position('zero')
ax.spines['right'].set_color('none')
ax.spines['bottom'].set_position('zero')
ax.spines['top'].set_color('none')

ax.set_xlim(-6,6)
ax.set_ylim(-20,20)

text(-3.0, 12,
     r"$f(x)=(1/4)*X^3+(3/4)*X^2-(3/2)*X-2$", horizontalalignment='center',
     fontsize=8)

plt.title("How to implement the Newton's method using python \n for finding the zeroes of a real-valued function",fontsize=10)

plt.xticks(fontsize=8)
plt.yticks(fontsize=8)

plt.savefig('NewtonMethodExemple.png')
show()

Note: with numpy it is also possible to find the root of a polynomial with root, example the following polynomial has 3 roots:

$$P(x)=\frac{x^3}{4}+\frac{3.x^2}{4}-\frac{3.x}{2}-2$$

returns:

>>> import numpy as np
>>> coeff = [1.0/4.0,3.0/4.0,-3.0/2.0,-2]
>>> np.roots(coeff)
array([-4.,  2., -1.])

References