# How to implement the iterative Newton–Raphson method to find roots of a function in Python

The Newton–Raphson method (commonly known as Newton’s method) is developed for finding roots of a given function or polynomial iteratively. We show two examples of implementing Newtons method using Python.

We often opt for the iterative methods for solving a number of problems in geosciences.

The goal of such an iterative scheme is to achieve convergence (for root finding or solving a system of linear equations) or divergence (applications using differential equations). In computer programming, we usually use a “for loop” to implement an iteration procedure.

Some the iterative methods I have covered so far are Monte Carlo methods and Genetic algorithm in the context of the earthquake location problem.

## What is the Newton–Raphson method?

The Newton–Raphson method (commonly known as Newton’s method) is developed for finding roots of a given function or polynomial iteratively.

Consider a non-linear equation, where we seek to find the root $x_r$ $$f(x_r) = 0$$

The Newton–Raphson method is an iterative scheme that relies on an initial guess, $x_0$, for the value of the root. From the initial guess, subsequent guesses are obtained iteratively until the scheme either converges to the root $x_r$ or the scheme diverges and we seek another initial guess. The sequence of guesses are obtained from the slope of the function.

$$slope = \frac{df(x_n)}{dx}= \frac{0-f(x_n)}{x_{n+1}-x_n}$$

This gives the Newton–Raphson iterative relation (Kutz, 2013): $$x_{n+1} = x_n - \frac{f(x_n)}{f’(x_n)}$$

Notice that in the above equation, the scheme fails if $f’(x_n) = 0$. In general, if the initial guess is sufficiently close to $x_r$, the scheme converges. See Burden and Faires (1997) for details on the conditions for convergence.

## Python Example 1

Let us now implement Newton’s method for finding root of the problem: $$f(x) = x^3 - 3x + 1 = 0$$

For this equation, we have $f(x) = x^3 - 3x + 1$ and $f’(x) = 3x^2 - 3$. Hence, the Newton’s formula becomes: $$x_{n+1} = x_n - \frac{x^3 - 3x + 1}{3x^2 - 3}$$

import numpy as np
import matplotlib.pyplot as plt
plt.style.use('seaborn')

x = np.zeros((1000, 1))

x[0] = 5.0  # initial guess

fig, ax = plt.subplots(1, 1)
for j in range(1000):
# x_{n+1} = x_n - \frac{x^3 - 3x + 1}{3x^2 - 3}
x[j+1] = x[j] - (x[j]**3 - 3*x[j] + 1)/(3*x[j]**2 - 3)

# f(x) = x^3 - 3x + 1
func = x[j+1]**3 - 3*x[j+1] + 1

ax.plot(x[j], func, 'o', color='b')

# termination criteria
if np.abs(func) < 10**-6:
break

plt.gca().invert_xaxis()
ax.set_xlabel('x')
ax.set_ylabel('f(x)')
plt.savefig('example1.png', dpi=300, bbox_inches='tight')
plt.close()
print(f"The root of the function is: {x[j+1]}")
print(f"Function value at the root: {func}")

The root of the function is: [1.5320889]
Function value at the root: [4.00197986e-08]


## Python Example 2

Let us now borrow a function used by Kutz, 2003.

$$f(x) = exp(x) - tan(x) = 0$$

The derivative of this function is $f’(x) = exp(x) - sec^2(x)$, which leads to the Newton’s formula:

$$x_{n+1} = x_n - \frac{exp(x_n) - tan(x_n)}{exp(x_n) - sec^2(x_n)}$$

import numpy as np
import matplotlib.pyplot as plt
plt.style.use('seaborn')

x = np.zeros((1000, 1))

x[0] = 10.0  # initial guess

fig, ax = plt.subplots(1, 1)
for j in range(1000):
# x_{n+1} = x_n - \frac{exp(x_n) - tan(x_n)}{exp(x_n) - sec^2(x_n)}
x[j+1] = x[j] - (np.exp(x[j]) - np.tan(x[j])) / \
(np.exp(x[j]) - (1/np.cos(x[j]))**2)

# f(x) = exp(x) - tan(x)
func = (np.exp(x[j+1]) - np.tan(x[j+1]))

ax.plot(x[j], func, 'o', color='b')

# termination criteria
if np.abs(func) < 10**-6:
break

plt.gca().invert_xaxis()
ax.set_xlabel('x')
ax.set_ylabel('f(x)')
plt.savefig('example2.png', dpi=300, bbox_inches='tight')
plt.close()
print(f"The root of the function is: {x[j+1]}")
print(f"Function value at the root: {func}")


The root of the function is: [-3.0964123]
Function value at the root: [-7.14244983e-11]


Now, let us run the above code for the initial guess of x[0] = 100.0.

## Conclusions

The Newton’s method is very fast to converge to the solution for a sufficiently close guess. However, if we have a bad guess then it does not converge at all.

## References

1. Kutz, J. N. (2013). Data-driven modeling & scientific computation: methods for complex systems & big data. Oxford University Press.
2. R. L. Burden and J. D. Faires, Numerical Analysis (Brooks/Cole, 1997).

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