Newton Raphson Method in Python


Newton-Raphson Method in Python

Newton's method, also known as the Newton-Raphson method, is a powerful numerical technique used for finding the roots of a real-valued function. Named after Sir Isaac Newton and Joseph Raphson, this method is based on the idea of linear approximation. In this blog post, we'll explore the Newton-Raphson method, its mathematical formulation, and how it can be implemented in Python.

Understanding the Newton-Raphson Method

Imagine you have a function () and you want to find the value of such that ()=0. The Newton-Raphson method starts with an initial guess 0 and iteratively refines this guess to get closer to the root. It uses the tangent line to the graph of () at the point to predict a better guess +1 for the root.

Mathematical Formulation

Given a function (), the Newton-Raphson method iteratively updates the current guess using the formula:



  • () is the derivative of () at the point .

This process is repeated until the difference between consecutive approximations is within a specified tolerance level.

Implementing the Newton-Raphson Method in Python

Code :

def newton_raphson(func, derivative, initial_guess, tol=1e-6, max_iter=100):


    Find the root of a function using the Newton-Raphson method.


    - func (callable): The function for which to find the root.

    - derivative (callable): The derivative of the function.

    - initial_guess (float): Initial guess for the root.

    - tol (float): Tolerance for the root approximation (default: 1e-6).

    - max_iter (int): Maximum number of iterations (default: 100).


    - float: Approximation of the root.


    x = initial_guess

    for _ in range(max_iter):

        fx = func(x)

        if abs(fx) < tol:

            return x

        x -= fx / derivative(x)

    raise ValueError("Newton-Raphson method did not converge")

# Example usage

def f(x):

    return x**2 - 4

def df(x):

    return 2*x

root = newton_raphson(f, df, 3)

print("Approximate root:", root)

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