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:

$${\mathrm{<span\; style="background-color:\; white;">�</span>}}_{\mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">1</span>}}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">�</span>}}_{\mathrm{<span\; style="background-color:\; white;">�</span>}}<span\; style="background-color:\; white;">-</span>\frac{\mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">�</span>}}_{\mathrm{<span\; style="background-color:\; white;">�</span>}}<span\; style="background-color:\; white;">)</span>}{{\mathrm{<span\; style="background-color:\; white;">�</span>}}^{\mathrm{<span\; style="background-color:\; white;">\prime </span>}}<span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">�</span>}}_{\mathrm{<span\; style="background-color:\; white;">�</span>}}<span\; style="background-color:\; white;">)</span>}$$

Where:

• ${�}^{\mathrm{\prime }}({�}_{�})$ 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.

    Args:

    - 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).

    Returns:

    - 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)