Jan 7, 2026

The Newton Raphson Method

What is the Newton Raphson Method and how do you use it?

Newton Raphson Method: Finding Roots with Calculus

The Newton Raphson Method is one of the most powerful and widely used numerical techniques for finding the roots of equations.

A root (or zero) of a function is a value of $x$ such that:

f(x) = 0

When equations become too difficult to solve algebraically, the Newton-Raphson method provides a fast and elegant way to approximate the solution using calculus.

What Is the Newton Raphson Method?

The Newton-Raphson method starts with an initial guess $x_0$ and repeatedly improves that guess by using the tangent line to the function.

This creates a sequence of increasingly accurate approximations:

x_0,\ x_1,\ x_2,\ x_3,\ \dots

Under the right conditions, the method converges extremely quickly to the true root.

The Newton Raphson Formula

x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}

Where:

$x_n$ is the current approximation.
$x_{n+1}$ is the next approximation.
$f(x_n)$ is the value of the function at $x_n$ .
$f'(x_n)$ is the derivative, or the slope of the tangent line.

Why This Formula Works

At the current estimate $x_n$ , we draw the tangent line to the graph of the function.

The tangent line equation is:

y = f(x_n) + f'(x_n)(x - x_n)

To estimate the root, we find where this tangent line crosses the x-axis by setting $y = 0$ :

0 = f(x_n) + f'(x_n)(x - x_n)

Solving for $x$ gives:

x = x_n - \frac{f(x_n)}{f'(x_n)}

This becomes the update formula used in the Newton-Raphson method.

Step by Step Algorithm

Define the function $f(x)$ .
Compute its derivative $f'(x)$ .
Choose an initial guess $x_0$ .
Calculate the next approximation using the Newton-Raphson formula.
Repeat until the estimates stop changing significantly.

A common stopping condition is:

|x_{n+1} - x_n| < 10^{-6}

Example: Finding $\sqrt{2}$

To compute $\sqrt{2}$ , we solve:

x^2 - 2 = 0

Step 1: Define the Function

f(x) = x^2 - 2

Step 2: Compute the Derivative

f'(x) = 2x

Step 3: Substitute into the Formula

x_{n+1} = x_n - \frac{x_n^2 - 2}{2x_n}

Step 4: Choose an Initial Guess

Let:

x_0 = 1.5

Successive tangent lines converging to the root of x² - 2 — Figure 1: Successive tangent lines converging to the root of $x^2 - 2$.

Iteration 1

x_1 = 1.5 - \frac{1.5^2 - 2}{2(1.5)}

x_1 = 1.4166666667

Iteration 2

x_2 = 1.4166666667 - \frac{1.4166666667^2 - 2}{2(1.4166666667)}

x_2 \approx 1.4142156863

Iteration 3

x_3 \approx 1.4142135624

Final Answer

\sqrt{2} \approx 1.4142135624

The actual value is:

\sqrt{2} = 1.414213562373095\ldots

After only three iterations, the approximation is accurate to many decimal places.

Iteration Table

Iteration	Approximation
$x_0$	1.5000000000
$x_1$	1.4166666667
$x_2$	1.4142156863
$x_3$	1.4142135624

Advantages of the Newton-Raphson Method

Extremely fast convergence near the root.
Simple formula to implement.
Widely used in mathematics, engineering, and computer science.
Forms the basis of many scientific computing algorithms.

Limitations

The method can fail if:

The derivative is zero or very close to zero.
The initial guess is too far from the root.
The function is not smooth.
The iterations diverge instead of converging.

Because of this, choosing a reasonable starting point is important.

Python Implementation

def newton_raphson(f, df, x0, tol=1e-6, max_iter=100):
    x = x0

    for i in range(max_iter):
        x_new = x - f(x) / df(x)

        if abs(x_new - x) < tol:
            return x_new

        x = x_new

    return x


# Example: sqrt(2)
f = lambda x: x**2 - 2
df = lambda x: 2 * x

root = newton_raphson(f, df, 1.5)
print(root)  # 1.4142135623730951