Result

Invalid Input: Please enter a positive number.

0

Convergence Graph

Iteration Steps

It #	Approx ($x_n$)	Delta ($\|x_n - x_{n-1}\|$)

Understanding the Newton-Raphson Method

The Newton-Raphson method is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function.

To find the square root of a number N, we are essentially solving for x in the equation:

x² = N \to f(x) = x² - N = 0

Using the derivative f'(x) = 2x, the iterative formula becomes:

x n+1 = ½ * (x n + N / x n)

Graph Interpretation: The graph above plots the approximation value ($x_n$) against the iteration count ($n$). Notice how quickly the curve flattens out. This "flattening" indicates convergence—where the change between steps becomes smaller than your selected precision tolerance.