Newton - Raphson

The Newton-Raphson method is an iterative algorithm widely used to approximate the roots of nonlinear functions. This method starts with an initial guess (\(x_0\)) and refines the estimate using the formula:

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

Here, \(f(x)\) is the function whose root is to be found, and \(f'(x)\) is its derivative. In each iteration, the algorithm uses the slope of the tangent at the current point \(x_n\) to estimate a new value closer to the root.

The process repeats until the difference between successive values is sufficiently small or \(f(x_n)\) is close to zero, indicating a solution with the desired accuracy has been reached.