The Newton-Raphson Algorithm

The Newton-Raphson algorithm is a root-finding algorithm which produces successively more accurate approximations to the roots of a real_valued function.

If \(x_0\) is an initial guess for a root of \(f\), then a better approximation than \(x_0\) to the root of \(f\) will be given by

\[ \boxed{ \begin{gathered} \\ \quad x_1=x_0-\frac{f(x_0)}{f'(x_0)}\quad\\ \\ \end{gathered} } \]

The above formula implies that the point \((x_1, 0)\) is the intercept of the \(x\) axis of the tangent of the curve of \(f\) at \((x_0, f)x_0))\), and \(x_1\) is the unique root of the linear approximation of \(f\) at the initial guessed value of \(x_0\). If this process is repeated, a general expression is obtained:

\[ \boxed{ \begin{gathered} \\ \quad x_n=x_{n-1}-\frac{f(x_{n-1})}{f'(x_{n-1})} \quad\\ \\ \end{gathered} } \]