# Modified Newton method

The Newton method is designed to find the root of a function, say $LaTeX: \nabla f(x)=0$, by the algorithm map $LaTeX: A(x) = x - \nabla^2 f(x)^{-1} \nabla f(x).$ This need not converge, so a modification is to use line search, resulting in the algorithm map:
$LaTeX: A(x) = x - s*\nabla^2 f(x)^{-1} \nabla f(x),$
where $LaTeX: s^* \in \arg\max \{f(x - s\nabla^2 f(x)^{-1} \nabla f(x)): s \ge 0\}.$ More generally, we could have another step size rule; as long as it is chosen to converge, the modified algorithm is sometimes called the damped Newton method.