Modified Newton method

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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.

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