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Newton's Method

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Searching for a zero point (root) of a function $ f$, one can use the Newton iteration for a numerical approximation thereof. The sequence $ x_0,\,x_1\,\ldots$ of the approxiamtion is gained by linearization. The approximation $ x_{l+1}$ is the intersection of the $ x$-axis with the tangent line in $ \left( x_l,f(x_l) \right)$:

$\displaystyle x_{l+1} = x_l - f(x_l)/f'(x_l)$

\includegraphics[width=0.6\linewidth]{Newton_Verfahren.eps}

For a simple root $ x_\star$ the Newton interation locally converges quadratically, that is

$\displaystyle \left\vert x_{l+1}-x_{\star} \right\vert \leq c\; \left\vert x_{l}-x_{\star} \right\vert^2 $

if the initial value $ x_0$ has been chosen sufficiently close to $ x_\star$.

\includegraphics[width=0.6\linewidth]{Newton_Verfahren.eps}

see also:

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  automatically generated 9/11/2008