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Newton's Method

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With Newton's method we can determine a zero $ x_\ast$ of a function $ f$ numerically. The sequence $ x_0,\,x_1\,\ldots$ of the approximations to $ x_\ast$ is obtained via linearization, i.e., $ x_{l+1}$ is the intersection of the tangent to $ f$ at $ \left( x_\ell,f(x_\ell) \right)$ with the $ x$-axis:

$\displaystyle x_{\ell+1} = x_\ell - f(x_\ell)/f^\prime(x_\ell)$

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

For a simple zero $ x_\ast$ $ (f^\prime(x_\ast)\ne0)$, Newton's iteration converges quadratically, i.e.,

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

if the initial approximation $ x_0$ is sufficiently close to $ x_\ast$.

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  automatically generated 6/22/2016