# Estimate the number of iterations of Newton's method needed to find a root of f(x)=cos(x)−x to within 10^(−100).

Estimate the number of iterations of Newton's method needed to find a root of $f\left(x\right)=\mathrm{cos}\left(x\right)-x$ to within ${10}^{-100}$.
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The idea behind the reasoning is the quadratic convergence of Newton's algorithm (if the zero is simple).
When you are near the zero $\alpha$, an iteration takes you from $\alpha +\delta$ to
$\left(\alpha +\delta \right)-\frac{f\left(\alpha +\delta \right)}{{f}^{\prime }\left(\alpha +\delta \right)}\approx \alpha +\delta -\frac{{f}^{\prime }\left(\alpha \right)\delta +{f}^{″}\left(\alpha \right)\frac{{\delta }^{2}}{2}}{{f}^{\prime }\left(\alpha \right)+{f}^{″}\left(\alpha \right)\delta }\approx \alpha +\frac{{f}^{″}\left(\alpha \right)}{2{f}^{\prime }\left(\alpha \right)}{\delta }^{2},$
so each step roughly doubles the number of correct digits.
If you start with approximately one correct digit, after seven steps, you have roughly ${2}^{7}=128$ correct digits.