Example 2

Computation of the errors for the integer power

It is known that, given a differentiable function $f:[a,b]\to {\bf R}$, $\vert x-y\vert\le \delta$ implies $\vert f(x)-f(y)\vert\le M\delta$, where $$M = \mathop{\rm sup}\limits_{x_0\in[a,b]} \left(\frac{\partial f}{\partial x}(x_0)\right)$$.

Let $r$ be a real number, $n$ natural and $r_0$ a known approximation of accuracy $\delta$.

Then the accuracy of $r_0^n$ as an approximation of $r^n$ is at least $\delta n (\vert r_0\vert+\delta)^{n-1}$.

If we want to get $r^n$ with an accuracy $\varepsilon$, then we need $r$ with accuracy:

$$\tilde\delta = \frac{\varepsilon}{n(\vert r_0\vert+\delta)^{n-1}}.$$