One-dimensional search techniques

In order to find the minimum of a function $f\colon\mathbb{R}\rightarrow\mathbb{R}$, we need to bracket him:

Definition 5.28 (Bracketing)   To bracket a minimum means to find a triple $\mathstrut a,b,c \in \mathbf{R}, a<b<c$, such that $f(b) < f(a)$ and $f(b) < f(c)$. This means that the minimum is in the interval $(a,c)$.

We show some algorithms, that are the most efficient in this field. First we introduce the family of sectioning algorithm, from which the the golden section search is probably the most suitable for our uses. Then we introduce the Brent's rule, a quadratic interpolation algorithm.