The section search

The algorithms of sectioning apply always the same policy: divide and conquer. The initial interval $[a,c]$ is reduced at each iteration to a smaller interval, already bracketing the minimum $x^\star$. We have so a series of encapsulated intervals (see figure 4.1)

$$x^\star\in[a_n,c_n]\subset[a_{n-1},c_{n-1}]\dots\subset[a,c].$$

Figure 4.1: Section search algorithm