I understand that the golden section search algorithm (for finding minimum points) is loosely based on the bisection method (for finding roots). In both methods, we first assign an upper and a lower bound ($x_u$, $x_l$, respectively) as an interval where we bracket our root (or minimum for the golden section search).
My question: why is it enough for the bisection method to only have one 'update equation', that is, finding the midpoint of the upper and lower bounds. In golden section search, we have two equations: $x_1 = x_l -d$ and $x_2 = x_u - d$, where $d=(\phi-1)(x_u-x_l)$.
The two equations above give a very 'asymmetric feel' to the update equations. What is going on here?
Suppose that we have points $x_{l} < x_{m} < x_{r}$ with $x_{m}=(x_{l}+x_{r})/2$ and $f(x_{m}) < f(x_{l})$ and $f(x_{m})<f(x_{r})$. We assume that $f$ is unimodal so that there must be a minimum somewhere between $x_{l}$ and $x_{r}$.
This information isn't enough to tell us whether the minimum point lies in the interval from $x_{l}$ to $x_{m}$ or in the interval from $x_{m}$ to $x_{r}$. This is fundamentally different from using bisection to find a root of $f$.
However, if we look at two points in between $x_{l}$ and $x_{r}$, with $x_{l} < x_{1} < x_{2} < x_{r}$, then depending on whether $f(x_{1}) < f(x_{2})$ or $f(x_{2}) < f(x_{1})$, we can determine whether the minimum is between $x_{l}$ and $x_{2}$ or between $x_{1}$ and $x_{r}$.
You could, for example, implement the search using $x_{1}=x_{l}+(1/3)(x_{r}-x_{l})$ and $x_{2}=x_{l}+(2/3)(x_{r}-x_{l})$.
The golden section search uses the golden ratio $\phi$ to determine the points $x_{1}$ and $x_{2}$ because when the interval is reduced in length, either $x_{1}$ or $x_{2}$ will end up as one of the two interior points in the next iteration.