Recipe

Run golden-section search on a bracket

Use when: you have a unimodal f and a valid bracket [a,b], and want its minimum to a stated tolerance without ever differentiating f

Goal: a bracket narrower than the required tolerance, guaranteed to trap the minimum

Taught in Golden-section search

1. 1

   Compute \rho=(3-\sqrt5)/2\approx0.381966, then x_1=a+\rho(b-a) and x_2=b-\rho(b-a).

   Example

   For [a,b]=[0,5]: \rho(5)\approx1.9098, so x_1\approx1.9098 and x_2\approx3.0902.

   Why: this fraction is the unique value making the surviving interior point automatically correct for the next, smaller bracket — the identity Step 3 relies on.

2. 2

   Compute f(x_1) and f(x_2).

   Example

   For f(x)=(x-2)^2+1: f(1.9098)\approx1.0081 and f(3.0902)\approx2.1885. Since f(x_1)<f(x_2), keep [0,3.0902] and carry x_1 forward.

   • f(x_1)<f(x_2) — the minimum lies in [a,x_2]; discard [x_2,b]; keep x_1 as the new bracket's interior point.
   • f(x_1)>f(x_2) — the minimum lies in [x_1,b]; discard [a,x_1]; keep x_2 as the new bracket's interior point.
   • f(x_1)=f(x_2) — the minimum lies in [x_1,x_2]; discard both outer pieces.

   Why: whichever side loses the comparison cannot contain the minimum, by the same unimodality argument Section 5 proved for any valid bracket.

3. 3

   In the new bracket, the carried-over point already sits at the correct golden-ratio fraction — confirm which side it's on (x_1 or x_2 of the new, smaller bracket) and compute only the other one fresh.

   ⚡ Shortcut

   Skip recomputing \rho(b-a) from scratch: the new bracket's missing point is just a_{\text{new}}+\rho(b_{\text{new}}-a_{\text{new}}) or b_{\text{new}}-\rho(b_{\text{new}}-a_{\text{new}}), using whichever formula matches the side that needs a fresh evaluation.

   Why: this is the entire efficiency payoff of using \rho specifically — any other split fraction would force recomputing BOTH interior points every iteration instead of one.

4. 4

   Repeat Steps 2-3 on the new, narrower bracket until b-a is smaller than the required tolerance \varepsilon; report the final bracket's midpoint as the estimated minimizer.

   Why: the bracket width shrinks by the fixed factor 1-\rho\approx0.618 every iteration, so a target tolerance translates directly into a predictable stopping point instead of an open-ended search.