6.2b Bisection labs

Source: Lab ideas from Computational Physics (Landau, Páez, Bordeianu), §6.1–6.2. Rewritten as original drills — not textbook Bisection.py. Continues 6.2 Bisection search.

Implement interval halving, probe the even square-well residual, and lock $E_{B}$. Stubs fail until filled in.

Use

$$g(E)=\sqrt{E}\,\cot\sqrt{V_0-E}-\sqrt{V_0-E}$$

with $\cot\theta=\cos\theta/\sin\theta$ (math.sin / math.cos). Hint only: guard against $\sin\kappa\approx 0$ if you leave the safe bracket.

Bisection convention for graded output: while $(b-a)/2>\varepsilon$ with $\varepsilon=10^{-14}$, set $m=(a+b)/2$; if $f(a)\,f(m)\le 0$ then $b\leftarrow m$, else $a\leftarrow m$ (update the stored $f(a)$). Return the final midpoint $(a+b)/2$.

Formula refresh

Warm up: pick the matching equation among lookalikes (same walk as the theory lesson).

Labs

Activity: Bisect a cubic

Find a root of $h(x)=x^{3}-x-2$ on $[1,2]$ by bisection ($\varepsilon=10^{-14}$). Print the root with 10 decimals.

No input.

Write-ups

0/3 solved

6.2b.W1

Using “sign”, “bracket”, and “half”, explain:

In one sentence: what does bisection do with the midpoint when $f(x_{-})f(x)<0$?

Minimum 3 words. Your answer must include: “sign”, “bracket”, “half”.

0 words/ 3 min

Meet the writing requirements, then you can open the model answer.

Explorers

Next: 6.3 Newton–Raphson search — faster local root-finding (then labs, then §6.5 Lagrange).