6.2 Bisection search

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §6.2. Rewritten in original wording — not a paste of Bisection.py. Continues 6.1 Quantum bound states. Labs: 6.2b Bisection labs.

Interval halving

Suppose $f$ is continuous on $[x_{-},x_{+}]$. The intermediate-value theorem plus repeated discarding of the unsigned half traps a root. Pick the sign condition and the midpoint update among lookalikes.

Stop when $(x_{+}-x_{-})/2<\varepsilon$ (or $|f(x)|$ is tiny). The reported root is the final midpoint.

Implementation notes

• Pick $\varepsilon$ from the physics (energy to $10^{-10}$ or better), not from “as small as possible.”
• Poles of $\tan$ / $\cot$ can fake sign changes — use the equivalent residual

$$g(E)=\sqrt{E}\,\cot\sqrt{10-E}-\sqrt{10-E}$$

and a bracket already known to have opposite signs (e.g. $[8.0,8.8]$ for $V_0=10$).

• Bisection needs only the sign of $f$, so mild evaluation noise is usually tolerable.

Warm-up polynomial

Before the well, practice on

$$h(x)=x^{3}-x-2=0,\qquad x\in[1,2].$$

$h(1)=-2$, $h(2)=4$, so a root is trapped; bisection yields $x\approx 1.5213797068$.

Back to the well

With $g(E)$ from §6.1 on $[8.0,8.8]$, the same loop returns

$$E_{B}\approx 8.5927852752.$$

Check $|g(E_{B})|$ in scientific notation — it should sit near machine round-off for a tight $\varepsilon$.

Quiz

Bisection

1 / 4

Bisection requires…

Pick an answer to see if you are right.

Next: 6.2b Bisection labs — implement generic bisection, probe $g(E)$, and lock $E_{B}$. Then 6.3 Newton–Raphson.