6.3 Newton–Raphson search

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §6.3–6.3.1. Rewritten in original wording — not a paste of NewtonCD.py. Continues 6.2 Bisection search. Labs: 6.3b Newton–Raphson labs.

Tangent intercept

Bisection is reliable but only halves the bracket each step. Newton–Raphson is usually faster when you already have a rough guess near a root of $f(x)=0$.

Graphically (Fig. 6.2): at the current guess $x_0$, draw the tangent line to $f$, then take the intercept of that line with the $x$-axis as the next guess. If $f$ were exactly linear, one step would finish; otherwise you repeat.

Write the update as an old guess plus a correction, Taylor-expand to first order, and force the linear model through zero. Pick each equation among lookalikes.

Repeat with the improved $x$ until $|f(x)|$ or $|\Delta x|$ falls below a chosen $\varepsilon$.

Derivative, failure modes, and backtracking

If you know $f'$ in closed form, use it. For messy residuals (including the square-well $g(E)$), approximate $f'$ by a forward difference, and backtrack when $|f|$ grows after a step. Pick each formula among lookalikes.

Newton assumes the local picture is nearly linear. Fig. 6.3 shows two classic traps:

Flat $f'$ blows up $|\Delta x|$; cycling searches escape the nearly-linear neighborhood. A practical hybrid: bisect until the bracket is small, then switch to Newton for the last digits.

Square-well reminder

Use the even residual from §6.1,

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

with a close start (e.g. $x_0=8.5$ for $V_0=10$, or $x_0=6.0$ for the first even root at $V_0=20$). Far starts can land near $\cot$ poles — plot first, or start with bisection.

Quiz

Newton–Raphson

1 / 4

The Newton correction is…

Pick an answer to see if you are right.

Next: 6.3b Newton–Raphson labs — cubic and square-well roots, iteration counts vs bisection, backtracking. Then 6.4 Magnetization search.