Numerical Methods · Design background
One transcendental equation, five root-finding methods, and a real lesson about which fixed-point rearrangement to trust.
Heat flows radially outward through a cylindrical pipe wall, then off the outer surface by convection. Balancing conduction through the wall against convection at the surface — the same kind of energy balance behind the Heated Copper Rod lab — produces a relationship between the inner radius $r_1$, the outer radius $r_2$, and the known physical constants of the problem:
Given $r_1=0.1575$ m, the question is: how thick does the wall need to be, i.e. what is $r_2-r_1$?
$r_2$ appears three separate times — once on the left, twice on the right, and one of those is nested inside another logarithm. There's no sequence of algebraic moves that isolates it. This is exactly the situation root-finding methods exist for: turn $f(r_2)=0$ and search for the root numerically instead of solving for it symbolically.
Fixed-point iteration needs the equation in the form $r_2 = g(r_2)$, then just repeats $r_2 \leftarrow g(r_2)$. The most obvious rearrangement — add $r_2$ to both sides of $f(r_2)=0$ — gives a $g$ that diverges immediately: starting from $r_2=0.4$, it shoots to 2.1, then 11.4, then 27, then 46 within five steps. A second rearrangement, solving for the innermost $r_2$ in the nested logs, gives a different $g$ that converges — but slowly, taking 88 iterations to reach a tolerance of 0.001.
Switch to "Fixed-Point" and watch the iteration count — this is the convergent rearrangement, already chosen for you. The lesson is that which rearrangement you pick matters enormously, and there's no general rule for finding a good one besides trying it and checking $|g'(r_2)|<1$ near the root. Open the lab →
Steffensen's method doesn't need a new rearrangement — it takes the exact same convergent $g$ and speeds it up using Aitken's $\Delta^2$ process. From a current estimate $p_0$, compute two ordinary fixed-point steps $p_1=g(p_0)$ and $p_2=g(p_1)$, then extrapolate past both:
This is the same idea as recognizing a slowly-converging sequence is approximately geometric, and jumping straight to its limit instead of waiting for it. Applied here, it cuts the iteration count from 88 down to 3 — without ever computing a derivative, which is exactly why it's worth knowing alongside Newton's method.
| Method | Iterations to tol=0.001 | Needs a derivative? |
|---|---|---|
| Bisection | 8 | No |
| Fixed-point | 88 | No |
| Steffensen's | 3 | No |
| Newton's | 4 | Yes (or estimated numerically) |
| Secant | 4 | No |
Bisection is the slow-but-unconditionally-safe baseline. Newton and secant are fast because they use local slope information, one way or another. Plain fixed-point is fast to set up but can be painfully slow to run — and Steffensen's method shows that's fixable without giving up the no-derivative property that makes fixed-point appealing in the first place.
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