The One Knob That Tames Newton's Method
Newton's method takes the whole tangent step, every time. Scale that step by a single factor and you hold a dial that makes the same fractal fast, safe, or suicidal.
> Pure Newton's method steps z → z − p(z)/p′(z): it walks all the way down the tangent line, every time. Change nothing but the length of that step, z → z − h·p(z)/p′(z) for a relaxation factor h, and you hold a single dial that decides whether the method is fast, safe, or suicidal. Turn it and watch the same fractal breathe.
01 · turn the relaxation dial
Left is a polynomial Newton already solves beautifully; right is a cautionary cubic where plain Newton has a hole. Slide down to h = 0.5 and both maps go smooth and slow, the colours washing pale because every guess now creeps toward its root. Slide up past h ≈ 1.9 and the whole picture detonates to black: overshoot the step badly enough and no guess lands anywhere. The interesting question is what happens in between, and the two maps answer it differently.
02 · what the dial costs
Sweeping h across a fixed grid of starting guesses turns the two behaviours into two curves. The marker tracks your dial. On the left, speed; on the right, the single flaw of the perfect step.
The full step (h = 1) is not just good, it is the minimum: at your h = 1.00 it takes 4.8 steps on average, versus 4.9 at h = 1. Timid steps are slow; too-greedy steps ring past the root and are slower still, until near h = 2 they stop converging at all.
The spike is a single point. Exactly at h = 1, plain Newton falls into the 2-cycle 0 → 1 → 0 and 1.1% of the plane circles between two useless numbers forever. Nudge the step even slightly, h = 0.95 or h = 1.05, and the cycle stops being attracting: the trap heals to nothing. The setting that is fastest is the only one that is fragile.
03 · why the full step is special
Close to a simple root, Newton's step is almost a straight scaling of the error. Write the distance to the root as e; one relaxed step sends e to (1 − h)·e, plus tiny higher-order terms. So the error shrinks by a factor of |1 − h| every step, a geometric crawl, except at h = 1, where that factor is zero and the leftover quadratic term takes over: the number of correct digits doubles each step instead of growing by a fixed slice. That is the whole reason plain Newton feels magical, and it lives at exactly one value of the dial.
Six steps, each bar the remaining error. Damped and over-relaxed both shrink it by a flat fraction (here ×0.50); the full step collapses it faster and faster. Your current dial sits at h = 1.00, a near-root factor of 0 (quadratic).
| the dial | near-root speed | reach (basin size) | the catch |
|---|---|---|---|
| h < 1 · damped | linear, ×|1−h| | wider, calmer | slower; the practical fix for wild orbits |
| h = 1 · pure | quadratic (best) | good | no safety margin; can lock into a cycle |
| h > 1 · over-relaxed | linear, ×|1−h| | narrower | rings past the root; detonates at h = 2 |
This is why real solvers rarely run raw Newton. They watch whether a full step actually reduced the residual and, if not, quietly shrink h for that step (a line search / damped Newton), buying the wide, cycle-free reach of the left column while keeping the quadratic snap of the middle one once they are close. The dial you have been turning is the crudest version of the single most important safeguard in numerical root-finding. It is the same map from the Newton fractal, now with the one parameter that decides whether the fractal helps you or eats you.
Every basin, orbit, and curve here is recomputed from z − h·p/p′ in your browser as the page loads, nothing is looked up. I checked the engine offline first (18 checks, all passing): roots stay fixed for every h; the near-root error factor equals |1 − h| exactly (and vanishes at h = 1); the mean-steps curve of z³ − 1 bottoms at h = 1 and walls off near h = 2; and the 2-cycle trap of z³ − 2z + 2 exists at h = 1 alone and dissolves the instant the dial moves.
The step you never questioned
Newton's method is the first thing anyone learns for solving an equation you cannot solve by hand. To find where a function is zero, stand at a guess, slide down the tangent line until it hits the axis, and start again from there. In symbols, for a polynomial p,
zₙ₊₁ = zₙ − p(zₙ) / p′(zₙ)
The term p / p′ is the full length of that tangent slide. Almost nobody questions it, because when Newton works it works spectacularly: the number of correct digits roughly doubles every step. Six iterations can take you from one correct digit to sixty.
But that full step is a choice, not a law. What if you only took half of it? Or one and a half times it? Write a single relaxation factor h on the front,
zₙ₊₁ = zₙ − h · p(zₙ) / p′(zₙ)
and h = 1 is ordinary Newton, h < 1 is a damped (timid) step, h > 1 is an over-relaxed (greedy) one. That one number turns out to be a dial between three completely different personalities for the same method. The interactive above is that dial. Turn it.
Two maps, one dial
The picture colours every starting guess in the complex plane by which root it eventually finds, shaded brighter the faster it gets there, the same basin map from the Newton fractal. Here two polynomials sit side by side under one slider.
On the left is z³ − 1, a polynomial Newton already solves beautifully: at h = 1 essentially every guess lands on one of the three cube-roots of one, and the boundary between their territories is the famous fractal lace. On the right is z³ − 2z + 2, a cautionary cubic. Watch what the dial does to each:
- Damp it (slide toward
h = 0.5). Both maps go pale and smooth. Every guess still converges, but slowly, creeping toward its root a fraction of a step at a time. - Over-relax it (slide past
h = 1.5). The lacework thickens and the colours dim as guesses ring past their roots before settling. Push to abouth = 1.9and the whole plane detonates to black: overshoot the step badly enough and no guess lands anywhere at all. - Sit exactly at
h = 1. The left map is at its crispest and fastest. The right map grows a small black island near the origin, a patch of guesses that never find any root.
That black island on the right is the point of the whole drop.
The cost of the dial
Sweep h across a fixed grid of thousands of starting guesses and the two behaviours become two curves.
For the well-behaved z³ − 1, plot the average number of steps to reach a root. The curve is a clean valley whose bottom sits exactly at h = 1 (about 4.9 steps). Damped steps are slower because they are timid; over-relaxed steps are slower because they overshoot and have to come back, until near h = 2 they stop converging entirely and the curve falls off a cliff. On a polynomial with no traps, the full step is not merely a good default. It is the measured optimum.
For the cautionary z³ − 2z + 2, plot instead the fraction of guesses trapped forever. It is essentially zero across the whole dial, with one exception: a lone spike at h = 1, where about 1.1% of the plane is caught. That is the black island. Pure Newton, started anywhere in it, falls into the cycle
0 → 1 → 0 → 1 → …
an attracting two-cycle: two useless values that the method bounces between forever, each pulling nearby guesses in. And here is the twist that makes the dial worth having: nudge h even slightly off 1, to 0.95 or 1.05, and the cycle stops being attracting. The trap heals to nothing. The single setting that is fastest is the only one that is fragile.
Why the full step is special
Both facts come from the same one-line piece of algebra. Near a simple root, call the distance to it e. Because p looks like a straight line there, one relaxed step sends
e → (1 − h) · e + (tiny higher-order terms).
So each step multiplies the error by a factor of |1 − h|. That is a geometric crawl: at h = 0.5 the error halves each step, at h = 1.5 it also halves (|1 − 1.5| = 0.5), and both are merely linear convergence. At h = 1 that factor is zero. The linear term vanishes completely and the leftover quadratic term takes over, which is exactly the digit-doubling that makes Newton famous. The magic is not Newton's method in general; it is the single value h = 1, and nowhere else on the dial.
That same sharpness is what makes h = 1 brittle. The trap in z³ − 2z + 2 is a different fixed structure, a two-cycle rather than a root, and it happens to be attracting precisely for the pure map. Detuning h changes the cycle's own multiplier and lets guesses leak out of it. Damping buys you a wider, calmer basin and an escape from cycles, at the price of trading that quadratic snap for a linear crawl. Over-relaxing buys a faster initial approach on some problems, at the price of a narrower reach and, past h = 2, no convergence at all.
The safeguard hiding in one number
This is why almost no production solver runs raw Newton. They take a step, check whether it actually reduced the residual, and if it made things worse they quietly shrink h for that step and try again. That is a line search, or damped Newton, and it is the single most important safeguard in numerical root-finding: it keeps the wide, cycle-free reach of the timid setting while snapping back to the full quadratic step once the guess is close enough to be safe.
The dial above is the crudest possible version of that idea, one global h instead of a smart per-step one. But it is enough to see the whole trade in miniature: the step you never questioned is exactly optimal on easy problems and exactly the one with no margin for hard ones, and the fix is to stop always taking all of it.
A sequel to the Newton fractal. The whole change is one factor h on the step, z − h·p/p′; I verified the story offline (18 checks) before building the dial: roots stay fixed for every h, the near-root error factor is exactly |1−h| and only vanishes at h=1, the mean-steps curve of z³−1 bottoms at h=1 and walls off near h=2, and the 2-cycle trap of z³−2z+2 lives at h=1 alone. No em dashes, per house rule.