The Universe of Discourse

Sun, 18 Oct 2020

Newton's Method and its instability

While messing around with Newton's method for last week's article, I built this Desmos thingy:

The red point represents the initial guess; grab it and drag it around, and watch how the later iterations change. Or, better, visit the Desmos site and play with the slider yourself.

(The curve here is !!y = (x-2.2)(x-3.3)(x-5.5)!!; it has a local maximum at around !!2.7!!, and a local minimum at around !!4.64!!.)

Watching the attractor point jump around I realized I was arriving at a much better understanding of the instability of the convergence. Clearly, if your initial guess happens to be near an extremum of !!f!!, the next guess could be arbitrarily far away, rather than a small refinement of the original guess. But even if the original guess is pretty good, the refinement might be near an extremum, and then the following guess will be somewhere random. For example, although !!f!! is quite well-behaved in the interval !![4.3, 4.35]!!, as the initial guess !!g!! increases across this interval, the refined guess !!\hat g!! decreases from !!2.74!! to !!2.52!!, and in between these there is a local maximum that kicks the ball into the weeds. The result is that at !!g=4.3!! the method converges to the largest of the three roots, and at !!g=4.35!!, it converges to the smallest.

This is where the Newton basins come from:

Julia set for the rational function
associated to Newton's method for ƒ:z→z³−1.  The plane is divided
symmetrically into three regions, colored red, green, and blue.  The
boundary of each two regions (red and green, say) is inhabited by a
series of leaf shapes of the third color (blue), and the boundaries
between the main regions (green, say) and the (blue) leaves are
inhabited by smaller leaves again of the other color (red), and so on
ad infinitum.  The boundaries are therefore an infinitely detailed
filigree of smaller and smaller leaves of all three colors.

Here we are considering the function !!f:z\mapsto z^3 -1!! in the complex plane. Zero is at the center, and the obvious root, !!z=1!! is to its right, deep in the large red region. The other two roots are at the corresponding positions in the green and blue regions.

Starting at any red point converges to the !!z=1!! root. Usually, if you start near this root, you will converge to it, which is why all the points near it are red. But some nearish starting points are near an extremum, so that the next guess goes wild, and then the iteration ends up at the green or the blue root instead; these areas are the rows of green and blue leaves along the boundary of the large red region. And some starting points on the boundaries of those leaves kick the ball into one of the other leaves…

Here's the corresponding basin diagram for the polynomial !!y = (x-2.2)(x-3.3)(x-5.5)!! from earlier:

Julia set for
the function (x-2.2)(x-3.3)(x-5.5).  There is a large pink region on
the left, a large yellow region on the right, and in between a large
blue hourglass-shaped region.  The boundary of yellow region is decorated
with pink bobbles that protrude into the blue region, and similarly
the boundary of the pink region is decorated with yellow
bobbles. Further details are in the text below.

The real axis is the horizontal hairline along the middle of the diagram. The three large regions are the main basins of attraction to the three roots (!!x=2.2, 3.3!!, and !!5.5!!) that lie within them.

But along the boundaries of each region are smaller intrusive bubbles where the iteration converges to a surprising value. A point moving from left to right along the real axis passes through the large pink !!2.2!! region, and then through a very small yellow bubble, corresponding to the values right around the local maximum near !!x=2.7!! where the process unexpectedly converges to the !!5.5!! root. Then things settle down for a while in the blue region, converging to the !!3.3!! root as one would expect, until the value gets close to the local minimum at !!4.64!! where there is a pink bubble because the iteration converges to the !!2.2!! root instead. Then as !!x!! increases from !!4.64!! to !!5.5!!, it leaves the pink bubble and enters the main basin of attraction to !!5.5!! and stays there.

If the picture were higher resolution, you would be able to see that the pink bubbles all have tiny yellow bubbles growing out of them (one is 4.39), and the tiny yellow bubbles have even tinier pink bubbles, and so on forever.

(This was generated by the Online Fractal Generator at; the labels were added later by me. The labels’ positions are only approximate.)

[ Addendum: Regarding complex points and !!f : z\mapsto z^3-1!! I said “some nearish starting points are near an extremum”. But this isn't right; !!f!! has no extrema. It has an inflection point at !!z=0!! but this doesn't explain the instability along the lines !!\theta = \frac{2k+1}{3}\pi!!. So there's something going on here with the complex derivative that I don't understand yet. ]

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