julia_plot sometimes does not draw the entire julia set

[DaveWitteMorris]

The algorithm for julia_plot (and fast_julia_plot) erroneously assumes that if |z| > 2, then z is not in the Julia set of the function f(z) = z² + c, because the orbit of z is unbounded under iteration of f(z). (To see that this is not always true, let c = -6. Then z = 3 is a fixed point, so its orbit is obviously not unbounded.) Looking on the bright side, the algorithm does give correct results whenever |c| <= 2 (which includes all points in the Mandelbrot set), so most users will never see a problem.

To fix this, the bound 2 should be replaced with an appropriate bound R that depends on c. By the quadratic formula, the largest (positive) fixed point of the function f*(x) = x² - |c| is R = (1 + sqrt(1 + 4|c|))/2. It is then easy to see that f*(x) > x for x > R. Then, by the triangle inequality, we have |z² + c| > |z| whenever |z| > R. So the orbit of z tends to infinity whenever |z| is greater than this value R.

I have uploaded a PR that replaces 2 with this value of R, but I have not done this before, and I apologize if I goofed up anything about this ticket.

Component: dynamics

Keywords: complexdynamics

Author: Dave Morris

Branch/Commit: a5a0408

Reviewer: Markus Wageringel

Issue created by migration from https://trac.sagemath.org/ticket/28557

[DaveWitteMorris]

Branch: u/gh-DaveWitteMorris/fix_julia_radius

[DaveWitteMorris]

Commit: f3bff2a

[mwageringel]

Reviewer: Markus Wageringel

[mwageringel]

comment:3

Ok, thank you. This change looks correct to me. Just in the documentation of julia_plot, I think you accidentally defined R_c as the square of the intended value. I changed it to \bigl(1 + \sqrt{1 + 4|c|}\bigr)/2 as you explained in the ticket description. Is this correct?

I also rebased and squashed your commits and removed some trailing whitespace. If you agree with my changes, please set this ticket to positive review.

---

New commits:

9310d79	28557: fixed the radius in julia_plot
9adac17	28557: fix R_c in documentation and remove whitespace

[mwageringel]

Changed commit from f3bff2a to 9adac17

[mwageringel]

Changed branch from u/gh-DaveWitteMorris/fix_julia_radius to u/gh-mwageringel/28557

[DaveWitteMorris]

comment:4

You are right. Thanks for the review, for correcting the docstring, and for fixing my newbie mistakes.

[mwageringel]

Changed stopgaps from none to none

[mwageringel]

comment:5

There is a merge conflict with the latest beta.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

b098431

Merge tag '9.0.beta9' into 28557

[sagetrac-git]

Changed commit from 9adac17 to b098431

[mwageringel]

comment:7

Since #23740, Sage can now plot julia sets of more general polynomials. I merged in the latest beta, but this branch will only affect the case of polynomials of the form f(z) = z^2 + c. As this is the most common case, this seems acceptable to me.

@DaveWitteMorris: Please check whether this is ok, or change the implementation if possible.

[DaveWitteMorris]

comment:8

Thanks for doing the merge. I'm sorry not to have done anything, but I moved to a different computer last month and have not been able to get git and sagemath working properly yet. Now that the semester is coming to an end, I should have more time and expect to be able to get my installation straightened out enough to work on this ticket later in the week.

[DaveWitteMorris]

comment:9

I fixed two minor pyflakes complaints in mandel_julia.py. (DynamicalSystem was imported twice, and QQ was imported but never used.) I am happy with the state of this patch, and I recommend a positive review if the patchbots are green.

However, I also think that julia_plot needs further work, as explained in the new metaticket #28923 (which tracks this ticket).

@mwageringel : Thanks again for your help with this ticket.

[DaveWitteMorris]

Changed branch from u/gh-mwageringel/28557 to u/gh-DaveWitteMorris/28557

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

a5a0408

fix pyflakes failures

[sagetrac-git]

Changed commit from b098431 to a5a0408

[mwageringel]

comment:12

Thanks for the updates.

[mwageringel]

comment:14

Replying to @DaveWitteMorris:

This branch was merged into #28772 (to deal with a merge conflict). My intention was for this ticket to be merged first, and then #28772. On the other hand, if #28772 is merged first, then I think this ticket would be redundant, so it could be closed without being merged.

No problem, as this ticket is now a dependency of #28772, so this one gets merged first. It is still useful to merge this ticket individually, as it makes for a cleaner history which makes it easier to figure out where a change is coming from, later on. Try for example git log --first-parent --oneline develop.

[vbraun]

Changed branch from u/gh-DaveWitteMorris/28557 to a5a0408