Fractal Related Discussion > Fractal Mathematics And New Theories

Newton-Raphson zooming

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gerrit:
Here's another hopefully interesting picture.
Top-left all pixels for which ball method found a period.
Top-right what is left over after verifying with Newton method that there is a nucleus.

For distance estimate definition $$d_e = |z_n|\log|z_n|/|z'_n|$$ we have
$$d_e/2 \leq d \leq 2d_e$$ with $$d$$ the true distance to the boundary.
If R is half the pixel distance it follows that if
$$R>2d_e$$ there is a nucleus in the pixel
$$R<d_e/(2\sqrt{2})$$ there is no nucleus in the pixel
otherwise could be either way.

Middle-left shows the yes and maybe pixels. Middle right where DE is sure there is a nucleus but ball+NR didn't find it.
Bottom-left periods missed by the ball method alone, bottom-right ball false positives.

claude:

--- Quote from: gerrit on December 06, 2017, 01:55:16 AM ---Why is the minimal separation between period p roots $$2^{-p}$$? Found it in the mandelroots code.

--- End quote ---

I think the two period P roots nearest the tip of the antenna are approximately 2-2P apart, at least that is what this short numerical experiment indicates:

--- Code: ---#include <mandelbrot-numerics.h>

int main()
{
mpfr_t distance;
mpfr_init2(distance, 53);
for (int period = 4; period <= 65536; period <<= 1)
{
mpc_t nucleus[3];
mpc_t delta;
mpc_init2(nucleus[0], 4 * period);
mpc_init2(nucleus[1], 4 * period);
mpc_init2(nucleus[2], 4 * period);
mpc_init2(delta, 4 * period);
mpc_set_si(nucleus[0], -2, MPC_RNDNN);
mpc_set_si(nucleus[1], -2, MPC_RNDNN);
m_r_nucleus(nucleus[0], nucleus[0], period, 64);
m_r_nucleus(nucleus[1], nucleus[1], period + 1, 64);
mpc_sub(delta, nucleus[0], nucleus[1], MPC_RNDNN);
m_r_nucleus(nucleus[2], nucleus[2], period + 1, 64);
mpc_sub(delta, nucleus[1], nucleus[2], MPC_RNDNN);
mpc_abs(distance, delta, MPFR_RNDN);
mpfr_log2(distance, distance, MPFR_RNDN);
mpfr_div_si(distance, distance, period, MPFR_RNDN);
mpfr_printf("%6d\t%Re\n", period, distance);
mpc_clear(nucleus[0]);
mpc_clear(nucleus[1]);
mpc_clear(nucleus[2]);
mpc_clear(delta);
}
return 0;
}
--- End code ---

output:

--- Code: ---     4  -7.5096990924173856e-01
8  -1.3888901895836767e+00
16  -1.6945028256046299e+00
32  -1.8472514137604439e+00
64  -1.9236257068802221e+00
128  -1.9618128534401109e+00
256  -1.9809064267200556e+00
512  -1.9904532133600277e+00
1024  -1.9952266066800139e+00
2048  -1.997613303340007e+00
4096  -1.9988066516700034e+00
8192  -1.9994033258350017e+00
16384  -1.9997016629175008e+00
--- End code ---

I have a hunch that the two period P roots closes to the tip of the antenna are the closest pair of period P roots, though I don't have a proof.

gerrit:
Thanks Claude, that looks like an excellent conjecture.
I found a writeup on roots here
--- Code: ---ftp://nozdr.ru/biblio/kolxo3/Cs/CsCa/Yap%20C.K.%20Fundamental%20problems%20in%20algorithmic%20algebra%20(web%20draft,%202000)(O)(550s)_CsCa_.pdf
--- End code ---
which in "lecture 6" also discusses bounds on basin of attraction for Newton's method.
Probably too much for me to digest but maybe you get something out of it if you don't know it already.