Ok.
The first step is to find the period of the central minibrot. You can do this by iterating the 4 c corners of a box around the view, and seeing at which iteration count the 4 z points surround the origin. See
http://www.mrob.com/pub/muency/period.html for a more detailed description of the method, I have an implementation which you can read here:
http://code.mathr.co.uk/mandelbrot-numerics/blob/HEAD:/c/lib/m_d_box_period.c (there's also an arbitrary precision version in the same directory)
Applying this algorithm to your point and radius gives a period of 48:
$ m-box-period 53 -0.8691524744 0.2556487868 1.25e-5 1000
48
Now you can apply Newton's method. For each step of Newton's method you need the period'th iteration of z -> z^2+c starting from z = 0 gives 0, c, c^2 + c, ... and at the same time you calculate the running derivative dz -> 2 z dz + 1. Pseudocode to make it explicit:
function F(c, period) {
z = 0, dz = 0
for (i = 0; i < period; ++i) {
dz = 2 * z * dz + 1
z = z * z + c
}
return z, dz
}
Now you can calculate the Newton step
:
function Newton(c, period, maxsteps) {
for (i = 0; i < maxsteps; ++i) {
z, dz = F(c, period)
c = c - z / dz
}
return c
}
You need to start from a "guessed" c value, the center of the box in which you found the period is usually good enough (the basins of attraction are fractal, in fact). In real code you can stop if dz is 0 to avoid exploding, or when z/dz gets small enough, the maxsteps is just a safety measure - when it starts to converge it tends to get very close really quickly (quadratic convergence). My implementation is here
http://code.mathr.co.uk/mandelbrot-numerics/blob/HEAD:/c/lib/m_d_nucleus.c (and there's an arbitrary precision version in the same directory).
Applying Newton's method to the center of the view with period 48 gives:
$ m-nucleus 53 -0.8691524744 0.2556487868 48 64
-8.6915874972342078e-01 2.5565708568620021e-01
To find the size of the minibrot I use the algorithm described here
http://ibiblio.org/e-notes/MSet/windows.htm my implementation is here
http://code.mathr.co.uk/mandelbrot-numerics/blob/HEAD:/c/lib/m_d_size.c (arbitrary precision version in the same directory) which I'll paste here as it's very short:
extern double _Complex m_d_size(double _Complex nucleus, int period) {
double _Complex l = 1;
double _Complex b = 1;
double _Complex z = 0;
for (int i = 1; i < period; ++i) {
z = z * z + nucleus;
l = 2 * z * l;
b = b + 1 / l;
} return 1 / (b * l * l);
}
It gives the size as a complex number, the magnitude is the size relative to the period 1 continent and the phase / argument is the angle of rotation. Applying this to your period 48 nucleus gives:
$ m-size 53 -8.6915874972342078e-01 2.5565708568620021e-01 48
1.1864737161932281e-08 -5.9691937849660148e-01
my program splits it into magnitude and phase (in radians) for convenience, so the minibrot is 1.1865e-8 times the size of the period 1 continent, so multiply the view size of your initial view by this number to get the minibrot to appear about the same size.
I attached an image of the minibrot (rendered with distance estimation).
June 06, 2011, 10:31:48 AM
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