Title: Analytic Formula for Deriving Mandelbrot?
Post by: kram1032 on December 05, 2009, 01:57:36 AM
Maybe things change due to the properties of Mandelbulb or even already in the standard Mandelbrot-set itself...
But I found the following:
The chain-rule gives:
d/dx f(g(x) = g'(x) * f'(g(x))
further:
d/dx f(g(h(x))) = h'(x) * g'(h(x)) * f'(g(h(x)))
and so on...
using f(x)=g(x)=h(x) gives
d/dx f'_iterated(x) = f'(x)*f'(f(x))*f'(f(f(x)))*...
for the real case x²+a that would mean:
 = x^2+a<br />f'(x) = 2x<br />)
*2*((x^2+a)^2+a)*...<br />f_i_t_e_r_a_t_i_o_n_n = 2^n*x*(x^2+a)*((x^2+a)^2+a)*...)
until you reach the nth iteration.
So, the derivative for the nth iteration would simply be the initial c value * each new iteration value * 2n, if I got it right.
(if you'd use the inital z-value, which is (0,0), the whole thing doesn't make any sense, but if I'm not mistaken, the result in case of the Mandelbrot doesn't change for the case of using z=c as starting value, except it's one iteration ahead:
0²+c = c
c²+c
(c²+c)+c²....
for aribitary (real) powers k (which are not dependend on z themselves), the formula just would be
initial c value * each new iteration value * kn
I hope that makes sense...
And I hope you know how to extend that to complex or triplex numbers and/or polar/spherical coordinates... - I looked into that and it seemed quite simple and my primitive tests got me thinking that the formula doesn't really change if you use complex rather than real numbers...
Any mistakes here?
But I found the following:
The chain-rule gives:
d/dx f(g(x) = g'(x) * f'(g(x))
further:
d/dx f(g(h(x))) = h'(x) * g'(h(x)) * f'(g(h(x)))
and so on...
using f(x)=g(x)=h(x) gives
d/dx f'_iterated(x) = f'(x)*f'(f(x))*f'(f(f(x)))*...
for the real case x²+a that would mean:
until you reach the nth iteration.
So, the derivative for the nth iteration would simply be the initial c value * each new iteration value * 2n, if I got it right.
(if you'd use the inital z-value, which is (0,0), the whole thing doesn't make any sense, but if I'm not mistaken, the result in case of the Mandelbrot doesn't change for the case of using z=c as starting value, except it's one iteration ahead:
0²+c = c
c²+c
(c²+c)+c²....
for aribitary (real) powers k (which are not dependend on z themselves), the formula just would be
initial c value * each new iteration value * kn
I hope that makes sense...
And I hope you know how to extend that to complex or triplex numbers and/or polar/spherical coordinates... - I looked into that and it seemed quite simple and my primitive tests got me thinking that the formula doesn't really change if you use complex rather than real numbers...
Any mistakes here?
Title: Re: Analytic Formula for Deriving Mandelbrot?
Post by: kram1032 on December 05, 2009, 03:20:44 PM
I tried to aply this to a standard mandelbrot and came up with this:
*x_o_l_d<br />y -> 2*(2*x*y+b)*y_o_l_d)
respectively, for
, it would be
^n^/^2 cos(n*arg(x+i y))+a)*x_o_l_d<br />y->n*((x^2+y^2)^n^/^2 sin(n*arg(x+i y))+b)*y_o_l_d)
and the z-coordinate in case of the mandelbulb would behave similarly...
is it really that simple?
(different formulae like z^n+z^(n-1)+...+z^2+c have more complex derivates but it might be possible to simplify them like that, too...)
respectively, for
and the z-coordinate in case of the mandelbulb would behave similarly...
is it really that simple?
(different formulae like z^n+z^(n-1)+...+z^2+c have more complex derivates but it might be possible to simplify them like that, too...)
Title: Re: Analytic Formula for Deriving Mandelbrot?
Post by: JosLeys on December 05, 2009, 07:30:46 PM
I posted something on using the derivative to find the distance estimate some time ago:
http://www.fractalforums.com/3d-fractal-generation/true-3d-mandlebrot-type-fractal/msg8346/#msg8346
http://www.fractalforums.com/3d-fractal-generation/true-3d-mandlebrot-type-fractal/msg8346/#msg8346
Title: Re: Analytic Formula for Deriving Mandelbrot?
Post by: kram1032 on December 05, 2009, 11:12:48 PM
Yeah, just found out that the distance estimation is similar to this...
I kinda didn't realise that.
So Im guess:
"Scrap that, old knowledge" ^^
btw: did anyone yet take a look at the derivative of the Mandelbrot?
(http://www.deviantart.com/download/145777895/derivation_of_Mandelbrot_by_kram1032.png)
I kinda didn't realise that.
So Im guess:
"Scrap that, old knowledge" ^^
btw: did anyone yet take a look at the derivative of the Mandelbrot?
(http://www.deviantart.com/download/145777895/derivation_of_Mandelbrot_by_kram1032.png)
Title: Re: Analytic Formula for Deriving Mandelbrot?
Post by: David Makin on December 05, 2009, 11:45:35 PM
Methinks you did something wrong, here's what the derivative of complex z^2+c looks like with a bailout of x^2+y^2>65536:
Title: Re: Analytic Formula for Deriving Mandelbrot?
Post by: David Makin on December 06, 2009, 12:48:43 AM
And here's the derivative of the "-sine" version of the z^8+c Mandelbulb:
(http://www.fractalforums.com/gallery/1/141_06_12_09_12_47_01.jpg)
(http://www.fractalforums.com/gallery/1/141_06_12_09_12_47_01.jpg)
Title: Re: Analytic Formula for Deriving Mandelbrot?
Post by: kram1032 on December 06, 2009, 01:01:23 AM
hmmm... interesting.
Actually no visible change in the result at all...
now I wonder about the difference of the derivation and the original. It seems to tend toward zero as iterations go infinite...
I'm doing a standard Mandelbrot of what I did the Buddhabrot before right now. Using similar bailout and 65536 iterations which is way higher than what I used for the Buddhabrot render (although it probably doesn't do that much of a difference...)
Actually no visible change in the result at all...
now I wonder about the difference of the derivation and the original. It seems to tend toward zero as iterations go infinite...
I'm doing a standard Mandelbrot of what I did the Buddhabrot before right now. Using similar bailout and 65536 iterations which is way higher than what I used for the Buddhabrot render (although it probably doesn't do that much of a difference...)
Title: Re: Analytic Formula for Deriving Mandelbrot?
Post by: David Makin on December 06, 2009, 01:29:10 AM
hmmm... interesting.
Actually no visible change in the result at all...
now I wonder about the difference of the derivation and the original. It seems to tend toward zero as iterations go infinite...
I'm doing a standard Mandelbrot of what I did the Buddhabrot before right now. Using similar bailout and 65536 iterations which is way higher than what I used for the Buddhabrot render (although it probably doesn't do that much of a difference...)
Actually no visible change in the result at all...
now I wonder about the difference of the derivation and the original. It seems to tend toward zero as iterations go infinite...
I'm doing a standard Mandelbrot of what I did the Buddhabrot before right now. Using similar bailout and 65536 iterations which is way higher than what I used for the Buddhabrot render (although it probably doesn't do that much of a difference...)
On the "normal" Buddhabrots when using either my delta DE or using analytical DE I found that for accuracy a minimum bailout value of the square of the magnitude for accuracy was arouind 128, less than that and both methods are subject to errors (the same goes for normal bicomplex fractals but quaternions are usually OK with lower bailouts).