This morning I heard the old song that talked about some guy who was meaner than a junk yard dog. Well, there must have been some ESP working here, because tonight I just got a FractoFax from Dr. J, bragging about his new dog. The dog is supposed to guard Dr. J's new laboratory during construction. He looks very mean and no doubt is the fractal space equivalent of the Junk Yard Dog. If you have difficulty with the picture, well the dog has bug eyes on the sides of his face, two small ears at the top, and a split nose. Well, maybe it is not a dog.....
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Figure 1. Dr. J's Junkyard Dog.
A little math. Paul Derbyshire offered a Mandelbrot set form of an iteration formula. Here I have replaced his constant with a parameter, p, and the negative 2 exponent with negative 3.
1) z := ez+c(z+p)-3
Since we want a Mandelbrot set, we require the determination of critical points for the initial value of z. Let the iteration be denoted as f.
2) f=ez + c/(z+p)3
The equation for the critical point is df/dz = 0. That is calculus for the artists in the crowd. So
3) f' = df/dz = ez - 3c/(z+p)4 = 0
We can solve 3) using Newton's method (as tried by Paul). zn+1 = zn - f'/f". Here f" is the second derivative.
4) f" = df'/dz = ez + 12c/(z+p)5
Another method is to rearrange eqn 3)
5) ez = 3c/(z+p)4
Now we isolate one z and make it the next estimate in terms of the other. When convergence occurs, the new z will equal the old.
Of the two ways to isolate the z in 5) only one converges, the one on the right. So, take logarithms of both sides
z = log(3c) - 4 log(z+p)
log(z+p) = (log(3c)-z)/4
z+p = exp((log(3c)-z)/4)
and finally we get
6) z = -p + exp((log(3c)-z)/4)
A few iterations of 6) followed by a single Newton iteration gets us the critical point for a given c.
Stay warm,
Jay
frm:Explode_M_p3j { ; by Jay Hill, 1998
; z=exp(z)+c/(z+p)^3
c=pixel, p=p1; p1 is a constant in the formula
z=0 ; find critical
point
z=.2*z+.8*(-p+exp((log(3*c)-z)/4))
z=.2*z+.8*(-p+exp((log(3*c)-z)/4))
z=.2*z+.8*(-p+exp((log(3*c)-z)/4))
z=.2*z+.8*(-p+exp((log(3*c)-z)/4))
z=.2*z+.8*(-p+exp((log(3*c)-z)/4))
z=.2*z+.8*(-p+exp((log(3*c)-z)/4))
z=.2*z+.8*(-p+exp((log(3*c)-z)/4))
z=.2*z+.8*(-p+exp((log(3*c)-z)/4))
z=.2*z+.8*(-p+exp((log(3*c)-z)/4))
z=.2*z+.8*(-p+exp((log(3*c)-z)/4))
z=.2*z+.8*(-p+exp((log(3*c)-z)/4))
z=.2*z+.8*(-p+exp((log(3*c)-z)/4))
z=.2*z+.8*(-p+exp((log(3*c)-z)/4))
z=.2*z+.8*(-p+exp((log(3*c)-z)/4))
z=.2*z+.8*(-p+exp((log(3*c)-z)/4))
:
z=exp(z)+c/sqr(z+p)/(z+p),
real(z)<=900000
}
junkyard_dog { ; (c) Jay Hill, 1998
reset=1960 type=formula
formulafile=explode.frm
formulaname=explode_m_p3j
center-mag=-10.1936/0/0.1168224/1/-90 params=2.22/0 float=y
maxiter=25600 inside=bof60
outside=2
colors=FAZAwA<40>181161151131121000<143>yy0zz0xx0<29>220000000<29>000
}
