Over at Fractal Forums, which I can't reach at present, there's an
attempt to integrate the Mandelbrot series via the Euler method or similar.
It occurs to me that one way to check the validity of this approach is to apply
the integration method to the derivative of the Mandelbrot series.
This is my initial attempt, feel free to make your own formulas from this.
Try p2 between 0.01 and 1
For d--mand, p1 should be large
For ed-mand , p1 0.1, p2 0.01 , or similar , this is examining the region where integral
of the derivative is mostly the same as the iterated function.
d--mand.frm
; Euler integration method, y[n+1] = y[n] + h*f(tn,y[n])
; For Mandelbrot set, f(tn,y[n]) = y[n]^2 + c
; y[n+1]=y[n] + h*(y[n]^2 + c)
e---mand {; Ed Mont, (c) 2023
z = c = pixel, u = p2*z :
z = z^2 + c;
u = u + p2*z
|u|<p1
}
; First derivative of the Mandelbrot series .
; z[n+1]' = ( 2*z[n]'*z[n] ) + 1
d---mand {; Ed Mont, (c) 2023
z = c = pixel, u=1:
u = (2*u*z) + 1
z = z^2 + c;
|u|<p1
}
; First derivative of the Mandelbrot series .
; z[n+1]' = ( 2*z[n]'*z[n] ) + 1
; Then Eulers method .
ed---mand {; Ed Mont, (c) 2023
z = c = pixel, u=1, w = p2*u,v=c:
z=v
u = (2*u*z) + 1
w = w + p2*u
v = z^2 + c
|w-z| < p1
}
