Householder Fractal
For Householder's Method documentation visit here. See Newton Fractal for additional fractal information.
The video covering Householder Fractals can be found here https://youtu.be/F9DFewL0mho. Code to generate these fractals is given in HouseholderFractal.plt written in gnuplot. To run the program have gnuplot installed (gnuplot.info) then type gnuplot HouseholderFractal.plt in the terminal in the directory where HouseholderFractal.plt is saved. Alternatively, open the gnuplot app then go to File > Open and select the HouseholderFractal.plt file, or from gnuplot cd to correct directory and then type load 'HouseholderFractal.plt'. The fractal will be generated and stored in HouseholderFractal.png and can take a while depending on the number of iterations, function complexity, and resolution. You can edit the .plt file using any text editor though I would recommend VS Code with the gnuplot syntax highlighting extension.
The following fractals were created using the following functions and settings normally centered at 0 + 0i with r of 2 and a 16:9 ratio.
p(z) = z ** 3 - 1
dp(z) = 3 * z ** 2
ddp(z) = 6*z
dddp(z) = 6
a = 0.5
a = complex(0.5,0.5)
p(z) = z**8 + 15*z**4 - 16
dp(z) = 8*z**7 + 60*z**3
ddp(z) = 56*z**6 + 180*z**2
dddp(z) = 336*z**5 + 360*z
a = 0.5
a = complex(0.5,0.5)
r = .1
r = .01
r = .001
r = .0001
r = .0001
p(z) = atan(z)
dp(z) = 1 / (z**2 +1)
ddp(z) = (-2*z) / ((z**2+1)**2)
dddp(z) = (8*z**2)/((z**2+1)**3) - 2/((z**2+1)**2)
a = 0.5
a = complex(0.5,0.5)
p(z) = sin(z)
dp(z) = cos(z)
ddp(z) = -sin(z)
dddp(z) = -cos(z)
a = 0.5
a = complex(0.5,0.5)
See Newton Fractal documentation for details on how to isolate the portion that converges on a single root. Then use your favorite photo editor to remove the backgrounds, recolor, and then combine.
