Laguerre's Method

The video covering Laguerre's Method can be found here https://youtu.be/blOARV4lnIM. Example code is provided written in Perl and can be found at Laguerre.pl. You can run the program locally by typing perl Laguerre.pl or run the program online using CodingGround.

Reference links:

• "Racines d'une Équation Algébrique" by Laguerre Hathi Trust Digital Library
• "On Laguerre's Method" by K. A. Redish Taylor & Francis Online
• "Numerical Methods That Work" by Forman S. Acton https://books.google.com

Laguerre's Fractals

Code to generate these fractals is given in LaguerreFractal.plt written in gnuplot. To run the program have gnuplot installed (gnuplot.info) then type gnuplot LaguerreFractal.plt in the terminal in the directory where LaguerreFractal.plt is saved. Alternatively, open the gnuplot app then go to File > Open and select the LaguerreFractal.plt file, or from gnuplot cd to correct directory and then type load 'LaguerreFractal.plt'. The fractal will be generated and stored in LaguerreFractal.png and can take a while depending on the 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.

z³-1

a = {1,0} #1+0*i
p(z) = z**3-1
dp(z) = 3*z**2
ddp(z) = 6*z
deg = 3

a = {.5,0}

a = {.5,.5}

z⁸+15z⁴-16

a = {1,0} #1+0*i
p(z) = z**8 + 15*z**4 - 16
dp(z) = 8*z**7 + 60*z**3
ddp(z) = 56*z**6 + 180*z**2
deg = 8

a = {.5,0}

a = {.5,.5}

r = .15

Coloring by Root

See documentation for Newton Fractal and apply the same approach.

z³-1

z⁸+15z⁴-16