Editing Julia set (section)

=== Pseudocode for multi-Julia sets ===
: <math>f(z) = z^n + c</math>
<syntaxhighlight lang="perl">
R = escape radius #  choose R > 0 such that R**n - R >= sqrt(cx**2 + cy**2)

for each pixel (x, y) on the screen, do:
{
    zx = scaled x coordinate of pixel; # (scale to be between -R and R)
    zy = scaled y coordinate of pixel; # (scale to be between -R and R)
  
    iteration = 0;
    max_iteration = 1001;
  
    while (zx * zx + zy * zy < R**2  AND  iteration < max_iteration) 
    {
        xtmp = (zx * zx + zy * zy) ^ (n / 2) * cos(n * atan2(zy, zx)) + cx;
	    zy = (zx * zx + zy * zy) ^ (n / 2) * sin(n * atan2(zy, zx)) + cy;
	    zx = xtmp;
    
        iteration = iteration + 1;
    } 
    if (iteration == max_iteration)
        return black;
    else
        return iteration;
}   
</syntaxhighlight>

Another recommended option is to reduce color banding between iterations by using a renormalization formula for the iteration. <ref name="Renormalizing the Mandelbrot Escape">{{cite web |last1=Vepstas |first1=Linas |title=Renormalizing the Mandelbrot Escape |url=https://linas.org/art-gallery/escape/escape.html |website=linas.org |publisher=Creative Commons |access-date=5 November 2023 |ref=ps1}} </ref> 

Such formula is given to be,
:<math>mu = k + 1 - \frac{\log{\log{|z_{k}|}}}{\log{n}}</math>
:<math>\forall f_{c,n}(z) = z^{n} + \text{lower power terms} + c</math>

where <math>k</math> is the escaping iteration, bounded by some <math>K</math> such that <math>0 \leq k < K</math> and <math> K \in \mathbb{N}</math>, and <math>|z_{k}|</math> is the magnitude of the last iterate before escaping.

This can be implemented, very simply, like so:
<syntaxhighlight lang="perl">
# simply replace the last 4 lines of code from the last example with these lines of code:

if(iteration == max_iteration)
    return black;
else    
    abs_z = zx * zx + zy * zy;
    return iteration + 1 - log(log(abs_z))/log(n);

</syntaxhighlight>

The difference is shown below with a Julia set defined as <math>f_{c, 2}(z)</math> where <math> c = -0.835 - 0.321i </math>.

{{Gallery
|title=Julia set with color banding and without
|width=320|height=144
|align=center
|File:JuliasetColorband.png
 |With color banding
 |alt1=Julia set with color banding
|File:JuliasetNoColorband.png
 |Without color banding
 |alt2=Julia set without color banding

}}