🚀 go-pugleaf

I recently read the book FRACTAL PROGRAMMING IN PASCAL and have a question
relating to the generation of Newton's Method fractals.  The following code
fragment iterates the equation z^3-1=0 :
xsquare= x * x
ysquare = y * y
denom = 3 * ((xsquare - ysquare) * (xsquare - yswaure) - 4 * xsquare *
ysquare)
IF denom = 0 then denom = 0.0000001
x = (2/3) * x + (xsquare - ysquare) / denom
y = (2/3) * y - (2 * x * y) / denom

This does, of course, work beautifully.  However, with my high school calculus
skills, I am unable to understand WHY.  Could someone please e-mail or post a
step by step explanation of each part of the above fragment?  Any help is
appreciated.

Patrick Kane