This is a good post explaining a lot of interesting concepts, with a section that has surprisingly bad algebra.

> The exact steps to go from the first form to this one are numerous to say the least, but I've included it in full for completeness.

The algebra included after that line has many unnecessary steps, as well as multiple cancelling sign errors. Most notably the second line to the third line does not distribute the negative sign correctly. Picking up after the second line, I would simply have:

y_n+1 = y_n + (1 - x * y_n^2) / y_n^2 * (y_n^3 / 2)

y_n+1 = y_n + (1 - x * y_n^2) * (y_n / 2)

y_n+1 = y_n + y_n / 2 - x * y_n^3/2

y_n+1 = 3/2 * y_n - x * y_n^3/2

y_n+1 = y_n (3/2 * y_n - x * y_n^2 / 2)

y_n+1 = y_n (1.5 * y_n - 0.5 * x * y_n * y_n)

This is fewer than 1/3 of the included steps (between the second line and the end), and has the advantage of being correct during the intermediate steps. I don't think any of my steps are anything other than self-evident to someone who understands algebra, but I'd be happy to entertain competing perspectives.