The equation for newton-raphson iterations looks like this:

First let's consider what an implementation of a 2-dimensional Newton-Raphson method might look like:

For the W4 method, the key differences are that instead of the inverse jacobian, we use the inverse of the upper- and lower triangular jacobians. The p-term is calculated the iteration before the current one. This makes it fundamentally quite different from newton-raphson. This is described in the following equation:

However, I used the following equation, which seems to be a better fit for my 2-dimensional case (omitting the diagonal matrix term)

Finally, this is what a possible implementation would look like.

Biggest takeaway? There's a lot of extra pages in math papers.