I am trying to understand "Computing PageRank using Power Extrapolation" by Taher Haveliwala, Sepandar Kamvar, Dan Klein, Chris Manning, and Gene Golub from Stanford University.
The paper I'm referring to can be found here.
I don't understand how their Algorithm 1: Computing $\mathbf y =\mathbf A\mathbf x$
$$\begin{align*} \mathbf y&=c\mathbf P^\top\mathbf x\\ w&=\|\mathbf x\|_1-\|\mathbf y\|_1\\ \mathbf y&=\mathbf y+w\mathbf v \end{align*}$$
How is this computing $\mathbf y =\mathbf P^\prime{}^\top\mathbf x$?