Suppose I start with a $n \times n$ matrix of zeros and ones:
$$ \begin{bmatrix} 0 & 0 & 0 & 1 & 1\\ 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1\\ \end{bmatrix} $$
Then I normalize each row such that it sums to $1$:
$$\begin{bmatrix} 0.& 0.& 0.& 0.5& 0.5\\ 0.2& 0.2& 0.2& 0.2& 0.2\\ 0.2& 0.2& 0.2& 0.2& 0.2\\ 0.2& 0.2& 0.2& 0.2& 0.2\\ 0.2& 0.2& 0.2& 0.2& 0.2\\ \end{bmatrix} $$
And then do the same for each column:
$$\begin{bmatrix} 0. & 0. & 0. & 0.384615 & 0.384615\\ 0.25& 0.25& 0.25& 0.153846& 0.153846\\ 0.25& 0.25& 0.25& 0.153846& 0.153846\\ 0.25& 0.25& 0.25& 0.153846& 0.153846\\ 0.25& 0.25& 0.25& 0.153846& 0.153846\\ \end{bmatrix}$$
Repeat this process 15 times, and I have:
$$\begin{bmatrix} 0. & 0. & 0. & 0.5 & 0.5\\ 0.25& 0.25& 0.25& 0.125& 0.125\\ 0.25& 0.25& 0.25& 0.125& 0.125\\ 0.25& 0.25& 0.25& 0.125& 0.125\\ 0.25& 0.25& 0.25& 0.125& 0.125\\ \end{bmatrix}$$
Assuming that the original matrix is such that this process is stable, each row and column in the final matrix should sum to $1$.
My questions:
Is there a name for what this algorithm converges to, or something closely related?
What algorithm will produce the same result but converge faster?