I want to compute a matrix of the form
$$(D+AA^{\prime})^{-1} AA^{\prime} \tag{1}$$
where $D$ is a diagonal matrix and $A$ is a $p \times k$ matrix with $p \gg k$. I got this expression after applying Woodbury matrix identity to another matrix, and it will not help to use it again.
I was wondering if there exists some approximation to the matrix of the above form so that I do not have to calculate the inverse directly. Thanks!