Given the matrix
$M = ( I + \alpha D P )$,
where $I$ is the nxn identity, $D$ is nxn symmetric and invertible, $P$ is nxn symmetric but not always invertible, and $\alpha$ is a scalar, is there a way to updatae $M^{-1}$ when $\alpha$ changes without performing a new matrix inverse calculation?
I need to calculate this inverse many times, so I'm trying to find a formula that doesn't require a new matrix inverse every time $\alpha$ changes. I've tried manipulations with the Woodbury identity, but they all include a matrix inverse that's a function of $\alpha$.
Let me edit this to backup a step and add some more information:
I'm solving for $s$ in this equation $s=s^{tr}-\alpha D P s$, where $s$ is a $n \times 1 $ vector and $s^{tr}$ is a constant $n \times 1 $ vector. Then the solution is then $s=(I+\alpha D P)^{-1}s^{tr}$. This is part of an iterative solution to solve for $s$ and $\alpha$, where $\alpha$ depends on $\| s\|$.