Let $P$ (symmetric) and $S$ be two matrices of size $N \times N$, and $y$ be a vector of size $N \times 1$. I would like to compute $PSP^{-1} y$ without computing $P^{-1}$ because, in my case, $P$ is an ill-conditioned matrix. I have tried to scale the matrix to reduce its condition number but it did not help. Since $P$ contains element ranging from $10^{-20}$ to $1$, approaches like LU and singular value decomposition also were not useful.
I am looking for a hint to approach this problem in a different way.