Upper bound for spectral radius of matrix multiplications

Question

For Chebyshev iteration, I want to find an upper bound for the highest eigenvalue of a matrix. I have a library in C++ to find eigenvalues for symmetric matrices, but for Chebyshev I need to find the upper bound for the highest eigenvalue of $D^{-1}A$, in which $D$ is the diagonal of A and A is symmetric. $D^{-1}A$ is in general non-symmetric. I wanted to do just like this answer, but I think it requires the multiplication to be symmetric too, doesn't it? Any ideas how to do it?