In particle physics, we use the Seesaw formula to compute effective masses for neutrinos. It reads $$ M_\mathrm{eff} = - M_D . M_N^{-1} . M_D^T. $$ As I learned the hard way, Matrix inversion is not very stable numerically.
I have to compute $M_\mathrm{eff}$ numerically, to as high precision as possible. I need both the eigenvalues and the eigenvectors. What alternatives do I have to inverting $M_N$, if the matrix $M_N$ is badly conditioned?