The rightmost eigenvalue (that is, the eigenvalue with largest real part) plays a critical role in linear stability analysis of dynamical systems. In this talk, I will first discuss how to compute this eigenvalue in a reliable manner for a large matrix. Methods that compute the complete set of eigenvalues are often not feasible for the matrices of interest. Iterative eigenvalue solvers, on the other hand, is likely to find the wrong eigenvalue; a proper “preconditioner” (or “filter”) that enhances the rightmost eigenvalue turns out to be necessary. I will review existing preconditioners and present a new technique based on the exponential transformation. Applying this preconditioner requires the action of matrix exponential on vectors, which I will talk about as well. Since the presence of noises in the matrix entries is inevitable, the rightmost point of the e-pseudospectrum (where e is the level of noises) is considered to be a more robust measurement of stability. I will present a new iterative approach for identifying this point based on perturbation theory.