Title: Reliable computation of exterior eigenvalues through matrix functions

Abstract: Exterior eigenvalues of large sparse matrices are needed for
various applications, such as linear stability analysis. These
eigenvalues are difficult to compute efficiently and reliably if they
are much smaller than the dominant eigenvalues in modulus. Traditional
spectral transformations such as Cayley transform are far from reliable.
In this talk, we discuss a simple idea of spectral transformation based
on functions of matrices that maps the desired exterior eigenvalues to
dominant ones. Approximations of the action of matrix functions on
vectors is fundamental for this approach, which can be performed by
rational Krylov subspace methods (RKSM). Numerical experiments for
linear and nonlinear eigenvalue problems demonstrate the reliability of
this method.