  We propose an iterative method to find pointwise growth exponential growth
rates in linear problems posed on essentially one-dimensional domains. Such
pointwise growth rates capture pointwise stability and instability in extended
systems and arise as spectral values of a family of matrices that depends
analytically on a spectral parameter, obtained via a scattering-type problem.
Different from methods in the literature that rely on computing determinants of
this nonlinear matrix pencil, we propose and analyze an inverse power method
that allows one to locate robustly the closest spectral value to a given
reference point in the complex plane. The method finds branch points,
eigenvalues, and resonance poles without a priori knowledge.
