  We present a new computational approach for a class of large-scale nonlinear
eigenvalue problems (NEPs) that are nonlinear in the eigenvalue. The
contribution of this paper is two-fold. We derive a new iterative algorithm for
NEPs, the tensor infinite Arnoldi method (TIAR), which is applicable to a
general class of NEPs, and we show how to specialize the algorithm to a
specific NEP: the waveguide eigenvalue problem. The waveguide eigenvalue
problem arises from a finite-element discretization of a partial differential
equation (PDE) used in the study waves propagating in a periodic medium. The
algorithm is successfully applied to accurately solve benchmark problems as
well as complicated waveguides. We study the complexity of the specialized
algorithm with respect to the number of iterations m and the size of the
problem n, both from a theoretical perspective and in practice. For the
waveguide eigenvalue problem, we establish that the computationally dominating
part of the algorithm has complexity O(nm^2 + sqrt(n) m^3). Hence, the
asymptotic complexity of TIAR applied to the waveguide eigenvalue problem, for
n that goes to infinity, is the same as for Arnoldi's method for standard
eigenvalue problems.
