The emerging needs to solve large-scale nonlinear eigenvalue problems arising in many engineering applications have come into notice. More researches have been conducted on efficient algorithm development and computational theory. However, the nonlinearity varying greatly from problem to problem results in a challenging computational task. Instead of considering arbitrary nonlinear eigenvalue problems, we consider a certain type of problems for robust and efficient algorithm developments. This particular type nonlinear eigenvalue problems consist of a dominated linear and positive definite pencil and a ``small'' nonlinear component. A number of applications give rise of nonlinear eigenvalue problems of such type. Examples include vibration study of fluid solid structures and eigencomputation problem from fiber optic design.

In this talk, a nonlinear eigenvalue problem we particular interested in is from the finite element analysis of the resonant frequencies and external Q of a waveguide loaded cavity, as currently be studied by researchers for next-generation accelerator design. We study iterative subspace projection methods, such as nonlinear Arnoldi method. We focus on the critical stages of algorithms, such as the choice of initial projection subspace, and the expansion and the refinement of projection subspace. We present a notable improvement over the early iterative projection methods in our case study.