ABSTRACT
There are needs for computing eigenvalues of a nonlinear and/or non-Hermitian eigenvalue problem that lie in a given region in the complex plane. For example, a conforming finite element discretization of the fourth order reformulation of the transmission eigenvalue problem (6.19) leads to a quadratic matrix eigenvalue problem. In general, only a small fraction of interior eigenvalues are of interest. Other than some crude qualitative estimates, no spectral information is available. In addition, the distribution of the eigenvalues is very complicated in general (see Fig. 10.1). Most existing eigenvalue solvers are not suitable for these problems. Traditional methods such as shift and invert Arnoldi are handicapped by the lack of a priori eigenvalue estimate.
