An Arnoldi-Based Iterative Scheme for Nonsymmetric Matrix Pencils Arising in Finite Element Stability Problems
A method for computing the desired eigenvalues and corresponding eigenvectors of a large-scale, nonsymmetric, complex generalized eigenvalue problem is described. This scheme is primarily intended for the normal mode analysis and the stability characterization of the stationary states of parameterized time-dependent partial differential equations, in particular, when a finite element method is used for the numerical discretization. The algorithm, which is based on the previous work of Saad, may be succintly described as a multiple shift-and-invert, restarted Arnoldi procedure which uses reorthogonalization and automatic shift selection to provide stability and convergence, while minimizing the overall computational effort. The application and efficiency of the method is illustrated using two representative test problems.