Eigenstates of Large Profiled Matrices

Abstract

The coefficient matrices of large systems of equations frequently have a profile structure. For dynamic and stability analyses, a considerable number of eigenstates of such matrices must be determined, for instance the eigenstates with the smallest eigenvalues. The algorithm of the eigenstate solver should be capable of handling multiple eigenvalues, which are common in symmetric structural systems. Conventional algorithms, such as algorithms based on the method of Lanczos, do not yield the eigenvalues in sequence and require special handling of multiple eigenvalues. This chapter presents a new algorithm for the special eigenvalue problem that preserves convex matrix profiles during iteration, yields the eigenvalues in ascending order, and does not require special precautions for multiple eigenvalues. The algorithm is applied to dam vibration problems. Multiple eigenvalues do not require special treatment. Eigenvalues of equal magnitude, but opposite sign, require special consideration.