Matrix Iteration for Large Symmetric Eigenvalue Problems

Eigenvalue problems are common in engineering tasks. In particular the prediction of structural stability and dynamic behavior leads to large symmetric real matrices with profile structure, for which a set of successive eigenvalues and the corresponding eigenvectors must be determined. In this paper, a new method of solution for the eigenvalue problem for large real symmetric matrices with profile structure is presented. This method yields the eigenstates in the sequence of the absolute values of their eigenvalues. The profile structure is preserved during iteration, thus reducing the storage requirements and the computational effort. Deflation of the matrix in combination with spectral shifts and repeated preconditioning are used to accelerate the iteration. The method is capable of handling multiple eigenvalues and eigenvalues of equal magnitude but opposite sign. For large matrices, less than one decomposition of the matrix is required for each desired eigenvalue. The determination of the eigenvector corresponding to a given eigenvalue requires one decomposition of the matrix.