Fast modal analysis of large, sparse but unstructured symmetric matrices

Many engineering and scientific applications require the computation of eigenvalues of large, sparse but unstructured symmetric matrices. Existing algorithms can be used to compute a few extreme eigenvalues (and corresponding eigenvectors). Using ideas from optimization theory and abandoning classical requirements of global orthogonality, we construct a Lanczos algorithm for computing not just the extreme eigenvalues of such matrices, but in fact all of the eigenvalues. The storage requirements are small, and this procedure has worked well on matrices with various eigenvalue distributions.