Algorithms for large eigenvalue problems in vibration and buckling analyses

Abstract

The eigenvalue problem plays a central role in the dynamic and buckling analyses of engineering structures. In practice, one is interested in only a few dozens of the eigenmodes of a system of thousands of degrees of freedom within a particular eigenvalue range. For linear symmetric eigenproblems, [K]{x} = λ[M]{x}, the eigensolutions are well behaved. The recommendations are subspace iteration or the Lanczos method working with [A] = [K-λ₀ M]⁻¹ where λ₀ is the middle of the eigenvalue range of interest. Subspace iteration gets both eigenvalues and eigenvectors. Lanczos gives the approximate eigenvalues which can easily be improved by inverse iteration to obtain the eigenvectors as by-products. For real nonsymmetric or complex symmetric linear eigenprohlems and polynomial eigenproblems, the eigensolutions may be defective. All classical methods, including subspace iteration fail. We recommend to use the Lanczos method to obtain the approximate eigenvalues of interest and to improve them by a new variance of inverse iteration, one vector at a time, and to get the independent generalised vectors as hy-products. We develop solution method for the special case that the approximate eigenvalue is indeed exact rendering a set of singular linear equations which can not be solved by existing algorithms.