An "industrial strength" algorithm for solving sparse symmetric generalized eigenproblems is described. The algorithm has its foundations in known techniques in solving sparse symmetric eigenproblems, notably the spectral transformation of Ericsson and Ruhe and the block Lanczos algorithm. However, the combination of these two techniques is not trivial; there are many pitfalls awaiting the unwary implementor. The focus of this paper is on identifying those pitfalls and avoiding them, leading to a "bomb-proof" algorithm that can live as a black box eigensolver inside a large applications code. The code that results comprises a robust shift selection strategy and a block Lanczos algorithm that is a novel combination of new techniques and extensions of old techniques.

A Shifted Block Lanczos Algorithm for Solving Sparse Symmetric Generalized Eigenproblems

Generalized eigenvalue problems are often solved by a combination of inverse iteration and the Rayleigh–Ritz procedure. In this paper we show that significant advantages can be obtained in this context by applying the Rayleigh–Ritz procedure to an inverted operator, either explicitly while using subspace iteration or implicitly by applying the Lanczos algorithm to the inverted operator. Since the Lanczos algorithm is much more powerful than subspace iteration it should be used whenever possible.

The Advantages of Inverted Operators in Rayleigh–Ritz Approximations

On the effectiveness of the Lanczos method for the solution of large eigenvalue problems

Thesis (Sc.D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1980.

Towards optimal solution techniques for large eigenproblems in structural mechanics

Performance of eigensolvers in modal analyses of nuclear plant building structures

The development of the tridiagonal reduction method and its implementation in NASTRAN are described for real eigenvalue analysis as typified by structural vibration and buckling problems. This method is an automatic matrix reduction scheme whereby the eigensolutions in the neighborhood of a specified point in the eigenspectrum can be accurately extracted from a tridiagonal eigenvalue problem whose order is much lower than that of the full problem. The process is effected without orbitrary lumping of masses or other physical quantities at selected node points and thus avoids one of the basic weaknesses of other techniques.

https://ntrs.nasa.gov/api/citations/19760024845/downloads/19760024845.pdf

P. F. Flanagan

Papers

Eigenvalue extraction in NASTRAN by the tridiagonal reduction (FEER) method: Real eigenvalue analysis