Lanczos algorithm with selective reorthogonalization for eigenvalue extraction in structural dynamic and stability analysis

Abstract

The Lanczos algorithm has been gaining increasing popularity as the eigensolver for structural analysis in recent years. The notorious numerical instability problem associated with the algorithm as originally proposed was overcome by performing full reorthogonalization among the Lanczos vectors serving as the bases of the solution subspace. Recent studies have shown that reorthogonalization is not required unless convergence to a true eigenvector from the Lanczos subspace is imminent. Hence, the reorthogonalization process performed among the Lanczos vectors may be significantly reduced, and the efficiency of the Lanczos algorithm may be further improved. The current article is the result from the implementation of this “selective reorthogonalization” principle. In this paper, the basics of the Lanczos algorithm and the full reorthogonalization procedure are first reviewed, and then the selective reorthogonalization algorithm is outlined. Next the implementation aspects are discussed, and finally verification as well as comparison examples are given. It is demonstrated that in most cases the selective reorthogonalization approach to the partial solution of an eigensystem is not only numerically stable but also is more efficient in computer time than that of the full reorthogonalization, and only comparable memory space is required.