BIB-VERSION:: CS-TR-v2.0
          ID:: STAN//NA-M-92-11
       ENTRY:: January 28, 1996
ORGANIZATION:: Stanford University, Department of Computer Science, Numerical
               Analysis Project
       TITLE:: A new approach for solving perturbed symmetric eigenvalue
               problems
        TYPE:: Manuscript
      AUTHOR:: Carey, Cheryl M. M.
      AUTHOR:: Chen, Hsin-Chu
      AUTHOR:: Golub, Gene H.
      AUTHOR:: Sameh, Ahmed H.
        DATE:: September 1992
       PAGES:: 20
    ABSTRACT:: In this paper, we present a new approach for the solution to
               a series of slightly perturbed symmetric eigenvalue problems
               $(A + BS_{i}B^{T}) x = \lambda\ x, 0 \leq\ i \leq\ m$, where
               $A = A^T\ \in\ R^{n\times n}, B \in\ R^{n\times p}$, and
               $S_i\ = S_{i}^{T}\ \in\ R^{p\times p}, p \ll\ n$. The matrix
               $B$ is assumed to have full column rank. The main idea of our
               approach lies in a specific choice of starting vectors used
               in the block Lanczos algorithm so that the effect of the
               perturbations is confined to lie in the first diagonal block
               of the block tridiagonal matrix that is produced by the block
               Lanczos algorithm. Subsequently, for the perturbed eigenvalue
               problems under our consideration, the block Lanczos scheme
               needs be applied to the original (unperturbed) matrix only
               once and then the first diagonal block updated for each
               perturbation so that for low-rank perturbations, the
               algorithm presented in this paper results in significant
               savings. Numerical examples based on finite element vibration
               analysis illustrated the advantages of this approach.
       NOTES:: [Adminitrivia V1/Prg/19960128]
         END:: STAN//NA-M-92-11