  A thorough backward stability analysis of Hotelling's deflation, an explicit
external deflation procedure through low-rank updates for computing many
eigenpairs of a symmetric matrix, is presented. Computable upper bounds of the
loss of the orthogonality of the computed eigenvectors and the symmetric
backward error norm of the computed eigenpairs are derived. Sufficient
conditions for the backward stability of the explicit external deflation
procedure are revealed. Based on these theoretical results, the strategy for
achieving numerical backward stability by dynamically selecting the shifts is
proposed. Numerical results are presented to corroborate the theoretical
analysis and to demonstrate the stability of the procedure for computing many
eigenpairs of large symmetric matrices arising from applications.
