Backward Stability of Explicit External Deflation for the Symmetric Eigenvalue Problem

TL;DR

This paper provides a comprehensive backward stability analysis of Hotelling's deflation method for symmetric eigenvalue problems, including bounds, conditions, and a dynamic shift strategy to ensure numerical stability.

Contribution

It introduces a detailed stability analysis, derives bounds and conditions, and proposes a dynamic shift strategy to enhance the stability of external deflation in eigenvalue computations.

Findings

Bounds on orthogonality loss and backward error derived

Conditions for backward stability identified

Numerical results confirm stability for large matrices

Abstract

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.