New backward error bounds of Rayleigh–Ritz projection methods for quadratic eigenvalue problem

Abstract

Rayleigh–Ritz projection is one of the efficient procedures to project a large-scale quadratic eigenvalue problem (QEP) into a small-scale QEP by properly choosing a low-dimensional subspace. One common way for solving the projected QEP is to recast it via linearization. In this paper, we establish bounds for backward errors of approximate eigenpairs of QEP relative to those of a linearization. These bounds give useful information to predict the numerical stability of Rayleigh–Ritz projection eigensolvers followed by a linearization. We present results of numerical experiments that support the predictions of the backward error analysis.

Keywords:

• Quadratic eigenvalue problem
• Rayleigh–Ritz projection
• linearization
• backward error

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work is supported by the National Natural Science Foundation of China [grant numbers 11961048, 12001262, and 11801258], Natural Science Foundation of Jiangxi Province [grant number 20181ACB20001]. Academic and Technical Leaders Training Plan of Jiangxi Province [grant number 20212BCJ23027].