Perturbation analysis for the hyperbolic QR factorization

Abstract

The hyperbolic QR factorization is a generalization of the classical QR factorization and can be regarded as the triangular case of the indefinite QR factorization proposed by Sanja Singer and Saša Singer. In this paper, the perturbation analysis for this factorization is considered using the classical matrix equation approach, the refined matrix equation approach, and the matrix–vector equation approach. The first order and rigorous normwise perturbation bounds with normwise or componentwise perturbations in the given matrix are derived. The obtained first order bounds can be much tighter than the corresponding existing ones. Each of the obtained rigorous bounds is composed of a small constant multiple of the corresponding first order bound and an additional term with simple form. In particular, for square matrix, the rigorous bounds for the factor R are just the 6+3 multiple of the corresponding first order bounds. These rigorous bounds can be used safely for all cases in comparison to the first order bounds.