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Componentwise Perturbation Analysis of the Schur Decomposition of a Matrix

The paper presents a rigorous perturbation analysis of the QR decomposition A=QR of an n×m matrix A using the method of splitting operators. New asymptotic componentwise perturbation bounds are derived for the elements of Q and R and the subspaces spanned by the first p≤m columns of A. The new bounds are less conservative than the known bounds and are significantly better than the normwise bounds. An iterative scheme is proposed to determine global componentwise bounds in the case of perturbations for which such bounds are valid. Several numerical results are given that illustrate the analysis and the quality of the bounds obtained.