An incremental method for computing dominant singular spaces
Computational information retrieval
Certification of the QR factor R and of lattice basis reducedness
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
On the Error in the Product QR Decomposition
SIAM Journal on Matrix Analysis and Applications
Rigorous Perturbation Bounds of Some Matrix Factorizations
SIAM Journal on Matrix Analysis and Applications
Perturbation analysis for the hyperbolic QR factorization
Computers & Mathematics with Applications
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This paper gives perturbation analyses for $Q_1$ and $R$ in the QR factorization $A=Q_1R$, $Q_1^TQ_1=I$ for a given real $m\times n$ matrix $A$ of rank $n$ and general perturbations in $A$ which are sufficiently small in norm. The analyses more accurately reflect the sensitivity of the problem than previous such results. The condition numbers here are altered by any column pivoting used in $AP=Q_1R$, and the condition number for $R$ is bounded for a fixed $n$ when the standard column pivoting strategy is used. This strategy also tends to improve the condition of $Q_1$, so the computed $Q_1$ and $R$ will probably both have greatest accuracy when we use the standard column pivoting strategy.First-order perturbation analyses are given for both $Q_1$ and $R$. It is seen that the analysis for $R$ may be approached in two ways---a detailed "matrix--vector equation" analysis which provides a tight bound and corresponding condition number, which unfortunately is costly to compute and not very intuitive, and a simpler "matrix equation" analysis which provides results that are usually weaker but easier to interpret and which allows the efficient computation of satisfactory estimates for the actual condition number. These approaches are powerful general tools and appear to be applicable to the perturbation analysis of any matrix factorization.