Demmel, James W. and Higham, Nicholas J.
(1993)
*Improved error bounds for underdetermined system solvers.*
SIAM Journal On Matrix Analysis And Applications, 14 (1).
pp. 1-14.
ISSN 1095-7162

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## Abstract

The minimal 2-norm solution to an underdetermined system $Ax = b$ of full rank can be computed using a QR factorization of $A^T $ in two different ways. One method requires storage and reuse of the orthogonal matrix $Q$, while the method of seminormal equations does not. Existing error analyses show that both methods produce computed solutions whose normwise relative error is bounded to first order by $c\kappa_2 ( A )u$, where $c$ is a constant depending on the dimensions of $A$, $\kappa_2 ( A ) = \| A^ + \|_2 \| A \|_2 $ is the 2-norm condition number, and $u$ is the unit roundoff. It is shown that these error bounds can be strengthened by replacing $\kappa_2(A)$ by the potentially much smaller quantity ${\operatorname{cond}}_2 ( A ) = \| \,| A^ + | \cdot | A |\, \|_2 $, which is invariant under row scaling of $A$. It is also shown that ${\operatorname{cond}}_2 ( A )$ reflects the sensitivity of the minimum norm solution $x$ to row-wise relative perturbations in the data $A$ and $b$. For square linear systems $Ax = b$ row equilibration is shown to endow solution methods based on LU or QR factorization of $A$ with relative error bounds proportional to ${\operatorname{cond}}_\infty ( A )$, just as when a QR factorization of $A^T $ is used. The advantages of using fixed precision iterative refinement in this context instead of row equilibration are explained.

Item Type: | Article |
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Uncontrolled Keywords: | underdetermined system, seminormal equations, QR factorization, rounding error analysis, backward error, componentwise error bounds, iterative refinement, row scaling |

Subjects: | MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis |

Depositing User: | Ms Lucy van Russelt |

Date Deposited: | 03 Jul 2006 |

Last Modified: | 20 Oct 2017 14:12 |

URI: | https://eprints.maths.manchester.ac.uk/id/eprint/352 |

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