# Backward error bounds for constrained least squares problems

Cox, Anthony J. and Higham, Nicholas J. (1999) Backward error bounds for constrained least squares problems. BIT Numerical Mathematics, 39 (2). pp. 210-227. ISSN 1572-9125

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## Abstract

We derive an upper bound on the normwise backward error of an approximate solution to the equality constrained least squares problem $\min_{Bx=d}\|b-Ax\|_2$. Instead of minimizing over the four perturbations to $A$, $b$, $B$ and $d$, we fix those to $B$ and $d$ and minimize over the remaining two; we obtain an explicit solution of this simplified minimization problem. Our experiments show that backward error bounds of practical use are obtained when $B$ and $d$ are chosen as the optimal normwise relative backward perturbations to the constraint system, and we find that when the bounds are weak they can be improved by direct search optimization. We also derive upper and lower backward error bounds for the problem of least squares minimization over a sphere: $\min_{\|x\|_2\le\alpha}\|b-Ax\|_2$.

Item Type: Article Equality constrained least squares problem - least squares minimization over a sphere - null space method - elimination method - method of weighting - backward error - backward stability MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theoryMSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis Ms Lucy van Russelt 28 Jun 2006 20 Oct 2017 14:12 http://eprints.maths.manchester.ac.uk/id/eprint/330

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