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Backward Error Bounds for Constrained Least Squares Problems
, 1999
"... . We derive an upper bound on the normwise backward error of an approximate solution to the equality constrained least squares problem minBx=d kb \Gamma Axk2 . 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 ..."
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. We derive an upper bound on the normwise backward error of an approximate solution to the equality constrained least squares problem minBx=d kb \Gamma Axk2 . 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 kxk 2 ff kb \Gamma Axk2 . Key words: Equality constrained least squares problem, least squares minimization over a sphere, null space method, elimination method, method of weighting, backward error, backward stability AMS subject classific...
Rowwise backward stable elimination methods for the equality constrained least squares problem. Numerical Analysis Report No
 Manchester Centre for Computational Mathematics
, 1998
"... the equality constrained least squares problem ..."
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ROWWISE BACKWARD STABLE ELIMINATION METHODS FOR THE EQUALITY CONSTRAINED LEAST SQUARES PROBLEM
"... Abstract. It is well known that the solution of the equality constrained least squares (LSE) problem minBx=d kb¡Axk2 is the limit of the solution of the unconstrained weighted least squares problem min x °°°hdb i ¡ hBA ix°°°2 as the weight tends to innity, assuming that [BT AT]T has full rank. We d ..."
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Abstract. It is well known that the solution of the equality constrained least squares (LSE) problem minBx=d kb¡Axk2 is the limit of the solution of the unconstrained weighted least squares problem min x °°°hdb i ¡ hBA ix°°°2 as the weight tends to innity, assuming that [BT AT]T has full rank. We derive a method for the LSE problem by applying Householder QR factorization with column pivoting to this weighted problem and taking the limit analytically, with an appropriate rescaling of rows. The method obtained is a type of direct elimination method. We adapt existing error analysis for the unconstrained problem to obtain a rowwise backward error bound for the method. The bound shows that, provided row pivoting or row sorting is used, the method is wellsuited to problems in which the rows of A and B vary widely in norm. As a byproduct of our analysis, we derive a rowwise backward error bound of precisely the same form for the standard elimination method for solving the LSE problem. We illustrate our results with numerical tests. Key words. constrained least squares problem, weighted least squares problem, Householder QR factorization, Gaussian elimination, elimination method, rounding error analysis, backward sta