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## 2006.171: Row-wise backward stable elimination methods for the equality constrained least squares problem

2006.171: Anthony J. Cox and Nicholas J. Higham (1999) Row-wise backward stable elimination methods for the equality constrained least squares problem. SIAM Journal On Matrix Analysis And Applications, 21 (1). pp. 313-326. ISSN 1095-7162

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## Abstract

It is well known that the solution of the equality constrained least squares (LSE) problem min Bx=d ||b-Ax||2 is the limit of the solution of the unconstrained weighted least squares problem

$$\min_x\left\| \bmatrix{ \mu d \cr b } - \bmatrix{\mu B \cr A } x \right\|_2$$

as the weight $\mu$ tends to infinity, assuming that $\bmatrix{B^T & A^T \cr}^T$ 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 row-wise backward error bound for the method. The bound shows that, provided row pivoting or row sorting is used, the method is well-suited to problems in which the rows of A and B vary widely in norm. As a by-product of our analysis, we derive a row-wise 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.

Item Type: Article constrained least squares problem, column pivoting, weighted least squares problem, Householder QR factorization, Gaussian elimination, elimination method, rounding error analysis, backward stability, row pivoting, row sorting MSC 2000 > 15 Linear and multilinear algebra; matrix theoryMSC 2000 > 65 Numerical analysis 2006.171 Miss Louise Stait 03 July 2006