Row-Wise Backward Stable Elimination Methods for the Equality Constrained Least Squares Problem

Abstract

It is well known that the solution ofthe equality constrained least squares (LSE) problem minBx=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.