ROW-WISE 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 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 ob-tained 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. Key words. constrained least squares problem, weighted least squares problem, Householder QR factorization, Gaussian elimination, elimination method, rounding error analysis, backward sta