On the Accuracy of the Karlson--Waldén Estimate of the Backward Error for Linear Least Squares Problems

Abstract

Abstract. We consider the backward error associated with a given approximate solution of a linear least squares problem. The backward error can be very expensive to compute, as it involves the minimal singular value of certain matrix that depends on the problem data and the approximate solution. An estimate based on a regularized projection of the residual vector has been proposed in the literature and analyzed by several authors. Although numerical experiments in the literature suggest that it is a reliable estimate of the backward error for any given approximate LS solution, to date no satisfactory explanation for this behavior had been found. We derive new bounds which confirm this experimental observation.

Linearization estimates of the backward errors for least squares problems

“…Grcar, Saunders, and Su show numerically that this lower bound is a robust approximation to the backward error. Several new results in the recent technique report conform to this experimental observation. For large sparse problems, Malyshev and Sadkane suggest using Lanczos bidiagonalization to evaluate Waldén, Karlson, and Sun's formula.…”

“…However, these conditions are not so useful in practice. A recent result [, Theorem 3.1] seems to indicate that these assumptions might be improved or even removed, and this is a topic under investigation.…”

LSMR: An Iterative Algorithm for Sparse Least-Squares Problems

“…and give the following Matlab script for computing the "economy size" sparse QR factorization K = QR and c ≡ Q T v (for which c = Ky ) and thence µ(x): In our experiments we use this script to compute µ(x k ) for each LSQR and LSMR iterate x k . We refer to this as the optimal backward error for x k because it is provably very close to the true µ(x k ) [7].…”

Estimation of Backward Perturbation Bounds for Linear Least Squares Problem

“…So how to estimate and effectively is worth further researching. Many authors including Wald n, Karlson, and Sun have paid much attention on the problem to derive explicit approximation or find upper and lower bounds for and [3][4][5][6][7][8][9][10][11][12][13]16,17]. The estimates and derived by Karlson and Wald n [6] in particular have been studied by several authors.…”

“…Gratton [12] gave tight bounds on which involve the estimate Grcar [8] proved that asymptotically equals in the sense that where is an exact solution of the LS problem (1). The above results showed that is an excellent estimate of .…”