Estimating the Backward Error in LSQR

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

“…In particular, it is known that ν is a good approximation to the LS backward error µ provided the approximate solution x is sufficiently close to a LS solutionx. Numerical experience, however, indicates that ν is a good approximation to the LS backward error µ for any given x; see [9,18,13]. In Section 3 we derive new bounds on µ in terms of the estimate ν of Karlson and Waldén, and prove that ν is always within a constant factor of the backward error µ.…”

“…For instance, it is used for testing fast, but potentially unstable, algorithms; see, e.g., [10]. It is used to monitor the convergence of iterative solution methods and to design reliable stopping criteria for these methods; see, e.g., [17,1,4,13,6]. In this context the approximate solution x is an iterate from any chosen iterative method, and we stop the iteration and accept x as a valid computed solution when the backward error (or an estimate of the backward error) is smaller than a chosen (relative) tolerance.…”

“…This estimate involves a regularized projection of the residual vector r = b−Ax. Methods for computing ν were considered by Grcar, Saunders, and Su [9] (see also the thesis of Su [18]) and its efficient computation in the iterative method LSQR [17,16] was proposed in [13].…”

Estimation of Backward Perturbation Bounds For Linear Least Squares Problem

“…The coefficient matrix A is"well1850.mtx", a 1850  712 matrix from the Matrix Market [7] with 8758 non-zeros entries, Applying the iterative method LSQR proposed in [6] to the least square problems (1) generates a sequence of approximate solutions { k x }. …”

“…It is also used to monitor the convergence of iterative solution methods and to design reliable stopping criteria for these methods [6]. Waldén, Karlson, and Sun [1] provided the following explicit expressions.…”