Bounds for the least squares distance using scaled total least squares

Summary.

The standard approaches to solving overdetermined linear systems $Bx \approx c$ construct minimal corrections to the data to make the corrected system compatible. In ordinary least squares (LS) the correction is restricted to the right hand side c, while in scaled total least squares (STLS) [14,12] corrections to both c and B are allowed, and their relative sizes are determined by a real positive parameter $\gamma$. As $\gamma \rightarrow 0$, the STLS solution approaches the LS solution. Our paper [12] analyzed fundamentals of the STLS problem. This paper presents a theoretical analysis of the relationship between the sizes of the LS and STLS corrections (called the LS and STLS distances) in terms of $\gamma$. We give new upper and lower bounds on the LS distance in terms of the STLS distance, compare these to existing bounds, and examine the tightness of the new bounds. This work can be applied to the analysis of iterative methods which minimize the residual norm, and the generalized minimum residual method (GMRES) [15] is used here to illustrate our theory.