Accurate Solution of Structured Least Squares Problems via Rank-Revealing Decompositions

Abstract

Least squares problems $\min_x \|b - Ax\|_2$ where the matrix $A\in \mathbb{C}^{m\times n}$ ($m\geq n$) has some particular structure arise frequently in applications. Polynomial data fitting is a well-known instance of problems that yield highly structured matrices, but many other examples exist. Very often, structured matrices have huge condition numbers $\kappa_2 (A) = \|A\|_2 \, \|A^\dagger\|_2$ ($A^\dagger$ is the Moore--Penrose pseudoinverse of $A$) and therefore standard algorithms fail to compute accurate minimum 2-norm solutions of least squares problems. In this work, we introduce a framework that allows us to compute minimum 2-norm solutions of many classes of structured least squares problems accurately, i.e., with errors $\| \widehat{x}_0 - x_0 \|_2 / \|x_0 \|_2 = O({\tt u})$, where ${\tt u}$ is the unit roundoff, independently of the magnitude of $\kappa_2 (A)$ for most vectors $b$. The cost of these accurate computations is $O(n^2 m)$ flops, i.e., roughly the same cost as standard algorithm...