Condition numbers of the least squares problems with multiple right-hand sides

Abstract

In this paper, we investigate the normwise, mixed and componentwise condition numbers of the least squares problem $\min\limits_{X\in \mathbb{R}^{n\times d}}\|AX-B\|_F$, where $A\in \mathbb{R}^{m\times n}$ is a rank-deficient matrix and $B\in \mathbb{R}^{m\times d}$. The closed formulas or upper bounds for these condition numbers are presented, which extend the earlier work for the least squares problem with single right-hand side (i.e. $B\equiv b$ is an $m$-vector) of several authors.