Solvability of the Core Problem with Multiple Right-Hand Sides in the TLS Sense

Abstract

Recently it was shown how necessary and sufficient information for solving an orthogonally invariant linear approximation problem $AX\approx B$ with multiple right-hand sides can be revealed through the so-called core problem reduction; see [I. Hnětynkova, M. Plesinger, and Z. Strakos, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 917--931]. The total least squares (TLS) serves as an important example of such approximation problem. Solvability of TLS was discussed in the full generality in [I. Hnětynkova et al., SIAM J. Matrix Anal. Appl., 32 (2011), pp. 748--770]. This theoretical study investigates solvability of core problems with multiple right-hand sides in the TLS sense. It is shown that, contrary to the single right-hand side case, a core problem with multiple right-hand sides may not have a TLS solution. Further possible internal structure of core problems is studied. Outputs of the classical TLS algorithm for the original problem $AX\approx B$ and for the core problem within $AX\approx B$ are compared.