The Core Problem within a Linear Approximation Problem Ax ≈ B with Multiple Right-hand Sides∗

This paper focuses on total least squares (TLS) problems AX ≈ B with multiple right-hand sides. Existence and uniqueness of a TLS solution for such problems was analyzed in the paper [I. Hnětynková, M. Plešinger, D. M. Sima, Z. Strakoš, and S. Van Huffel (2011)]. For TLS problems with single right-hand sides the paper [C. C. Paige and Z. Strakoš (2006)] showed how necessary and sufficient information for solving Ax ≈ b can be revealed from the original data through the so-called core problem concept. In this paper we present a theoretical study extending this concept to problems with multiple right-hand sides. The data reduction we present here is based on the singular value decomposition of the system matrix A. We show minimality of the reduced problem; in this sense the situation is analogous to the single right-hand side case. Some other properties of the core problem, however, can not be extended to the case of multiple right-hand sides.