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On solution of total least squares problems with multiple right-hand sides. (English) Zbl 1396.65090
Summary: Consider a linear approximation problem \(AX\approx B\) with multiple right-hand sides. When errors in the data are confirmed both to \(B\) and \(A\), the total least squares (TLS) concept is used to solve this problem. Contrary to the standard least squares approximation problem, a solution of the TLS problem may not exist. For a single (vector) right-hand side, the classical theory has been developed by G. H. Golub and C. F. Van Loan [SIAM J. Numer. Anal. 17, 883–893 (1980; Zbl 0468.65011)], and S. van Huffel and J. Vandewalle [The total least squares problem: computational aspects and analysis. Philadelphia, PA: SIAM (1991; Zbl 0789.62054)], and then complemented recently by the core problem approach of C. C. Paige and the third author [Numer. Math. 91, No. 1, 117–146 (2002; Zbl 0998.65046); in: Total least squares and errors-in-variables modeling. Analysis, algorithms and applications. Dordrecht: Kluwer Academic Publishers. 25–34 (2002; Zbl 1001.65033); SIAM J. Matrix Anal. Appl. 27, No. 3, 861–875 (2006; Zbl 1097.15003)]. Analysis of the problem with multiple right-hand sides is still under development. In this short contribution we present conditions for the existence of a TLS solution.

MSC:
65F22 Ill-posedness and regularization problems in numerical linear algebra
65F20 Numerical solutions to overdetermined systems, pseudoinverses
15A06 Linear equations (linear algebraic aspects)
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