# zbMATH — the first resource for mathematics

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)
Full Text: