Analysis and Solution of the Nongeneric Total Least Squares Problem

Total least squares (TLS) is one method of solving overdetermined sets of linear equations AX ≈ B that is appropriate when there are errors in both the observation matrix B and the data matrix A. Golub and Van Loan (G. H. Golub and C. F. Van Loan, SIAM J. Numer. Anal., 17 (1980), pp. 883–893) introduced this method into the field of numerical analysis and developed an algorithm based on the singular value decomposition. However in some TLS problems, called nongeneric, their algorithm fails to compute a finite TLS solution. This paper generalizes their TLS computations in order to solve these nongeneric TLS problems. The authors describe the properties of those problems and prove that the proposed generalization remains optimal with respect to the TLS criteria for any number of observation vectors in B if additional constraints are imposed. Finally, the TLS computation is summarized in one algorithm which includes the proposed generalization.