THE ANALYSIS FOR THE TOTAL LEAST-SQUARES PROBLEM WITH MORE THAN ONE SOLUTION

This paper presents an analysis of the solutions of the total least squares problem (TLS) AX almost-equal-to B in cases where the matrix (A, B) may have multiple smallest singular values. General formulas for the minimum norm TLS solutions are given; the difference between the TLS and the LS solutions is obtained; the error bounds for the perturbed TLS solutions with or without minimal length are deduced. The analysis is useful especially for rank deficient problems and generalizes previous results of Golub and Van Loan, Van Huffel and Vandewalle, and Zoltowski. Numerical results for a practical application are also given to verify the error bounds.