Algebraic relations between the total least squares and least squares problems with more than one solution.

*(English)*Zbl 0761.65030This paper completes our discussion on the total least squares (TLS) and the least squares (LS) problems for the linear system \(AX=B\) which may contain more than one solution. It generalizes the work of G. H. Golub and C. F. Van Loan [SIAM J. Numer. Analysis 17, 883-893 (1980; Zbl 0468.63011)], and of S. Van Huffel and J. Vandewalle [Numer. Math. 55, No. 4, 431-449 (1989; Zbl 0663.65038)]. The TLS problem is extended to a more general case. The sets of the solutions and the squared residuals for the TLS and LS problems are compared. The concept of the weighted squared residuals is extended and the difference between the TLS and the LS approaches is derived. The connection between the approximate subspaces and the perturbation theories are studied.

It is proved that under some moderate conditions, all the corresponding quantities for the solution sets of the TLS and the modified LS problems are close to each other, while the quantities for the solution set of the LS problem are close to the corresponding ones of a subset of that of the TLS problem.

It is proved that under some moderate conditions, all the corresponding quantities for the solution sets of the TLS and the modified LS problems are close to each other, while the quantities for the solution set of the LS problem are close to the corresponding ones of a subset of that of the TLS problem.

Reviewer: M.Wei

##### MSC:

65F20 | Numerical solutions to overdetermined systems, pseudoinverses |

15A18 | Eigenvalues, singular values, and eigenvectors |

15A09 | Theory of matrix inversion and generalized inverses |

15A06 | Linear equations (linear algebraic aspects) |

##### Keywords:

singular value decomposition; overdetermined system; perturbation bound; rank deficient; algebraic relation; total least squares; weighted squared residuals**OpenURL**

##### References:

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