The Restricted Total Least Squares Problem: Formulation, Algorithm, and Properties

Abstract

The restricted total least squares (RTLS) problem, presented in this paper, is devised for solving overdetermined sets of linear equations $AX \approx B$ in which the data $[ A;B ]$ are perturbed by errors of the form $E^ * = DEC$. D and C are known matrices and E is an arbitrary but bounded matrix. By choosing D and C appropriately, the RTLS problem formulation can handle any weighted least squares (LS), generalized LS, total LS, and generalized total LS problem. Also, equality constraints can be imposed. In order to solve these problems, a computationally efficient and numerically reliable restricted TLS algorithm, based on the restricted singular value decomposition (RSVD), of the matrix triplet $( [ A;B ],D,C )$, is developed. This RSVD is a generalization of the ordinary SVD for triple matrix products. The matrices involved may be rank-deficient and the explicit formation of matrix inverses and products is avoided. Using the RSVD, some properties of the RTLS problem are proven.