Condition numbers for the truncated total least squares problem and their estimations

Abstract

AbstractIn this paper, we present explicit expressions for the mixed and componentwise condition numbers of the truncated total least squares (TTLS) solution of under the genericity condition, where A is a m × n real data matrix and is a real m‐vector. Moreover, we reveal that normwise, componentwise, and mixed condition numbers for the TTLS problem can recover the previous corresponding counterparts for the total least squares (TLS) problem when the truncated level of the TTLS problem is n. When A is a structured matrix, the structured perturbations for the structured truncated TLS (STTLS) problem are investigated and the corresponding explicit expressions for the structured normwise, componentwise, and mixed condition numbers for the STTLS problem are obtained. Furthermore, the relationships between the structured and unstructured normwise, componentwise, and mixed condition numbers for the STTLS problem are studied. We devise reliable condition estimation algorithms for the TTLS problem by utilizing small‐sample statistical condition estimation techniques. The proposed condition estimation algorithms employ the singular value decomposition (SVD) of the augmented matrix to reduce the computational complexity, where both unstructured and structured normwise, mixed, and componentwise condition estimations are considered. The proposed condition estimation algorithms can be integrated into the SVD‐based direct solver for the small and medium size TTLS problem to give the error estimation for the numerical TTLS solution. Numerical experiments are reported to illustrate the reliability of the proposed condition estimation algorithms.