Mixed and componentwise condition numbers for a linear function of the solution of the total least squares problem
For researchers in numerical linear algebra, this work provides more accurate condition numbers for TLS problems, particularly for structured data, though it is an incremental extension of existing condition analysis techniques.
This paper derives explicit expressions for mixed and componentwise condition numbers for a linear function of the solution to the total least squares problem, showing they provide sharper perturbation bounds than normwise condition numbers, which overestimate errors due to ignoring data sparsity and scaling.
In this paper, we consider the mixed and componentwise condition numbers for a linear function of the solution to the total least squares (TLS) problem. We derive the explicit expressions of the mixed and componentwise condition numbers through the dual techniques. The sharp upper bounds for the derived mixed and componentwise condition numbers are obtained. For the structured TLS problem, we consider the structured perturbation analysis and obtain the corresponding expressions of the mixed and componentwise condition numbers. We prove that the structured ones are smaller than their corresponding unstructured ones based on the derived expressions. Moreover, we point out that the new derived expressions can recover the previous results on the condition analysis for the TLS problem. The numerical examples show that the derived condition numbers can give sharp perturbation bounds, on the other hand normwise condition numbers can severely overestimate the relative errors because normwise condition numbers ignore the data sparsity and scaling. Meanwhile, from the observations of numerical examples, it is more suitable to adopt structured condition numbers to measure the conditioning for the structured TLS problem.