  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.
