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In this paper, we consider the mixed and componentwise condition numbers fora linear function of the solution to the total least squares (TLS) problem. Wederive the explicit expressions of the mixed and componentwise conditionnumbers through the dual techniques. The sharp upper bounds for the derivedmixed and componentwise condition numbers are obtained. For the structured TLSproblem, we consider the structured perturbation analysis and obtain thecorresponding expressions of the mixed and componentwise condition numbers. Weprove that the structured ones are smaller than their correspondingunstructured ones based on the derived expressions. Moreover, we point out thatthe new derived expressions can recover the previous results on the conditionanalysis for the TLS problem. The numerical examples show that the derivedcondition numbers can give sharp perturbation bounds, on the other handnormwise condition numbers can severely overestimate the relative errorsbecause normwise condition numbers ignore the data sparsity and scaling.Meanwhile, from the observations of numerical examples, it is more suitable toadopt structured condition numbers to measure the conditioning for thestructured TLS problem.

Mixed and componentwise condition numbers for a linear function of the solution of the total least squares problem