In this paper, we present explicit expressions for the mixed and componentwise condition numbers of the truncated total least squares (TTLS) solution of $A\mathit{x}\approx \mathit{b}$ under the genericity condition, where A is a m × n real data matrix and $\mathit{b}$ 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 $[A\mathit{b}]$ 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.

中文翻译：

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截断总最小二乘问题的条件数及其估计

在本文中，我们给出了截断总最小二乘 (TTLS) 解的混合和分量条件数的显式表达式 $\mathrm{一种}\mathit{X}\approx \mathit{乙}$在通用性条件下，其中A是一个m  ×  n实数数据矩阵，并且$\mathit{乙}$是一个实数m向量。此外，我们揭示了当 TTLS 问题的截断级别为n 时，TTLS 问题的规范、组件和混合条件数可以恢复总最小二乘 (TLS) 问题的先前对应对应项。当A是结构化矩阵，研究了结构化截断 TLS (STTLS) 问题的结构化扰动，并获得了 STTLS 问题的结构化范数、分量和混合条件数的相应显式表达式。此外，研究了 STTLS 问题的结构化和非结构化规范、组件和混合条件数之间的关系。我们利用小样本统计条件估计技术为 TTLS 问题设计了可靠的条件估计算法。所提出的条件估计算法采用增广矩阵的奇异值分解 (SVD)$[\mathrm{一种}\mathit{乙}]$以降低计算复杂度，其中考虑了非结构化和结构化规范、混合和组件条件估计。所提出的条件估计算法可以集成到基于 SVD 的中小型 TTLS 问题的直接求解器中，以给出数值 TTLS 解的误差估计。数值实验报告说明所提出的条件估计算法的可靠性。