Perturbation analysis and condition numbers for the Tikhonov regularization of total least squares problem and their statistical estimation

Abstract

Condition number plays an important role in perturbation analysis, the latter is a tool to judge whether a numerical solution makes sense, especially for ill-posed problems. In this paper, perturbation analysis of the Tikhonov regularization of total least squares problem (TRTLS) is considered. The explicit expressions of normwise, mixed and componentwise condition numbers for the TRTLS problem are first presented. With the intermediate result, i.e. normwise condition number, we can recover the upper bound of TRTLS problem. To improve the computational efficiency in calculating the normwise condition number, a new compact and tight upper bound of the TRTLS problem is introduced. In addition, we also derive the normwise, mixed and componentwise condition numbers for TRTLS problem when the coefficient matrix, regularization matrix and right-hand side vector are all perturbed. We choose the probabilistic spectral norm estimator and the small-sample statistical condition estimation method to estimate these condition numbers with high reliability. Numerical experiments are provided to verify the obtained results.