Structured Condition Numbers for the Tikhonov Regularization of Discrete Ill-Posed Problems

Abstract

The possibly most popular regularization method for solving the least squares problem $\mathop{\rm min}\limits_x$$||Ax-b||_2$ with a highly ill-conditioned or rank deficient coefficient matrix $A$ is the Tikhonov regularization method. In this paper we present the explicit expressions of the normwise, mixed and componentwise condition numbers for the Tikhonov regularization when $A$ has linear structures. The structured condition numbers in the special cases of nonlinear structure i.e. Vandermonde and Cauchy matrices are also considered. Some comparisons between structured condition numbers and unstructured condition numbers are made by numerical experiments. In addition, we also derive the normwise, mixed and componentwise condition numbers for the Tikhonov regularization when the coefficient matrix, regularization matrix and right-hand side vector are all perturbed, which generalize the results obtained by Chu et al. [Numer. Linear Algebra Appl., 18 (2011), 87-103].