Year: 2017
Author: Lingsheng Meng, Bing Zheng
Journal of Computational Mathematics, Vol. 35 (2017), Iss. 2 : pp. 169–186
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].
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.1608-m2015-0279
Journal of Computational Mathematics, Vol. 35 (2017), Iss. 2 : pp. 169–186
Published online: 2017-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 18
Keywords: Tikhonov regularization Discrete ill-posed problem Structured least squares problem Structured condition number.