Perturbation analysis and condition numbers of scaled total least squares problems
type: Item
instance of: scholarly article
title: Perturbation analysis and condition numbers of scaled total least squares problems
zbMATH Open document ID: 1171.65031
author: Liangmin Zhou
author: Lijing Lin
author: Sanzheng Qiao
author: Yimin Wei
published in: Numerical Algorithms
publication date: 2009-06-30T00:00:00Z
review text: Scaled total least squares (STLS) is a generalization of total least squares (TLS). The problem is to approximate the solution \(x\) of the linear system \(Ax=b\) by \(y\), satisfying \((A+E)y=\lambda b-r\). Here \([r~E]\) is the minimal perturbation (minimal in Frobenius norm \(\|r~E\|_F\)), needed to make the latter system solvable. The scaling parameter \(\lambda\) is positive. TLS is a special case corresponding to \(\lambda=1\). First properties of TLS are recalled and used to show when STLS has a unique solution. Then a sharp, but impractical, relative condition number (in 2-norm) for the STLS problem is derived. However, it implies that the gap \(\hat{\sigma}_n-\sigma_{n+1}\) (assumed positive) with \(\hat{\sigma}_n\) and \(\sigma_{n+1}\) the smallest singular values of \(A\) and \([A~\lambda b]\) respectively is the essential determining element. The larger the gap, the better the condition. Similarly relative condition numbers in \(\infty\)-norm, both mixed (ratio of norms) and componentwise (norm of componentwise ratios) are obtained.
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 65F20
Mathematics Subject Classification ID: 65F22
zbMATH DE Number: 5571648
zbMATH Keywords: scaled total least squares
zbMATH Keywords: condition number
zbMATH Keywords: perturbation analysis
zbMATH Keywords: error bound
zbMATH Keywords: scaling
zbMATH Keywords: singular values
describes a project that uses: mctoolbox
describes a project that uses: VanHuffel
MaRDI profile type: Publication
full work available at URL: s11075-009-9269-0
OpenAlex ID: W2052261993
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DOI: 10.1007/S11075-009-9269-0
description: scientific article; zbMATH DE number 5571648