Perturbation analysis and condition numbers of scaled total least squares problems (Q1027795)

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.