Estimation of the backward errors for scaled total least squares problems

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Abstract

In this paper, we prove that both the extended minimal backward errors and the true minimal backward errors of the scaled total least squares (STLS) problem are identical to those of its core problem. This conclusion also holds for the asymptotic estimates of the extended minimal backward errors. Benefiting from its lower dimensionality, the core problem can effectively reduce the computational cost of backward error evaluation. We further derive practical, low-cost computable backward error estimates for a STLS problem using the Lanczos bidiagonalization process, and we demonstrate that our results can be readily extended to standard least squares and data least squares problems. To improve the efficiency of the Lanczos bidiagonalization-based TLS algorithm, we propose practical stopping criteria for the iterative numerical solution of STLS problems. Numerical experiments fully validate the computational advantages of our proposed approaches.

Keywords

scaled total least squares problem / core problem / Krylov subspace / Lanczos bidiagonalization / backward error

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