Estimation of the backward errors for scaled total least squares problems

Zhanshan Yang; Bing Zheng; Tiexiang Li; Zhanshan Yang; Bing Zheng; Tiexiang Li

doi:10.3934/era.2026359

Electronic Research Archive

2026, Volume 34, Issue 11: 8488-8514. doi: 10.3934/era.2026359

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Estimation of the backward errors for scaled total least squares problems

1.
School of Mathematics and Statistic, Qinghai Minzu University, Xining 810007, China
2.
School of Mathematics and Statistic, Lanzhou University, Lanzhou 730000, China
3.
School of Mathematics, Southeast University, Nanjing 210096, China

Received: 09 July 2026 Revised: 03 September 2026 Accepted: 11 September 2026 Published: 08 October 2026

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.
- scaled total least squares problem,
- core problem,
- Krylov subspace,
- Lanczos bidiagonalization,
- backward error
Citation: Zhanshan Yang, Bing Zheng, Tiexiang Li. Estimation of the backward errors for scaled total least squares problems[J]. Electronic Research Archive, 2026, 34(11): 8488-8514. doi: 10.3934/era.2026359

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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.

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