# On the condition number of the total least squares problem

@article{Jia2011OnTC,
title={On the condition number of the total least squares problem},
author={Zhongxiao Jia and Bingyu Li},
journal={Numerische Mathematik},
year={2011},
volume={125},
pages={61-87}
}
• Published 2 October 2011
• Mathematics
• Numerische Mathematik
This paper concerns singular value decomposition (SVD)-based computable formulas and bounds for the condition number of the total least squares (TLS) problem. For the TLS problem with the coefficient matrix $$A$$ and the right-hand side $$b$$, a new closed formula is presented for the condition number. Unlike an important result in the literature that uses the SVDs of both $$A$$ and $$[A,\ b]$$, our formula only requires the SVD of $$[A,\ b]$$. Based on the closed formula, both lower and upper…
• Mathematics, Engineering
• 2016
In this paper, within a unified framework of the condition number theory we present the explicit expression of the projected condition number of the equality constrained indefinite least squares
• Mathematics
Linear and Multilinear Algebra
• 2022
It is shown that the condition numbers and perturbation bound of the TLS problem are unified in the ones of the MTLS problem.
• Mathematics
• 2016
In this paper, based on a novel definition of the condition number, the expression of the relative condition number of equality constrained indefinite least squares problem is first presented when
• Mathematics
Numerical Algorithms
• 2022
This paper is devoted to condition numbers of the total least squares problem with linear equality constraint (TLSE). With novel limit techniques, closed formulae for normwise, mixed and
• Mathematics
Numer. Algorithms
• 2022
For TLSE problems with equilibratory input data, numerical experiments illustrate that normwise condition number-based estimate is sharp to evaluate the forward error of the solution, while for sparse and badly scaled matrices, mixed and componentwise condition numbers-based estimations are much tighter.
• Mathematics
SIAM J. Matrix Anal. Appl.
• 2013
An explicit expression for the condition number of the truncated total least squares (TLS) solution of $Ax \approx b$ is presented using the notion of the Frechet derivative, and upper bounds are given, which are simple to compute and interpret.
• Mathematics
ArXiv
• 2020
Konecker-product-based formulae for normwise, mixed and componentwise condition numbers of the minimum Frobenius norm TLSE solution are given and Compact upper bounds of these condition numbers are provided to reduce the storage and computation cost.
• Mathematics
Calcolo
• 2018
In this paper, we show that the normwise condition number of the scaled total least squares problem can be transformed into a new and compact form. Considering the relationship between the scaled
• Mathematics
Calcolo
• 2018
In this paper, we show that the normwise condition number of the scaled total least squares problem can be transformed into a new and compact form. Considering the relationship between the scaled

## References

SHOWING 1-10 OF 42 REFERENCES

• Computer Science
SIAM J. Matrix Anal. Appl.
• 2000
The RQI-PCGTLS method is further developed, the choice of initial approximation and termination criteria are discussed, andumerical results confirm that the given algorithm achieves rapid convergence and good accuracy.
• Mathematics
Numerische Mathematik
• 2002
The theory reveals the necessary and sufficient condition for preserving the smallest singular value of a matrix while appending (or deleting) a column, which represents a basic matrix theory result for updating the singular value decomposition.
• Mathematics
SIAM J. Matrix Anal. Appl.
• 2011
Close formulas are derived for the condition number of a linear function of the total least-squares solution that can be computed using the singular values and the right singular vectors of [A,b] and A and an upper bound is provided that requires the computation of the largest and the smallest singular value of A.
• Mathematics
Milestones in Matrix Computation
• 2007
An algorithm for solving the TLS problem is proposed that utilizes the singular value decomposition and which provides a measure of the underlying problem''s sensitivity.
• Mathematics
Numerical Algorithms
• 2009
A perturbation analysis of the STLS problem, which is a generalization of the TLS problem, is presented, and a normwise condition number for theSTLS problem is given, which follows immediately from the authors' STLS results.
• Mathematics
SIAM J. Matrix Anal. Appl.
• 2000
Upper perturbation bounds of weighted projections associated with the WLS and WLSE problems when W ranges over the set $\cal D$ of positive diagonal matrices are derived and applied to the cases whenW ranges over a subset of real symmetric positive semidefinite matrices.
• Computer Science
SIAM J. Matrix Anal. Appl.
• 2011
This paper revisits the analysis of the total least squares (TLS) problem AX≈B with multiple right-hand sides given by Van Huffel and Vandewalle in the monograph and proposes a new classification based on properties of the singular value decomposition of the extended matrix [B|A].