Backward perturbation analysis for scaled total least‐squares problems

Chang, Xio-Wen; Titley-Péloquin, David

doi:10.1002/nla.640

Cited by 13 publications

(19 citation statements)

References 15 publications

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“…It can easily be shown that µ = 0 if and only if x =x; see, e.g., [5,Corollary 3.1]. Thus as long as x =x, the matrix M in (2.2) has full column rank and σ min (M ) > 0.…”

mentioning

confidence: 99%

On the Accuracy of the Karlson--Waldén Estimate of the Backward Error for Linear Least Squares Problems

Gratton¹,

Jiránek²,

Titley-Péloquin³

2012

SIAM J. Matrix Anal. & Appl.

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Abstract. We consider the backward error associated with a given approximate solution of a linear least squares problem. The backward error can be very expensive to compute, as it involves the minimal singular value of certain matrix that depends on the problem data and the approximate solution. An estimate based on a regularized projection of the residual vector has been proposed in the literature and analyzed by several authors. Although numerical experiments in the literature suggest that it is a reliable estimate of the backward error for any given approximate LS solution, to date no satisfactory explanation for this behavior had been found. We derive new bounds which confirm this experimental observation.

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“…It can easily be shown that µ = 0 if and only if x =x; see, e.g., [5,Corollary 3.1]. Thus as long as x =x, the matrix M in (2.2) has full column rank and σ min (M ) > 0.…”

mentioning

confidence: 99%

On the Accuracy of the Karlson--Waldén Estimate of the Backward Error for Linear Least Squares Problems

Gratton¹,

Jiránek²,

Titley-Péloquin³

2012

SIAM J. Matrix Anal. & Appl.

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“…Suppose y is an approximate solution of the STLS problem . Chang and Titley‐Peloquin prove that y is the exact STLS solution for the data set [ A + E , b + f ] if and only if [ E , f ] is in the set

ϵ_{A MathClass-punc, b} (γ) MathClass-rel= ({, [E MathClass-punc, f] (|}, \begin{matrix} falsefalsearray \\ arrayleft {MathClass-open(A + E MathClass-close)}^{T} MathClass-open[b + f - MathClass-open(A + E MathClass-close) y MathClass-close] = - \frac{∥ b + f - MathClass-open(A + E MathClass-close) y ∥^{2}}{γ^{- 2} + ∥ y ∥^{2}} y, \\ arrayleft \frac{∥ b + f - MathClass-open(A + E MathClass-close) y ∥^{2}}{γ^{- 2} + ∥ y ∥^{2}} < σ_{min}^{2} MathClass-open(A + E MathClass-close) \end{matrix}) MathClass-punc.)

…”

Section: Backward Error For the Scaled Total Least Squares Problemmentioning

confidence: 99%

“…Chang and Titley‐Peloquin derive an expression for the extended backward error of the scaled total least squares (STLS) problem, which is an asymptotically tight lower bound on the true backward error. Furthermore, several results on the backward error results of OLS, the data least squares (DLS), and the total least squares (TLS) are then obtained from the backward error results for STLS in .…”

Section: Introductionmentioning

confidence: 99%

“…This idea is not new. Indeed, Higham and Higham , Grcar , and Chang and Titley‐Peloquin have suggested linearization estimate of the backward error for the OLS and the STLS problems.…”

Section: Introductionmentioning

confidence: 99%

“…The rest of the paper is organized as follows. In Section 2, we give further discussion on the relationship between the backward error and its linearization estimate for the STLS problem obtained by Chang and Titley‐Peloquin [, Theorem 3.1] and then study an asymptotic estimate of the backward error for the OLS problem and the DLS problem. Numerical tests are used to validate our results.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Linearization estimates of the backward errors for least squares problems

Liu

Zhao

2011

Numerical Linear Algebra App

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SUMMARY We study the linearization method to estimate the backward error of approximate solutions to several least squares problems, including the scaled total least squares (STLS) problem, the equality constrained least squares (LSE) problem, and the least squares problem over a sphere (LSS). For the STLS problem, we present several new results about the linearization estimate derived by Chang and Titley‐Peloquin. For the LSE and the LSS problems, we derive linearization estimates of the backward errors and compare them with existing backward error bounds by numerical tests. Our experiments show that the linearization estimates are good enough approximations of the backward errors as the approximate solution approaches the exact one. Copyright © 2011 John Wiley & Sons, Ltd.

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Randomized algorithms for total least squares problems

Xie

Xiang

Wei

2018

Numerical Linear Algebra App

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Motivated by the recently popular probabilistic methods for low-rank approximations and randomized algorithms for the least squares problems, we develop randomized algorithms for the total least squares problem with a single right-hand side. We present the Nyström method for the medium-sized problems. For the large-scale and ill-conditioned cases, we introduce the randomized truncated total least squares with the known or estimated rank as the regularization parameter. We analyze the accuracy of the algorithm randomized truncated total least squares and perform numerical experiments to demonstrate the efficiency of our randomized algorithms. The randomized algorithms can greatly reduce the computational time and still maintain good accuracy with very high probability.

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Backward perturbation analysis for scaled total least‐squares problems

Cited by 13 publications

References 15 publications

On the Accuracy of the Karlson--Waldén Estimate of the Backward Error for Linear Least Squares Problems

On the Accuracy of the Karlson--Waldén Estimate of the Backward Error for Linear Least Squares Problems

Linearization estimates of the backward errors for least squares problems

Randomized algorithms for total least squares problems

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