Towards A Backward Perturbation Analysis For Data Least Squares Problems

Chang, Xiao-Wen; Golub, Gene H.; Paige, Christopher C.

doi:10.1137/060668626

Cited by 3 publications

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References 28 publications

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“…Interested readers can find some references on backward perturbation analysis of linear systems (including least-squares problems) in [5].…”

Section: Introductionmentioning

confidence: 99%

Backward perturbation analysis for scaled total least‐squares problems

Chang¹,

Titley-Péloquin²

2009

Numerical Linear Algebra App

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The scaled total least-squares (STLS) method unifies the ordinary least-squares (OLS), the total leastsquares (TLS), and the data least-squares (DLS) methods. In this paper we perform a backward perturbation analysis of the STLS problem. This also unifies the backward perturbation analyses of the OLS, TLS and DLS problems. We derive an expression for an extended minimal backward error of the STLS problem. This is an asymptotically tight lower bound on the true minimal backward error. If the given approximate solution is close enough to the true STLS solution (as is the goal in practice), then the extended minimal backward error is in fact the minimal backward error. Since the extended minimal backward error is expensive to compute directly, we present a lower bound on it as well as an asymptotic estimate for it, both of which can be computed or estimated more efficiently. Our numerical examples suggest that the lower bound gives good order of magnitude approximations, while the asymptotic estimate is an excellent estimate. We show how to use our results to easily obtain the corresponding results for the OLS and DLS problems in the literature.

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“…Interested readers can find some references on backward perturbation analysis of linear systems (including least-squares problems) in [5].…”

Section: Introductionmentioning

confidence: 99%

Backward perturbation analysis for scaled total least‐squares problems

Chang¹,

Titley-Péloquin²

2009

Numerical Linear Algebra App

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“…The result of Theorem 5.1 is used in [2] for the backward perturbation analysis for the DLS problem.…”

Section: Minimal Backward Errors and Acceptablementioning

confidence: 99%

“…If we only had y ≈x then finding F as close as possible to A would tend to force (3.7) to hold. A good example of this is in [2] which deals with the minimum backward error for an approximate solution y to the DLS problem, see section 5. Using the notation in (5.4), in [2] we used…”

Section: Minimal Backward Errors and Acceptablementioning

confidence: 99%

“…But theoretical arguments and numerical experiments given in [2] have shown that when y is a reasonable approximation to the solution of the DLS problem for Ax ≈ b, the set N DLS+ can usually be used with no further constraints to obtain the minimal backward errors for the DLS problem. This is probably true for the STLS problem as well.…”

Section: Minimal Backward Errors and Acceptablementioning

confidence: 99%

“…There has been a lot of work on backward error problems, especially in recent years. For example for consistent linear systems (including structured problems), see [10], [12], [19], [24], [25], [30] and [33]; for unconstrained least squares problems, see [6], [8], [11]- [14], [20]- [23] and [31]; for constrained least squares problems, see [3], [13] and [14]; and for data least squares problems, see [2].…”

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Characterizing Matrices That Are Consistent with Given Solutions

Chang

Paige²,

Titley-Péloquin³

2009

SIAM J. Matrix Anal. & Appl.

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Abstract. For given vectors b ∈ C m and y ∈ C n we describe a unitary transformation approach to deriving the set F of all matrices F ∈ C m×n such that y is an exact solution to the compatible system F y = b. This is used for deriving minimal backward errors E and f such that (A+E)y = b+f when possibly noisy data A ∈ C m×n and b ∈ C m are given, and the aim is to decide if y is a satisfactory approximate solution to Ax = b. The approach might be different, but the above results are not new. However we also prove the apparently new result that two well known approaches to making this decision are theoretically equivalent, and discuss how such knowledge can be used in designing effective stopping criteria for iterative solution techniques. All these ideas generalize to the following formulations. We extend our constructive approach to derive a superset F ST LS+ of the set F ST LS of all matrices F ∈ C m×n such that y is a scaled total least squares solution to F y ≈ b. This is a new general result that specializes in two important ways. The ordinary least squares problem is an extreme case of the scaled total least squares problem, and we use our result to obtain the set F LS of all matrices F ∈ C m×n such that y is an exact least squares solution to F y ≈ b. This complements the original less-constructive derivation of Waldén, Karlson and Sun [Numerical Linear Algebra with Applications, 2:271-286 (1995)]. We do the equivalent for the data least squares problem-the other extreme case of the scaled total least squares problem. Not only can the results be used as indicated above for the compatible case, but the constructive technique we use could also be applicable to other backward problems-such as those for under-determined systems, the singular value decomposition, and the eigenproblem.Key words. matrix characterization, approximate solutions, iterative methods, linear algebraic equations, least squares, data least squares, total least squares, scaled total least squares, backward errors, stopping criteria.AMS subject classifications. 15A06, 15A29, 65F05, 65F10, 65F20, 65F25, 65G99. DOI. (Digital Object Identifier)1. Introduction. We will study a class of 'backward' problems for linear systems F y ≈ b. Specifically, given two vectors y and b we want to find the sets of all matrices F such that y is the exact solution (i.e., F y = b), the least squares (LS) solution, the data least squares (DLS) solution, and the scaled total least square (STLS) solution. We will propose a unified unitary transformation approach to handling these problems.Some of these problems have been investigated before. The result for the compatible case is well-known and the result for the least squares case was obtained elegantly by Waldén, Karlson and Sun in [31]. But while [31] presents, then proves, the least squares result, our approach shows how to derive a more general result in a fairly simple way, and we suspect that this constructive approach is not only easier to comprehend for non-mathematicians, but perhaps easier to apply to ot...

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Linearization estimates of the backward errors for least squares problems

Liu

Zhao

2011

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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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Towards A Backward Perturbation Analysis For Data Least Squares Problems

Cited by 3 publications

References 28 publications

Backward perturbation analysis for scaled total least‐squares problems

Backward perturbation analysis for scaled total least‐squares problems

Characterizing Matrices That Are Consistent with Given Solutions

Linearization estimates of the backward errors for least squares problems

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