Backward Error and Condition of Structured Linear Systems

Higham, Desmond J.; Higham, Nicholas J.

doi:10.1137/0613014

Cited by 102 publications

(64 citation statements)

References 16 publications

(13 reference statements)

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“…Structured condition analysis has been involved in a lot of recent work, e.g. [3][4][5]. As an example, Higham and Higham [3] define the structured componentwise backward error and apply it to symmetric matrices and Toeplitz matrices as examples.…”

Section: Sce For Structured Systemsmentioning

confidence: 99%

“…[3][4][5]. As an example, Higham and Higham [3] define the structured componentwise backward error and apply it to symmetric matrices and Toeplitz matrices as examples. Here, we provide more applications where SCE clearly reflects the structured conditions.…”

Section: Sce For Structured Systemsmentioning

confidence: 99%

“…For example, for n = 4000 we have max n (2) csce = 1.4×10 2 . On the other hand, the Skeel condition number is skeel = 2.9×10 3 . In fact, the problem is considered to be ill-conditioned by condition estimators such as MATLAB routines rcond (which uses LAPACK condition estimators xGECON based on Higham's modification of Hager's method [8]) and condest (which is based on Hager's method and a block generalization by Higham and Tisseur [9]).…”

Section: Examplementioning

confidence: 99%

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Applications of statistical condition estimation to the solution of linear systems

Laub¹,

Xia²

2008

Numerical Linear Algebra App

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SUMMARYThis paper discusses some applications of statistical condition estimation (SCE) to the problem of solving linear systems. Specifically, triangular and bidiagonal matrices are studied in some detail as typical of structured matrices. Such a structure, when properly respected, leads to condition estimates that are much less conservative compared with traditional non-statistical methods of condition estimation. Some examples of linear systems and Sylvester equations are presented. Vandermonde and Cauchy matrices are also studied as representative of linear systems with large condition numbers that can nonetheless be solved accurately. SCE reflects this. Moreover, SCE when applied to solving very large linear systems by iterative solvers, including conjugate gradient and multigrid methods, performs equally well and various examples are given to illustrate the performance. SCE for solving large linear systems with direct methods, such as methods for semi-separable structures, are also investigated. In all cases, the advantages of using SCE are manifold: ease of use, efficiency, and reliability.

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Section: Sce For Structured Systemsmentioning

confidence: 99%

Section: Sce For Structured Systemsmentioning

confidence: 99%

Section: Examplementioning

confidence: 99%

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Applications of statistical condition estimation to the solution of linear systems

Laub¹,

Xia²

2008

Numerical Linear Algebra App

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“…If we require A ∈ A we get what Bunch [17] calls strong stability. For a discussion of the difference between stability and strong stability for Toeplitz algorithms, see [46,79].…”

Section: Stability and Weak Stabilitymentioning

confidence: 99%

Stability analysis of a general toeplitz systems solver

1995

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We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that R T R is close to A T A. Thus, when the algorithm is used to solve the semi-normal equations R T Rx = A T b, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem min Ax − b 2 .

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“…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].…”

mentioning

confidence: 99%

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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Backward Error and Condition of Structured Linear Systems

Cited by 102 publications

References 16 publications

Applications of statistical condition estimation to the solution of linear systems

Applications of statistical condition estimation to the solution of linear systems

Stability analysis of a general toeplitz systems solver

Characterizing Matrices That Are Consistent with Given Solutions

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