## Stochastic Perturbation Theory (1988)

Citations: | 664 - 33 self |

### BibTeX

@MISC{Stewart88stochasticperturbation,

author = {G. W. Stewart},

title = {Stochastic Perturbation Theory},

year = {1988}

}

### Years of Citing Articles

### OpenURL

### Abstract

. In this paper classical matrix perturbation theory is approached from a probabilistic point of view. The perturbed quantity is approximated by a first-order perturbation expansion, in which the perturbation is assumed to be random. This permits the computation of statistics estimating the variation in the perturbed quantity. Up to the higher-order terms that are ignored in the expansion, these statistics tend to be more realistic than perturbation bounds obtained in terms of norms. The technique is applied to a number of problems in matrix perturbation theory, including least squares and the eigenvalue problem. Key words. perturbation theory, random matrix, linear system, least squares, eigenvalue, eigenvector, invariant subspace, singular value AMS(MOS) subject classifications. 15A06, 15A12, 15A18, 15A52, 15A60 1. Introduction. Let A be a matrix and let F be a matrix valued function of A. Two principal problems of matrix perturbation theory are the following. Given a matrix E, pr...

### Citations

2019 |
Perturbation Theory For Linear Operators
- Kato
- 1980
(Show Context)
Citation Context ...ate up to terms of thesrst-order; it is the uniquesrst order approximation that is linear in E. Notes and references. For general surveys of perturbation theory for matrices and linear operators, see =-=[25]-=-,[42]. The idea of usingsrst-order expansions of nonlinear functions of random variables is by no means new. Gauss [12]{[14], used the technique to approximate the variances of parameters from nonline... |

1170 |
The Algebraic Eigenvalue Problem
- Wilkinson
- 1965
(Show Context)
Citation Context ...ctor corresponding to . x4.2. A completely dierent approach to perturbation theory for Hermitian matrices is given by Davis and Kahan [8]. The Homan{Wielandt theorem appears in [20], and Wilkinson =-=[46]-=- gives an elementary proof. x4.4. A proof of Theorem 4.8 may be found in [39]. It is possible to develop perturbation bound for spaces of singular vectors corresponding to clusters of singular values ... |

680 | Approximation Theorems of Mathematical Statistics - Serfling - 1980 |

406 |
Measurement error models
- Fuller
- 1987
(Show Context)
Citation Context ...s|techniques that require a precise knowledge of . As usual, statisticians and numerical analysts have worked on the problem without consulting one another. Fuller's book on measurement error models =-=[11]-=- is the denitive source for the statistical approach (it contains much more than the model treated here). On the numerical side, Golub and Van Loan [16] (also see [43]) have created a technique based... |

200 |
Introduction to Mathematical Statistics
- Hogg, Craig
- 1995
(Show Context)
Citation Context ...iform e. Notes and references. The background for this section will be found in almost any probability or statistics book that treats multivariate distributions. Elementary treatments may be found in =-=[21]-=-,[30]. The notation G(M;) was suggested by the use of the letter G in queuing theory to stand for a general distribution, a practice started by Kendal [26]. x2.1. The material in this section appeare... |

189 |
An analysis of the total least squares problem
- Golub, Loan
- 1980
(Show Context)
Citation Context ...her. Fuller's book on measurement error models [11] is the denitive source for the statistical approach (it contains much more than the model treated here). On the numerical side, Golub and Van Loan =-=[16]-=- (also see [43]) have created a technique based on the singular value decomposition known as total least squares, which is closely related to the statisticians technique. When is small, total least ... |

184 |
Linear Regression Analysis
- Seber, Lee
- 2003
(Show Context)
Citation Context ...a underlying this section is due to David and Stewart [6]. The present treatment is take from [41]. For an introduction to regression analysis with a survey of the errors-in-the-variables problem see =-=[31]-=-. STOCHASTIC PERTURBATION THEORY 19 When is too large, one must have recourse to other techniques|techniques that require a precise knowledge of . As usual, statisticians and numerical analysts hav... |

98 |
The rotation of eigenvectors by a perturbation
- Davis, Kahan
- 1970
(Show Context)
Citation Context ... Theorem 4.4 is not the only justication of this choice of representation. It can be shown that the singular values of P are the tangents of the canonical angles between the subspaces X and ~ X (see =-=[8]-=-,[35] for denitions). Consequently, kPk = k ~ X Xk is a bound on the separation of the two subspaces. x4.1.5. Theorem 4.5 may be found in [8]. For generalization, see [24]. x4.1.6. A proof of Theore... |

73 |
Modern probability theory and its applications
- Parzen
- 1960
(Show Context)
Citation Context ... e. Notes and references. The background for this section will be found in almost any probability or statistics book that treats multivariate distributions. Elementary treatments may be found in [21],=-=[30]-=-. The notation G(M;) was suggested by the use of the letter G in queuing theory to stand for a general distribution, a practice started by Kendal [26]. x2.1. The material in this section appeared in ... |

62 |
The probability that a numerical analysis problem is difficult
- Demmel
- 1988
(Show Context)
Citation Context ...ical. Theorem 2.8 provides a resolution of this diculty. The perturbation theory developed in this paper should not be confused with results on the properties of random matrices. For example, Demmel =-=[9]-=- considers the distance of a random matrix from a manifold of degenerate problems. Here the random matrices are not small, and the concern is not with perturbations of a matrix function. The work of W... |

48 |
Stochastic processes occurring in the theory of queues and their analysis by the method of imbedded Markov chains
- Kendall
- 1953
(Show Context)
Citation Context ...ions. Elementary treatments may be found in [21],[30]. The notation G(M;) was suggested by the use of the letter G in queuing theory to stand for a general distribution, a practice started by Kendal =-=[26]-=-. x2.1. The material in this section appeared in some lecture notes by the author (c. 1982). Theorem 2.3 has been published by Neudecker and Wansbeek [28]. Although their paper treats normal matrices,... |

37 |
The variation of the spectrum of a normal matrix
- Hoffman, Wielandt
(Show Context)
Citation Context ... y, the left eigenvector corresponding to . x4.2. A completely dierent approach to perturbation theory for Hermitian matrices is given by Davis and Kahan [8]. The Homan{Wielandt theorem appears in =-=[20]-=-, and Wilkinson [46] gives an elementary proof. x4.4. A proof of Theorem 4.8 may be found in [39]. It is possible to develop perturbation bound for spaces of singular vectors corresponding to clusters... |

34 |
Perturbation theory for pseudo-inverse
- Wedin
- 1973
(Show Context)
Citation Context ...s and references. Perturbation theory for the least squares problem begins with Golub and Wilkinson [18], who producedsrst-order expansions for least squares solutions. For surveys of the theory, see =-=[44, 35, 42]-=-. For extensions to more general problems, see [10, 29]. In the statistics literature, Hodges and Moore [19] and Davies and Hutton [7] have usedsrst order approximations to least squares solutions to ... |

23 |
Residual bounds on approximate eigensystems of nonnormal matrices
- Kahan, Parlett, et al.
(Show Context)
Citation Context ...he subspaces X and ~ X (see [8],[35] for denitions). Consequently, kPk = k ~ X Xk is a bound on the separation of the two subspaces. x4.1.5. Theorem 4.5 may be found in [8]. For generalization, see =-=[24]-=-. x4.1.6. A proof of Theorem 4.6 may be found in [34]. x4.1.8. Equation (4.24) does not represent all possible normalizations. In [27], which treats only eigenvectors, the normalizing function is allo... |

23 |
Derivatives and perturbations of eigenvectors
- Meyer, Stewart
- 1988
(Show Context)
Citation Context ... x4.1.5. Theorem 4.5 may be found in [8]. For generalization, see [24]. x4.1.6. A proof of Theorem 4.6 may be found in [34]. x4.1.8. Equation (4.24) does not represent all possible normalizations. In =-=[27]-=-, which treats only eigenvectors, the normalizing function is allowed to be any dierentiable function. Theorem 4.7 appears to be new. x4.1.9. There is some ambiguity in the term \generalized Rayleigh... |

22 |
The acceptability of regression solutions: Another look at computational accuracy
- Beaton, Rubin, et al.
- 1976
(Show Context)
Citation Context ...conometrics. x3.1.2. The presence of bias in regression coecients with errors in the variables has long been known to statisticians under the name asymptotic inconsistency; e.g., see [23, x9.4], and =-=[1, 40]-=-. x3.2.1. As mentioned above, the perturbation expansion is due to Golub and Wilkinson [18]. x3.2.2. The quantities in (3.10) were derived in [36] as sensitivity coecients for linear regression. x3.2... |

18 | Note on the iterative refinement of least squares solutions - Golub, Wilkinson - 1966 |

18 |
Error bounds for approximate invariant subspaces of closed linear operators
- Stewart
- 1971
(Show Context)
Citation Context ...from its neighbors controls the sensitivity of its singular vector to perturbations. 30 G. W. STEWART Notes and references. The general approach to invariant subspaces taken here is due to the author =-=[33]-=-,[34]. For another view of the subject the reader reader is referred to Kato's work [25], which also treats the perturbation of operators in innite-dimensional settings. x4.1.2. The term \simple" ref... |

15 |
The selection of variates for use in prediction with some comments on the general problem of nuisance parameters
- Hotelling
- 1940
(Show Context)
Citation Context ... nonlinear functions of random variables is by no means new. Gauss [12]{[14], used the technique to approximate the variances of parameters from nonlinear least squaressts. Hotelling, writing in 1940 =-=[22]-=-, refers to a \method of dierentials," with the implication that the practice was widespread. Recently Chatelin [3]{[5] has usedsrst-order expansions and random matrices to analyze the eects of roun... |

14 |
Theoria Motus Corporum Coelestium
- GAUSS
- 1809
(Show Context)
Citation Context ...or general surveys of perturbation theory for matrices and linear operators, see [25],[42]. The idea of usingsrst-order expansions of nonlinear functions of random variables is by no means new. Gauss =-=[12]-=-{[14], used the technique to approximate the variances of parameters from nonlinear least squaressts. Hotelling, writing in 1940 [22], refers to a \method of dierentials," with the implication that t... |

11 |
Computer solution and perturbation analysis of generalized linear least squares problems
- Paige
- 1979
(Show Context)
Citation Context ... problem begins with Golub and Wilkinson [18], who producedsrst-order expansions for least squares solutions. For surveys of the theory, see [44, 35, 42]. For extensions to more general problems, see =-=[10, 29]-=-. In the statistics literature, Hodges and Moore [19] and Davies and Hutton [7] have usedsrst order approximations to least squares solutions to assess the eects of errors in the regression matrix|th... |

6 |
The effect of errors in the independent variables in linear regression. Biometrika
- Davis, Hutton
- 1975
(Show Context)
Citation Context ... least squares solutions. For surveys of the theory, see [44, 35, 42]. For extensions to more general problems, see [10, 29]. In the statistics literature, Hodges and Moore [19] and Davies and Hutton =-=[7]-=- have usedsrst order approximations to least squares solutions to assess the eects of errors in the regression matrix|the problem of errors in the variables as it is known. See also [2] for applicati... |

6 |
Data uncertainties and least squares regression
- Hodges, Moore
- 1972
(Show Context)
Citation Context ...edsrst-order expansions for least squares solutions. For surveys of the theory, see [44, 35, 42]. For extensions to more general problems, see [10, 29]. In the statistics literature, Hodges and Moore =-=[19]-=- and Davies and Hutton [7] have usedsrst order approximations to least squares solutions to assess the eects of errors in the regression matrix|the problem of errors in the variables as it is known. ... |

5 |
Analyse statistique de la qualite numerique et arithmetic de la resolution approchee d'equations par calcul sur ordinateur, Etude F.133, Centre Scienti de
- Chatelin
- 1988
(Show Context)
Citation Context ...ariances of parameters from nonlinear least squaressts. Hotelling, writing in 1940 [22], refers to a \method of dierentials," with the implication that the practice was widespread. Recently Chatelin =-=[3]-=-{[5] has usedsrst-order expansions and random matrices to analyze the eects of rounding error on numerical calculations. As we have pointed out, the chief diculty with this approach is that 4 G. W. ... |

5 | Econometric Methods, McGraw-Hill - Johnston, DiNardo - 1997 |

4 |
and perturbation bounds for subspaces associated with certain eigenvalue problems
- Error
- 1973
(Show Context)
Citation Context ...its neighbors controls the sensitivity of its singular vector to perturbations. 30 G. W. STEWART Notes and references. The general approach to invariant subspaces taken here is due to the author [33],=-=[34]-=-. For another view of the subject the reader reader is referred to Kato's work [25], which also treats the perturbation of operators in innite-dimensional settings. x4.1.2. The term \simple" referrin... |

3 |
Huffel, Analysis of the Total Least Squares Problem and Its Use
- Van
- 1987
(Show Context)
Citation Context ...ook on measurement error models [11] is the denitive source for the statistical approach (it contains much more than the model treated here). On the numerical side, Golub and Van Loan [16] (also see =-=[43]-=-) have created a technique based on the singular value decomposition known as total least squares, which is closely related to the statisticians technique. When is small, total least squares and lea... |

2 |
The asymptotic bias and mean-squared error of double K-class estimators when the distrubances are small
- Brown, Kadane, et al.
- 1974
(Show Context)
Citation Context ...es and Hutton [7] have usedsrst order approximations to least squares solutions to assess the eects of errors in the regression matrix|the problem of errors in the variables as it is known. See also =-=[2]-=- for applications to econometrics. x3.1.2. The presence of bias in regression coecients with errors in the variables has long been known to statisticians under the name asymptotic inconsistency; e.g.... |

2 | Hypothesis testing with errors in the variables
- David, Stewart
- 1988
(Show Context)
Citation Context ...(P ) and R( ~ P ) (see [35] for denitions). Thus the bounds estimate how far the column space of ~ A deviates from that of A. x3.4. The basic idea underlying this section is due to David and Stewart =-=[6]-=-. The present treatment is take from [41]. For an introduction to regression analysis with a survey of the errors-in-the-variables problem see [31]. STOCHASTIC PERTURBATION THEORY 19 When is too lar... |

1 | l'utilisation en calcul matriciel de modeles probabilistes pour la simulation des erreurs de calcul, Comptes Rendus de l'Academie des Sciences - De - 1988 |

1 | Perturbation theory for the least squares problem with linear equality constraints - en - 1980 |

1 |
A limit theorm for the norm of random matrices
- Geman
- 1980
(Show Context)
Citation Context ...s the dierence in our favor, since kEk F in (1.2) could be replaced by the spectral norm dened below by (1.5). However, a result on the limiting behavior of the spectral norm of stochastic matrices =-=[15]-=- shows that p 2n is a reasonable estimate of kEk, so that (1.2) will still be an overestimate. STOCHASTIC PERTURBATION THEORY 3 In principle, the answer to thesrst question is that given thesrst and ... |

1 |
Fourth-order properties of normally distributed random matrices
- Neudecker, Wansbeek
- 1987
(Show Context)
Citation Context ...al distribution, a practice started by Kendal [26]. x2.1. The material in this section appeared in some lecture notes by the author (c. 1982). Theorem 2.3 has been published by Neudecker and Wansbeek =-=[28]-=-. Although their paper treats normal matrices, their proof is quite general. x2.2. The formal use of the function E[trace(X T X)] as a norm on randommatrices appears to be new. Its major problem is th... |

1 |
Average condition number for solving linear equations
- Weis, Wasilkowski, et al.
- 1986
(Show Context)
Citation Context ...he distance of a random matrix from a manifold of degenerate problems. Here the random matrices are not small, and the concern is not with perturbations of a matrix function. The work of Weiss et al. =-=[45]-=- is closer to ours in that they assume their random errors are small enough to ignore higher-order terms; but their concern is with evaluating average condition numbers, not with perturbation theory a... |

1 |
en, Perturbation theory for the least squares problem with linear equality constraints
- Eld
- 1980
(Show Context)
Citation Context ... problem begins with Golub and Wilkinson [18], who producedsrst-order expansions for least squares solutions. For surveys of the theory, see [44, 35, 42]. For extensions to more general problems, see =-=[10, 29]-=-. In the statistics literature, Hodges and Moore [19] and Davies and Hutton [7] have usedsrst order approximations to least squares solutions to assess the eects of errors in the regression matrix|th... |

1 |
Note on the iterative re of least squares solution
- Golub, Wilkinson
- 1966
(Show Context)
Citation Context ...cient. Simulations suggest that least squares analysis can be trusted when is less than 0:3. Notes and references. Perturbation theory for the least squares problem begins with Golub and Wilkinson =-=[18]-=-, who producedsrst-order expansions for least squares solutions. For surveys of the theory, see [44, 35, 42]. For extensions to more general problems, see [10, 29]. In the statistics literature, Hodge... |