## Inversion Error, Condition Number, And Approximate Inverses Of Uncertain Matrices (2000)

Venue: | Inverses of Uncertain Matrices. Linear Algebra and its Applications |

Citations: | 6 - 1 self |

### BibTeX

@INPROCEEDINGS{Ghaoui00inversionerror,,

author = {Laurent El Ghaoui},

title = {Inversion Error, Condition Number, And Approximate Inverses Of Uncertain Matrices},

booktitle = {Inverses of Uncertain Matrices. Linear Algebra and its Applications},

year = {2000},

pages = {342--1}

}

### OpenURL

### Abstract

The classical condition number is a very rough measure of the effect of perturbations on the inverse of a square matrix. First, it assumes the perturbation is infinitesimally small. Second, it does not take into account the perturbation structure (e.g., Vandermonde). Similarly, the classical notion of inverse of a matrix neglects the possibility of large, structured perturbations. We define a new quantity, the structured maximal inversion error, that takes into account both structure and non necessarily small perturbation size. When the perturbation is infinitesimal, we obtain a "structured condition number". We introduce the notion of approximate inverse, as a matrix that best approximates the inverse of a matrix with structured perturbations, when the perturbation varies in a given range. For a wide class of perturbation structures, we show how to use (convex) semidefinite programming to compute bounds on on the structured maximal inversion error and structured condition number, and compute an approximate inverse. The results are exact when the perturbation is "unstructured"---we then obtain an analytic expression for the approximate inverse. When the perturbation is unstructured and additive, we recover the classical condition number; the approximate inverse is the operator related to the Total Least Squares (orthogonal regression) problem.

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Citation Context ...imation (chapter 14) and automatic error analysis (chapter 24). The present paper is also related to interval arithmetic computations, which is a large field of study, since its introduction by Moore =-=[15, 16].-=- We briefly comment on this connection in §8. The invertibility radius is related to the notion of nonsingularity radius (or distance to the nearest singular matrix). Most authors concentrated on the... |

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Citation Context ...ppendix A.sAPPROXIMATE INVERSES 3 1.3. Previous work. A complete bibliography on structured perturbations in linear algebra is clearly out of scope here. Many chapters of the excellent book by Higham =-=[14]-=- are relevant, especially the parts on error bounds for linear systems (pages 143-145), condition number estimation (chapter 14) and automatic error analysis (chapter 24). The present paper is also re... |

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Citation Context ...are discussed by Rohn in [21, 22, 23]. Alternative norms for measuring the error can be used, as pointed out by Hagher [12]. The structured condition number problem is addressed by Bartels and Higham =-=[2]-=- and by Gohberg and Koltracht [11]. The approach is based on the differentiation of a mapping describing the perturbation structure, which gives information on the effect of infinitesimal perturbation... |

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Citation Context ...rs concentrated on the case when the perturbation enters affinely in A(∆). Even in this case, computing this quantity is NP-hard, see Poljak and J. Rohn [20] and Nemirovskii [17]. Demmel [6] and Rum=-=p [24]-=- discuss bounds for the nonsingularity radius in this case. The bound proposed here is a variant of that given by Fan et al. in [9]. The maximal inversion error is closely related to systems of linear... |

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Citation Context ...information on the effect of infinitesimal perturbations. Matrix structures are described by a variety of tools. The displacement-rank model is one, see Gohberg, Kailath and Olshevsky [10] and Common =-=[5]-=-. The LFR models used here are classical in robust control (see e.g. Boyd et al. [4]). These models are used in the context of least squares problems with uncertain data by the authors in [7]. The res... |

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Citation Context ... 23]. Alternative norms for measuring the error can be used, as pointed out by Hagher [12]. The structured condition number problem is addressed by Bartels and Higham [2] and by Gohberg and Koltracht =-=[11]-=-. The approach is based on the differentiation of a mapping describing the perturbation structure, which gives information on the effect of infinitesimal perturbations. Matrix structures are described... |

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Citation Context ...d by LFR models). Exact (NP-hard) bounds on (interval) solutions to such systems are discussed by Rohn in [21, 22, 23]. Alternative norms for measuring the error can be used, as pointed out by Hagher =-=[12]-=-. The structured condition number problem is addressed by Bartels and Higham [2] and by Gohberg and Koltracht [11]. The approach is based on the differentiation of a mapping describing the perturbatio... |

1 |
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Citation Context ...imation (chapter 14) and automatic error analysis (chapter 24). The present paper is also related to interval arithmetic computations, which is a large field of study, since its introduction by Moore =-=[15, 16].-=- We briefly comment on this connection in §8. The invertibility radius is related to the notion of nonsingularity radius (or distance to the nearest singular matrix). Most authors concentrated on the... |