A Theory of Condition

Rice, John R.

doi:10.1137/0703023

Cited by 286 publications

(156 citation statements)

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“…The first two steps of this paradigm are quite standard [24], [35] ; only the third is new. This paradigm is best explained by applying it to matrix inversion:…”

Section: Tjp^w^f-ymentioning

confidence: 99%

The Probability That a Numerical Analysis Problem is Difficult

Demmel¹

1988

Mathematics of Computation

106

126

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Abstract.Numerous problems in numerical analysis, including matrix inversion, eigenvalue calculations and polynomial zerofinding, share the following property: The difficulty of solving a given problem is large when the distance from that problem to the nearest "ill-posed" one is small. For example, the closer a matrix is to the set of noninvertible matrices, the larger its condition number with respect to inversion. We show that the sets of ill-posed problems for matrix inversion, eigenproblems, and polynomial zerofinding all have a common algebraic and geometric structure which lets us compute the probability distribution of the distance from a "random" problem to the set. From this probability distribution we derive, for example, the distribution of the condition number of a random matrix. We examine the relevance of this theory to the analysis and construction of numerical algorithms destined to be run in finite precision arithmetic.

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“…The first two steps of this paradigm are quite standard [24], [35] ; only the third is new. This paradigm is best explained by applying it to matrix inversion:…”

Section: Tjp^w^f-ymentioning

confidence: 99%

The Probability That a Numerical Analysis Problem is Difficult

Demmel¹

1988

Mathematics of Computation

106

126

View full text Add to dashboard Cite

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“…To the best of our knowledge a general theory of condition numbers was first given by Rice in [12]. Let φ : R s → R t be a mapping, where R s and R t are the usual s-and t-dimensional Euclidean spaces equipped with some norms, respectively.…”

Section: General Considerationsmentioning

confidence: 99%

“…Let φ : R s → R t be a mapping, where R s and R t are the usual s-and t-dimensional Euclidean spaces equipped with some norms, respectively. If φ is continuous and Fréchet differentiable in the neighborhood of a 0 ∈ R s , then, according to [12], the relative normwise condition number of a 0 is given by cond(a 0 ) := lim ε→0 sup ∆a ≤ε…”

Section: General Considerationsmentioning

confidence: 99%

On mixed and componentwise condition numbers for Moore–Penrose inverse and linear least squares problems

2006

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Abstract. Classical condition numbers are normwise: they measure the size of both input perturbations and output errors using some norms. To take into account the relative of each data component, and, in particular, a possible data sparseness, componentwise condition numbers have been increasingly considered. These are mostly of two kinds: mixed and componentwise. In this paper, we give explicit expressions, computable from the data, for the mixed and componentwise condition numbers for the computation of the Moore-Penrose inverse as well as for the computation of solutions and residues of linear least squares problems. In both cases the data matrices have full column (row) rank.

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“…Notice that we have not specified what norms are used in this definition, but in principle one can use different norms in the data and solution spaces [38]. From this definition it is clear that the condition number κ[f (A)] is some sort of "derivative" of the function X = f (A) that we want to compute.…”

Section: Numerical Backgroundmentioning

confidence: 99%