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Further bounds for the smallest singular value and the spectral condition number. (English) Zbl 0947.15012
Let \({\mathbf A}\) be an \(n\times n\) complex matrix, and let \(\sigma _1({\mathbf A})\geq \sigma _2({\mathbf A})\geq\dots\geq \sigma _n({\mathbf A})\) be the singular values of \({\mathbf A}\). The author shows how to construct an increasing sequence of lower bounds for \(\sigma _n({\mathbf A})\) which improves the bounds of Y. Yu and D. Gu [Linear Algebra Appl. 253, 25-38 (1997; Zbl 0876.15015)]. The spectral condition number \(\kappa _2({\mathbf A})= \sigma _1({\mathbf A})/\sigma _n({\mathbf A})\) measures the sensibility of the solution of \({\mathbf A}{\mathbf x}={\mathbf b}\) to errors in the data or to round-off errors, and one can estimate \(\kappa _2({\mathbf A})\) using a lower bound for \(\sigma _n({\mathbf A})\) and an upper bound for \(\sigma _1({\mathbf A})\). New upper bounds for \(\kappa _2({\mathbf A})\) are derived in the last part of the article.

MSC:
15A42 Inequalities involving eigenvalues and eigenvectors
15A12 Conditioning of matrices
65F35 Numerical computation of matrix norms, conditioning, scaling
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[1] Hong, Y.P.; Pan, C.T., Lower bound for the smallest singular value, Linear algebra and its applications, 172, 27-32, (1992) · Zbl 0768.15012
[2] Yu, Y.-S.; Gu, D.-H., A note on a lower bound for the smallest singular value, Linear algebra and its applications, 253, 25-38, (1997) · Zbl 0876.15015
[3] Merikoski, J.K.; Urpala, U.; Virtanen, A.; Tam, T.Y.; Uhlig, F., Best upper bound for the 2-norm condition number of a matrix, Linear algebra and its applications, 254, 335-365, (1997) · Zbl 0877.15006
[4] Guggenheimer, H.W.; Edelman, H.W.; Johnson, C.R., A simple estimate of the condition number of a linear system, College mathematics journal, 26, 2-5, (1995) · Zbl 1291.15022
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