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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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##### References:
 [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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