Further lower bounds for the smallest singular value

Johnson, Charles R.; Szulc, Tomasz

doi:10.1016/s0024-3795(97)00330-3

Cited by 47 publications

(25 citation statements)

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“…In Section 4, we obtain the bounds that are complementary with the bounds from Section 3 and which can be applied to more general matrices. These bounds, as shown in an example, can also improve bounds obtained in [6] and [8]. The paper is closed with conclusions presented in Section 5.…”

supporting

confidence: 73%

“…Due to the relationship with our results, we also mention Gersgorin-based bounds given in [6], [7] and [9] and bounds based on some variants of Gersgorin theorem proposed in [8].…”

mentioning

confidence: 64%

“…We show that these bounds are sharper than those of [8] for the matrices considered therein. A limitation of the proposed bounds follows from the fact that they provide nontrivial results only for matrices which, by means of unitary transformations, become matrices whose Hermitian parts are strictly diagonally dominant.…”

mentioning

confidence: 76%

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Loewner matrix ordering in estimation of the smallest singular value

Peña¹,

Szulc²

2011

ELA

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Abstract. In this paper, some new lower bounds for the smallest singular value of a square complex matrix A are derived. A key tool to obtain these bounds is using some Hermitian matrices which are, in the sense of Loewner matrix ordering, below the Hermitian part of A or, more generally, below the Hermitian part of the unitary equivalences of A. Two types of new bounds are proposed. The first bound can be applied to matrices with positive diagonal entries and strictly diagonally dominant Hermitian parts. It is always at least as large (never worse) as pure Gersgorin-based bound due to C.R. Johnson [C.R. Johnson. A Gersgorin-type lower bound for the smallest singular value. Linear Algebra Appl., 112: 1-7, 1989.]. The other bound is complementary with the first one and it can be effectively applied to matrices whose Hermitian parts are very far from diagonal dominance.

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supporting

confidence: 73%

“…Due to the relationship with our results, we also mention Gersgorin-based bounds given in [6], [7] and [9] and bounds based on some variants of Gersgorin theorem proposed in [8].…”

mentioning

confidence: 64%

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Loewner matrix ordering in estimation of the smallest singular value

Peña¹,

Szulc²

2011

ELA

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“…The lower bound on the smallest eigenvalue for the matrix J may also be used to obtain a lower bound for the smallest singular value of a possibly nonsymmetric matrix. This lower bound seems to be much stronger than existing ones, see, e.g., [19,20,37].…”

Section: Introductionmentioning

confidence: 59%

“…Then, Johnson [19,Theorem 3] and (i, j) ∈ E(A) if and only if a i j 0 or a ji 0 . This bound is an improvement of a corresponding result in [20]. Further lower bounds for the smallest singular value may be found in [42], but they are based on the determinant of A which is expensive to compute in our case.…”

Section: Applicationmentioning

confidence: 67%

The semismooth Newton method for the solution of reactive transport problems including mineral precipitation-dissolution reactions

et al. 2010

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Optimal Gerschgorin‐type inclusion intervals of singular values

Li¹,

Huang²,

Li³

et al. 2006

Numerical Linear Algebra App

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SUMMARYIn this paper, some optimal inclusion intervals of matrix singular values are discussed in the set A of matrices equimodular with matrix A. These intervals can be obtained by extensions of the Gerschgorintype theorem for singular values, based only on the use of positive scale vectors and their intersections. Theoretic analysis and numerical examples show that upper bounds of these intervals are optimal in some cases and lower bounds may be non-trivial (i.e. positive) when PA is a H -matrix, where P is a permutation matrix, which improves the conjecture in Reference (Linear Algebra Appl. 1984; 56:105-119).

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Further lower bounds for the smallest singular value

Cited by 47 publications

References 1 publication

Loewner matrix ordering in estimation of the smallest singular value

Loewner matrix ordering in estimation of the smallest singular value

The semismooth Newton method for the solution of reactive transport problems including mineral precipitation-dissolution reactions

Optimal Gerschgorin‐type inclusion intervals of singular values

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