On Two-Sided Bounds Related to Weakly Diagonally Dominant <i>M</i>-Matrices with Application to Digital Circuit Dynamics

Shivakumar, P. N.; Williams, J. J.; Ye, Qiang; Marinov, Corneliu A.

doi:10.1137/s0895479894276370

Cited by 61 publications

(46 citation statements)

References 11 publications

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“…Among problems of interest are solving a linear system Ax = b and finding the smallest eigenvalue of A (corresponding to the Perron root of the inverse); see [2,12,21,22]. There are many well established numerical methods for solving such problems and they lead to a backward stable solution, which has an error depending on the condition of the problem.…”

Section: Introductionmentioning

confidence: 99%

Accurate computation of the smallest eigenvalue of a diagonally dominant $M$-matrix

Alfa

Xue

2001

Math. Comp.

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Abstract. If each off-diagonal entry and the sum of each row of a diagonally dominant M -matrix are known to certain relative accuracy, then its smallest eigenvalue and the entries of its inverse are known to the same order relative accuracy independent of any condition numbers. In this paper, we devise algorithms that compute these quantities with relative errors in the magnitude of the machine precision. Rounding error analysis and numerical examples are presented to demonstrate the numerical behaviour of the algorithms.

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Section: Introductionmentioning

confidence: 99%

Accurate computation of the smallest eigenvalue of a diagonally dominant $M$-matrix

Alfa

Xue

2001

Math. Comp.

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“…An idea that has played an important role in deriving strong perturbation bounds for diagonally dominant matrices is to reparameterize the matrix in terms of its diagonally dominant parts and off-diagonal entries (see [45] We note that the diagonally dominant parts v have been introduced as parameters to represent matrices in [2,3] for diagonally dominant M -matrices and in [45,46] for diagonally dominant matrices, but they have also been used previously in the literature for various other purposes (see [1,40,42,43]). Clearly, v ≥ 0 if and only if A is row diagonally dominant and its diagonal entries are nonnegative.…”

Section: Preliminaries and Examplementioning

confidence: 99%

Relative Perturbation Theory for Diagonally Dominant Matrices

Dailey¹,

Dopico²,

Ye³

2014

SIAM J. Matrix Anal. & Appl.

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Abstract. In this paper, strong relative perturbation bounds are developed for a number of linear algebra problems involving diagonally dominant matrices. The key point is to parameterize diagonally dominant matrices using their off-diagonal entries and diagonally dominant parts and to consider small relative componentwise perturbations of these parameters. This allows us to obtain new relative perturbation bounds for the inverse, the solution to linear systems, the symmetric indefinite eigenvalue problem, the singular value problem, and the nonsymmetric eigenvalue problem. These bounds are much stronger than traditional perturbation results, since they are independent of either the standard condition number or the magnitude of eigenvalues/singular values. Together with previously derived perturbation bounds for the LDU factorization and the symmetric positive definite eigenvalue problem, this paper presents a complete and detailed account of relative structured perturbation theory for diagonally dominant matrices.

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“…Note that any nonzero 1 1 matrix is irreducible. A matrix A D .a ij / 2 C n n is called a weakly chained diagonally dominant M -matrix [4] if A satisfies the following conditions: (i) For all i; j 2 N with i ¤ j , a ij Ä 0 and a i i > 0;…”

Section: Introductionmentioning

confidence: 99%

“…Estimating the bounds for the minimum eigenvalue .A/ of an M -matrix A is an interesting subject in matrix theory, and has important applications in many practical problems [4][5][6] …”

Section: Introductionmentioning

confidence: 99%