Outer Interval Solution of the Eigenvalue Problem under General Form Parametric Dependencies

Kolev, Lubomir V.

doi:10.1007/s11155-006-4875-1

Cited by 24 publications

(30 citation statements)

References 15 publications

(13 reference statements)

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“…Outer approximation was also presented by Beaumont [4]; he used a polyhedral approximation of eigenpairs and an iterative improvement. Kolev [20] developed an outer approximation algorithm for the general case with nonlinear dependencies.…”

mentioning

confidence: 99%

Bounds on Real Eigenvalues and Singular Values of Interval Matrices

Hladík¹,

Daney²,

Tsigaridas³

2010

SIAM J. Matrix Anal. & Appl.

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Abstract. We study bounds on real eigenvalues of interval matrices, and our aim is to develop fast computable formulae that produce as-sharp-as-possible bounds. We consider two cases: general and symmetric interval matrices. We focus on the latter case, since on the one hand such interval matrices have many applications in mechanics and engineering, and on the other hand many results from classical matrix analysis could be applied to them. We also provide bounds for the singular values of (generally nonsquare) interval matrices. Finally, we illustrate and compare the various approaches by a series of examples.

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mentioning

confidence: 99%

Bounds on Real Eigenvalues and Singular Values of Interval Matrices

Hladík¹,

Daney²,

Tsigaridas³

2010

SIAM J. Matrix Anal. & Appl.

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“…These methods referred to as exact methods [5,6] are based on the use of the so-called outer solutions [7,24,25] …”

Section: Remarkmentioning

confidence: 99%

“…System (23) can be solved using a method of polynomial complexity (such as the simple iteration method or the direct method from [7,Section 4].…”

Section: The Real Eigenvalue Casementioning

confidence: 99%

“…Following [1][2][3][4][5][6][7], the interval analysis approach is adopted in the present paper. According to this approach, the conventional eigenvalue problem (1) is replaced with the following 'perturbed' eigenvalue problem (e.g.…”

Section: Introductionmentioning

confidence: 99%

“…According to this approach, the conventional eigenvalue problem (1) is replaced with the following 'perturbed' eigenvalue problem (e.g. [7])…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Eigenvalue range determination for interval and parametric matrices

Kolev

2010

Circuit Theory & Apps

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SUMMARYStudy of the dynamic behaviour of linear limped parameter circuits or systems under parametric uncertainties is a well-established research area. The present paper addresses the problem of determining the exact (within rounding errors) ranges for the real and imaginary parts of an eigenvalue of real matrices whose elements are either independent intervals or linear (affine) functions of independent interval parameters. A unified method for solving the above problem is suggested in the paper. It is iterative and reduces to forming and solving, at each iteration, two real (incomplete) quadratic systems to find outer interval bounds on the set of the right and left eigenvectors associated with the eigenvalue considered. The method is applicable if: (i) the solutions of both quadratic systems involved are positive and (ii) the outer interval bounds on the eigenvectors satisfy certain constant sign conditions. It is shown that its numerical complexity is polynomial in the size n of the matrix studied. Numerical examples illustrating the applicability of the new method are provided.

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Determining the stability margin in linear interval parameter electric circuits via a DAE model

Kolev

2011

Circuit Theory & Apps

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SUMMARYThis paper addresses the problem of determining the stability margin M s in linear time-invariant electric circuits whose parameters R, L, C, M and controlled source coefficients are allowed to take on fixed but unknown values within corresponding preset intervals. A method for solving the above robust stability problem is suggested. Unlike previous results resorting to the state-space formulation, the new method is based on an appropriate description of the circuit considered using a specific model in semi-state form. The set of differential algebraic equations (DAEs) thus obtained contain the circuit parameters in a linear manner. The latter property permits M s to be determined in an efficient way using the right endpoint of the range related to the real part of the first (closest to the imaginary axis) eigenvalue of a corresponding interval generalized eigenvalue problem. The new method is shown to have polynomial numerical complexity with respect to the size of the generalized eigenvalue problem. Two numerical examples illustrating the new DAE approach are provided.

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Outer Interval Solution of the Eigenvalue Problem under General Form Parametric Dependencies

Cited by 24 publications

References 15 publications

Bounds on Real Eigenvalues and Singular Values of Interval Matrices

Bounds on Real Eigenvalues and Singular Values of Interval Matrices

Eigenvalue range determination for interval and parametric matrices

Determining the stability margin in linear interval parameter electric circuits via a DAE model

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