Hermitian matrices depending on three parameters: Coalescing eigenvalues

Dieci, Luca; Pugliese, Alessandro

doi:10.1016/j.laa.2012.01.009

Cited by 12 publications

(18 citation statements)

References 19 publications

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“…As proved in [8,Theorem 2.6], the solution U gives the same Schur form as the "minimum variation decomposition" (MVD) of [3] and-at the end of the loop γ-we will have (2.2) U (1) = U (0)Φ with Φ = diag e iαj , j = 1, . .…”

Section: Smooth Schur Decompositionmentioning

confidence: 92%

“…Our algorithm localizes generic coalescing points of the function A (see [8] for definition of generic coalescing points). The mathematical basis of our algorithmic developments is given by the work [8], which provides a rigorous and thorough mathematical justification to a remarkable insight of Stone [15].…”

Section: Localization Of Coalescing Pointsmentioning

confidence: 99%

“…The mathematical basis of our algorithmic developments is given by the work [8], which provides a rigorous and thorough mathematical justification to a remarkable insight of Stone [15]. The basic idea is as follows.…”

Section: Localization Of Coalescing Pointsmentioning

confidence: 99%

“…Next, let us consider a Hermitian matrix valued function, for which we still label the eigenvalues so that they are ordered as before, and continuous. In this case, it is well known (e.g., see [8]) that it is a generic property of smooth Hermitian functions of 1 and 2 real parameters to have simple eigenvalues. Indeed, it has been known since the work [16] that having a pair of coalescing eigenvalues for a Hermitian function is a (real) codimension 3 phenomenon.…”

Section: Introduction a Hermitian Matrix A ∈ C N×n A * = A Alwaysmentioning

confidence: 99%

“…Indeed, it has been known since the work [16] that having a pair of coalescing eigenvalues for a Hermitian function is a (real) codimension 3 phenomenon. Therefore, we should expect that if a Hermitian function of 3 real parameters has coalescing eigenvalues, they will occur at isolated points-called conical intersections-and the eigenvalues would be merely continuous functions (not even once differentiable) at the coalescing point (e.g., we refer to [8] for mathematical of Ω being a cube, and propose techniques which are highly parallelizable through a natural subdivision scheme. Further, we will develop a new 1-dimensional (1-d) solver which approximates the "minimum variation decomposition" (a specific smooth eigendecomposition) of A along a closed path, and renders an accurate approximation to the sought smooth phase matrix at negligible extra cost with respect to that of finding any eigendecomposition.…”

Section: Introduction a Hermitian Matrix A ∈ C N×n A * = A Alwaysmentioning

confidence: 99%

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Approximating Coalescing Points for Eigenvalues of Hermitian Matrices of Three Parameters

Dieci¹,

Papini²,

Pugliese³

2013

SIAM J. Matrix Anal. & Appl.

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We consider a Hermitian matrix valued function A(x) ∈ C n×n , smoothly depending on parameters x ∈ Ω ⊂ R 3 , where Ω is an open bounded region of R 3 . We develop an algorithm to locate parameter values where the eigenvalues of A coalesce: conical intersections of eigenvalues. The crux of the method requires one to monitor the geometric phase matrix of a Schur decomposition of A, as A varies on the surface S bounding Ω. We develop (adaptive) techniques to find the minimum variation decomposition of A along loops covering S and show how this can be used to detect conical intersections. Further, we give implementation details of a parallelization of the technique, as well as details relative to the case of locating conical intersections for a few of A's dominant eigenvalues. Several examples illustrate the effectiveness of our technique.

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Section: Smooth Schur Decompositionmentioning

confidence: 92%

Section: Localization Of Coalescing Pointsmentioning

confidence: 99%

Section: Localization Of Coalescing Pointsmentioning

confidence: 99%

Section: Introduction a Hermitian Matrix A ∈ C N×n A * = A Alwaysmentioning

confidence: 99%

Section: Introduction a Hermitian Matrix A ∈ C N×n A * = A Alwaysmentioning

confidence: 99%

See 3 more Smart Citations

Approximating Coalescing Points for Eigenvalues of Hermitian Matrices of Three Parameters

Dieci¹,

Papini²,

Pugliese³

2013

SIAM J. Matrix Anal. & Appl.

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Wind‐induced inter‐story drift analysis and equivalent static wind load for multiple targets of tall buildings

Zheng

et al. 2015

Structural Design Tall Build

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SUMMARYIn super high-rise buildings with varying story heights, the wind-induced inter-story drifts might violate the specified limit. However, these effects have seldom been concerned in wind-induced response analysis. The theory and application of equivalent static wind load (ESWL) for wind-induced inter-story drifts of super high-rise buildings were studied in this paper. A spectral decomposition method suitable for multi-point excitation problems was firstly proposed. The formula of ESWL targeting for largest inter-story drift was derived. For more reasonable structural design, the ESWL for multiple targets including displacement atop of building and inter-story drifts at all story levels is put forward, in which the dominant modal inertial forces are adopted as the based load vectors. The presented methods were finally verified by its application for the wind-induced response analysis for a tallest super tall building in Guangzhou. The researched results showed that the proposed spectral decomposition method not only has the same precision as the complete quadratic combination method but also possesses higher computation efficiency. The ESWL for multiple targets produces the same static responses for all the specified wind-induced response, so it is much more rational for wind-resistant structural design. Meanwhile, it is more reasonable to select the wind-induced responses in the same direction simultaneously as the targeted values for obtaining the required ESWLs; however, the ESWL targeting for the wind-induced responses in all degrees of freedom would generate more queer and unrealistic ESWLs distribution.

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Continuous Decompositions and Coalescing Eigenvalues for Matrices Depending on Parameters

Dieci

Papini

Pugliese

et al. 2013

Lecture Notes in Mathematics

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In this work, we consider symmetric positive definite pencils depending on two parameters. That is, we are concerned with the generalized eigenvalue problem A(x) − λB(x), where A and B are symmetric matrix valued functions in R n×n , smoothly depending on parameters x ∈ Ω ⊂ R 2 ; further, B is also positive definite. In general, the eigenvalues of this multiparameter problem will not be smooth, the lack of smoothness resulting from eigenvalues being equal at some parameter values (conical intersections). We first give general theoretical results on the smoothness of eigenvalues and eigenvectors for the present generalized eigenvalue problem, and hence for the corresponding projections, and then perform a numerical study of the statistical properties of coalescing eigenvalues for pencils where A and B are either full or banded, for several bandwidths. Our numerical study will be performed with respect to a random matrix ensemble which respects the underlying engineering problems motivating our study.

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Hermitian matrices depending on three parameters: Coalescing eigenvalues

Cited by 12 publications

References 19 publications

Approximating Coalescing Points for Eigenvalues of Hermitian Matrices of Three Parameters

Approximating Coalescing Points for Eigenvalues of Hermitian Matrices of Three Parameters

Wind‐induced inter‐story drift analysis and equivalent static wind load for multiple targets of tall buildings

Continuous Decompositions and Coalescing Eigenvalues for Matrices Depending on Parameters

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