On the sensitivities of multiple eigenvalues

Publication: Research - peer-reviewJournal article – Annual report year: 2011

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On the sensitivities of multiple eigenvalues. / Gravesen, Jens; Evgrafov, Anton; Nguyen, Dang Manh.

In: Structural and Multidisciplinary Optimization, Vol. 44, No. 4, 2011, p. 583-587.

Publication: Research - peer-reviewJournal article – Annual report year: 2011

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Author

Gravesen, Jens; Evgrafov, Anton; Nguyen, Dang Manh / On the sensitivities of multiple eigenvalues.

In: Structural and Multidisciplinary Optimization, Vol. 44, No. 4, 2011, p. 583-587.

Publication: Research - peer-reviewJournal article – Annual report year: 2011

Bibtex

@article{d04f3d6140f243aaaa57499314ca06fd,
title = "On the sensitivities of multiple eigenvalues",
keywords = "Multiple eigenvalues, Sensitivity analysis, Symmetric polynomials",
publisher = "Springer",
author = "Jens Gravesen and Anton Evgrafov and Nguyen, {Dang Manh}",
year = "2011",
doi = "10.1007/s00158-011-0644-9",
volume = "44",
number = "4",
pages = "583--587",
journal = "Structural and Multidisciplinary Optimization",
issn = "1615-147X",

}

RIS

TY - JOUR

T1 - On the sensitivities of multiple eigenvalues

A1 - Gravesen,Jens

A1 - Evgrafov,Anton

A1 - Nguyen,Dang Manh

AU - Gravesen,Jens

AU - Evgrafov,Anton

AU - Nguyen,Dang Manh

PB - Springer

PY - 2011

Y1 - 2011

N2 - We consider the generalized symmetric eigenvalue problem where matrices depend smoothly on a parameter. It is well known that in general individual eigenvalues, when sorted in accordance with the usual ordering on the real line, do not depend smoothly on the parameter. Nevertheless, symmetric polynomials of a number of eigenvalues, regardless of their multiplicity, which are known to be isolated from the rest depend smoothly on the parameter. We present explicit readily computable expressions for their first derivatives. Finally, we demonstrate the utility of our approach on a problem of finding a shape of a vibrating membrane with a smallest perimeter and with prescribed four lowest eigenvalues, only two of which have algebraic multiplicity one.

AB - We consider the generalized symmetric eigenvalue problem where matrices depend smoothly on a parameter. It is well known that in general individual eigenvalues, when sorted in accordance with the usual ordering on the real line, do not depend smoothly on the parameter. Nevertheless, symmetric polynomials of a number of eigenvalues, regardless of their multiplicity, which are known to be isolated from the rest depend smoothly on the parameter. We present explicit readily computable expressions for their first derivatives. Finally, we demonstrate the utility of our approach on a problem of finding a shape of a vibrating membrane with a smallest perimeter and with prescribed four lowest eigenvalues, only two of which have algebraic multiplicity one.

KW - Multiple eigenvalues

KW - Sensitivity analysis

KW - Symmetric polynomials

U2 - 10.1007/s00158-011-0644-9

DO - 10.1007/s00158-011-0644-9

JO - Structural and Multidisciplinary Optimization

JF - Structural and Multidisciplinary Optimization

SN - 1615-147X

IS - 4

VL - 44

SP - 583

EP - 587

ER -