On the sensitivities of multiple eigenvalues
Publication: Research - peer-review › Journal 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-review › Journal article – Annual report year: 2011
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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 -