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Reconstruction of Structured Quadratic Pencils from Eigenvalueson Ellipses and Parabolas

Published online by Cambridge University Press:  17 July 2014

R. Ibragimov  and
V. Vatchev
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Abstract

In the present paper we study the reconstruction of a structured quadratic pencil fromeigenvalues distributed on ellipses or parabolas. A quadratic pencil is a square matrixpolynomial

QP(λ) = M λ2+Cλ +K,

where M,C, andK are realsquare matrices. The approach developed in the paper is based on the theory of orthogonalpolynomials on the real line. The results can be applied to more general distribution ofeigenvalues. The problem with added single eigenvector is also briefly discussed. As anillustration of the reconstruction method, the eigenvalue problem on linearized stabilityof certain class of stationary exact solution of the Navier-Stokes equations describingatmospheric flows on a spherical surface is reformulated as a simple mass-spring system bymeans of this method.

Keywords

structured quadratic pencilinverse problemscomplex eigenvaluesorthogonal polynomials
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 5: Spectral problems , 2014 , pp. 138 - 147
Copyright
© EDP Sciences, 2014

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References

R. Askey. Linearization of the product of orthogonal polynomials. Problems in Analysis (R. Gunning, ed.), Princeton University Press, Princeton, N.J., (1970), 223–228.
Cai, Y., Kuo, Y., Lin, W., Xu, S.. Solutions to a quadratic inverse eigenvalue problem. Linear Algebra Appl., 430 (2008), 1590–1606. CrossRefGoogle Scholar
Chu, M., Golub, G.. Structured inverse eigenvalue problems? Acta Numer., 11 (2002), 1–71. CrossRefGoogle Scholar
Chu, M., Kuo, Y., Lin, W.. On inverse quadratic eigenvalue problems with partially prescribed eigenstructure. SIAM J. Matrix Anal. Appl., 25 (2004), 995–1020. CrossRefGoogle Scholar
Chu, M., Lin, M., Dong, B.. Semi-definite programming techniques for structured quadratic inverse eigenvalue problems. Numerical Algorithms, 53 (2010), no. 4, 419–437 . Google Scholar
Gasper, G.. Linearization of the product of Jacobi polynomials. I , II. Canad. J. Math., 22 (1970), 171–175, 582–593. CrossRefGoogle Scholar
Ibragimov, R., Pelinovsky, D.. Incompressible viscous fluid flows in a thin spherical shell. J. Math. Fluid. Mech., 11 (2009), 60–90. CrossRefGoogle Scholar
Mlotkowski, W., Szwarc, R.. Non-negative linearization for polynomials orthogonal with respect to discrete measures. Constr. Approx., 17 (2001), 413–429. CrossRefGoogle Scholar
Totik, V.. Orthogonal Polynomials, Surveys in Approximation Theory., 1 (2005), 70–125.Google Scholar