Decompositions and coalescing eigenvalues of symmetric definite pencils depending on parameters
Pages 1879 - 1910
Abstract
In this work, we consider symmetric positive definite pencils depending on two parameters. That is, we are concerned with the generalized eigenvalue problem , where A and B are symmetric matrix valued functions in , smoothly depending on parameters ; furthermore, 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). Our main goal is precisely that of locating parameter values where eigenvalues are equal. We first give general theoretical results for the present generalized eigenvalue problem, and then introduce and implement numerical methods apt at detecting conical intersections. Finally, we 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.
References
[1]
Berkolaiko G and Parulekar A Locating conical degeneracies in the spectra of parametric self-adjoint matrices SIAM J Matrix Anal Appl 2021 42 1 224-242
[2]
Chern JL and Dieci L Smoothness and periodicity of some matrix decompositions SIAM J. Matrix Anal. Appl. 2001 22 772-792
[3]
Dieci L and Eirola T On smooth decomposition of matrices SIAM J. Matrix Anal. Appl. 1999 20 800-819
[4]
Dieci L and Papini A Continuation of eigendecompositions Futur. Gener. Comput. Syst. 2003 19 1125-1137
[5]
Dieci L, Papini A, and Pugliese A Coalescing points for eigenvalues of banded matrices depending on parameters with application to banded random matrix functions Numerical Algorithms 2019 80 1241-1266
[6]
Dieci L and Pugliese A Two-parameter SVD: Coalescing singular values and periodicity SIAM J. Matrix Anal. Appl. 2009 31 375-403
[7]
Gingold H A method of global blockdiagonalization for matrix-valued functions SIAM J. Math. Anal. 1978 9-6 1076-1082
[8]
Golub, G.H., Van Loan, C.F.: Matrix computations Johns Hopkins University Press (1996)
[9]
van Hemmen JL and Ando T An inequality for trace ideals Commun. Math. Phys. 1980 76 143-148
[10]
Higham NJ Functions of matrices. Theory and Computation 2008 Philaelphia SIAM
[11]
Hirsch MW Differential Topology 1976 New York Springer-Verlag
[12]
Hsieh PF and Sibuya Y A global analysis of matrices of functions of several variables J. Math. Anal. Appl. 1966 14 332-340
[13]
Jarlebring E, Kvaal S, and Michiels W Computing all pairs (λ,μ) such that λ is a double eigenvalue of a + μB SIAM J. Matrix Anal. Appl. 2011 32 3 902-927
[14]
Kalinina E.A.: On multiple eigenvalues of a matrix dependent on a parameter. Computer Algebra in Scientific Computing, pp 305–314, 2016. Springer International Publishing
[15]
Kato T Perturbation Theory for Linear Operators 1976 2nd edn Berlin Springer-Verlag
[16]
Kaufman L An algorithm for the banded symmetric generalized matrix SIAM J. Matrix Anal. Appl. 1993 14 2 372-389
[17]
Krantz, S.G., Parks, H.R.: A primer of real analytic functions. birkhäuser Basel (2002)
[18]
Lancaster P On eigenvalues of matrices dependent on a parameter Numer Math 1964 6 377-387
[19]
Li K, Li T-Y, and Zeng Z An algorithm for the generalized symmetric tridiagonal eigenvalue problem Numer Algorithms 1994 8 269-291
[20]
Mehl C, Mehrmann V, and Wojtylak M Parameter-dependent rank-one perturbations of singular hermitian or symmetric pencils SIAM J. Matrix Anal. Appl. 2017 38 1 72-95
[21]
Rellich F Perturbation theory of eigenvalue problems, Courant Institute of Mathematical Sciences 1954 New York New York University
[22]
Srikantha Phani A, Woodhouse J, and Fleck NA Wave propagation in two-dimensional periodic lattices J. Acoust. Soc. Am. 2006 119 1995-2005
[23]
Soize, C.: Random matrix models and nonparametric method for uncertainty quantification. Handbook for Uncertainty Quantification, 1. In: Ghanem, R., Higdon, D., Owhadi, H. (eds.), pp. 219–287. Springer International Publishing, Switzerland (2017)
[24]
Tisseur F and Meerbergen K The quadratic eigenvalue problem SIAM Rev 2001 43-2 235-286
[25]
Uhlig F Coalescing eigenvalues and Crossing eigencurves of 1-parameter matrix flows SIAM J Coalescing Matrix Anal. Appl. 2020 41 4 1528-1545
[26]
Wilkinson JH Algebraic eigenvalue problem 1988 Oxford Clarendon Press. Oxford University Press
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Publication History
Published: 01 December 2022
Accepted: 02 May 2022
Received: 12 July 2021
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