On Nonlinearization of 3-parameter Eigenvalue Problems

Authors

DOI:

https://doi.org/10.26713/cma.v13i3.1816

Keywords:

Generalized eigenvalue problem, 3-parameter eigenvalue problem, Nonlinear eigenvalue problem, Condition number

Abstract

In this paper, the linear 3-parameter eigenvalue problem (3PEP) in terms of matrix equations is considered. Using the Rayleigh Quotient iteration method, any one of the three parameters can be fixed to transform the problem into a linear 2-parameter eigenvalue problem (2PEP). This admits a family nonlinear eigenvalue problems (NEP) in one parameter. The transformation results from the elimination of the second equation of respective 2PEP, which is re-arranged as a generalized eigenvalue problem (GEP) of the form \(Ey=\lambda Fy\), where \(E\) and \(F\) are matrices \(n\times n\) over \(\mathbb{C}\), \(y\in \mathbb{C}^n\) is a non-zero vector and \(\lambda\) is a scalar. A review of some results of the condition number of NEP is also presented in this paper.

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References

J. Asakura, T. Sakurai, H. Tadano, T. Ikegami and K. Kimura, A numerical method for nonlinear eigenvalue problems using contour integrals, Japan Society for Industrial and Applied Mathematics Letters 1 (2009), 52 – 55, DOI: 10.14495/JSIAML.1.52.

F. V. Atkinson, Multiparameter Eigenvalue Problems, Academic Press, New York (1972).

M. V. Barel and P. Kravanja, Nonlinear eigenvalue problems and contour integrals, Journal of Computational and Applied Mathematics 292 (2016), 526 – 540, DOI: 10.1016/j.cam.2015.07.012.

T. Betcke and H. Voss, A Jacobi–Davidson-type projection method for nonlinear eigenvalue problems, Future Generation Computer Systems 20(3) (2004), 363 – 372, DOI: 10.1016/j.future.2003.07.003.

F. Binkowski, L. Zschiedrich and S. Burger, A Riesz-projected-based method for non-linear eigenvalue problems, Journal of Computational Physics 419 (2020), 109678, DOI: 10.1016/j.jcp.2020.109678.

N. Bora, Applications of multiparameter eigenvalue problems, Nepal Journal of Science and Technology 19(2) (2021), 75 – 82, DOI: 10.3126/njst.v20i1.39434.

F. Chatelin, Eigenvalues of Matrices, John Wiley and Sons Ltd., Chichester (1973).

X. P. Chen and H. Dai, A modified Newton method for nonlinear eigenvalue problems, East Asian Journal on Applied Mathematics 8(1) (2018), 139 – 150, DOI: 10.4208/eajam.100916.061117a.

K. D. Cock and B. D. Moor, Multiparameter eigenvalue problems and shift-invariance, IFAC-PapersOnline 54(9) (2021), 159 – 165, DOI: 10.1016/j.ifacol.2021.06.071.

C. Conca, A. Osses and J. Planchard, Asymptotic analysis relating spectral models in fluid-solid vibrations, SIAM Journal on Numerical Analysis 35 (1998), 1020 – 1048, URL: https://www.jstor.org/stable/2587120.

H. Dai and Z.-Z. Bai, On smooth LU decompositions with applications to solutions of nonlinear eigenvalue problems, Journal of Computational Mathematics 28 (2010), 745 – 766, DOI: 10.4208/jcm.1004-m0009.

B. Dong, B. Yu and Y. Yu, A homotopy method for finding all solutions of a multiparameter eigenvalue problem, SIAM Journal on Matrix Analysis and Applications 37(2) (2016), 550 – 571, DOI: 10.1137/140958396.

M. El-Guide, A. Mie¸dlar and Y. Saad, A rational approximation method for acoustic nonlinear eigenvalue problems, Engineering Analysis and Boundary Elements 111 (2020), 44 – 54, DOI: 10.1016/j.enganabound.2019.10.006.

B. Gavin, A. Mie¸dlar and E. Polizzi, FEAST eigensolver for nonlinear eigenvalue problems, Journal of Computational Science 27 (2018), 107 – 117, DOI: 10.1016/j.jocs.2018.05.006.

A. Ghosh and R. Alam, Sensitivity and backward perturbation analysis of multiparameter eigenvalue problems, SIAM Journal on Matrix Analysis and Applications 39(4) (2018), 1750 – 1775, DOI: 10.1137/18M1181377.

S. Güttel and F. Tisseur, Nonlinear eigenvalue problems, Acta Numerica 26 (2017), 1 – 94, DOI: 10.1017/S0962492917000034.

M. E. Hochstenbach and B. Plestenjak, Computing several eigenvalues of nonlinear eigenvalue problems by selection, Calcolo 57 (2020), Article number 16, DOI: 10.1007/s10092-020-00363-9.

M. E. Hochstenbach, K. Meerbergen, E. Mengi and B. Plestenjak, Subspace methods for three-parameter eigenvalue problems, Numerical Linear Algebra with Applications 26(4) (2019), e2240, DOI: 10.1002/nla.2240.

M. E. Hochstenbach, T. Kosir and B. Plestenjak, A Jacobi-Davidson type method for two-parameter eigenvalue problem, SIAM Journal on Matrix Analysis and Applications 26(2) (2004), 477 – 497, DOI: 10.1137/S0895479802418318.

E. Jarlebring, A. Koskela and G. Mele, Disguised and quasi-Newton method for nonlinear eigenvalue problems, Numerical Algorithms 79 (2018), 311 – 355, DOI: 10.1007/s11075-017-0438-2.

E. Jarlebring, Broyden’s method for nonlinear eigenproblems, SIAM Journal on Scientific Computing 41(2) (2019), A989 – A1012, DOI: 10.1137/18M1173150.

E. Jarlebring, S. Kvaal and W. Michiels, Computing all pairs (λ,µ) such that λ is a double eigenvalue of A + µB, SIAM Journal on Matrix Analysis and Applications 32(3) (2011), 902 – 927, DOI: 10.1137/100783157.

E. Jarlebring, W. Michiels and K. Meerbergen, A linear eigenvalue algorithm for the nonlinear eigenvalue problem, Numerische Mathematik 122(1) (2012), 169 – 195, DOI: 10.1007/s00211-012-0453-0.

X. Ji, A two-dimensional bisection method for solving two-parameter eigenvalue problems, SIAM Journal on Matrix Analysis and Applications 13(4) (1992), 1085 – 1093, DOI: 10.1137/0613065.

T. Košir, Finite-dimensional multiparameter spectral theory: the nonderogatory case, Linear Algebra and its Applications 212-213 (1994), 45 – 70, DOI: 10.1016/0024-3795(94)90396-4.

D. Kressner, A block Newton method for nonlinear eigenvalue problems, Numerische Mathematik 114(2) (2009), 355 – 372, , DOI: 10.1007/s00211-009-0259-x.

J. Lampe and H. Voss, A survey on variational characterizations for nonlinear eigenvalue problems, Electronic Transactions on Numerical Analysis 55 (2022), 1 – 75, DOI: 10.1553/etna_vol55s1.

V. Mehrmann and H. Voss, Nonlinear eigenvalue problems: A challenge for modern eigenvalue methods, GAMM-Mitteilungen 27 (2004), 121 – 152, DOI: 10.1002/gamm.201490007.

A. Neumaier, Residual inverse iteration for the nonlinear eigenvalue problem, SIAM Journal on Numerical Analysis 22 (1985), 914 – 923, DOI: 10.1137/0722055.

J. Ortega and W. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Classics in Applied Mathematics series, SIAM, xxviii + 358 (2000), DOI: 10.1137/1.9780898719468.

B. Plestenjak, A continuation method for a right definite two-parameter eigenvalue problem, SIAM Journal on Matrix Analysis and Applications 21(4) (2000), 1163 – 1184, DOI: 10.1137/S0895479898346193.

B. Plestenjak, Numerical methods for nonlinear two-parameter eigenvalue problems, BIT Numerical Mathematics 56 (2016), 241 – 262, DOI: 10.1007/s10543-015-0566-9.

E. Ringh and E. Jarlebring, Nonlinearizing two-parameter eigenvalue problems, SIAM Journal on Matrix Analysis and Applications 42(2) (2021), 775 – 799, DOI: 10.1137/19M1274316.

J.-I. Rodriguez, J.-H. Du, Y. You and L.-H. Lim, Fiber product homotopy method for multiparameter eigenvalue problems, Numerische Mathematik 148 (2021), 853 – 888, DOI: 10.1007/s00211-021-01215-6.

K. Ruymbeek, Projection Methods for Parametrized and Multiparameter Eigenvalue Problems, PhD thesis, KU Leuven, Belgium (2021), URL: https://people.cs.kuleuven.be/~wim.michiels/thesis-Koen.pdf.

B. D. Sleeman, Multiparameter spectral theory in Hilbert space, Journal of Mathematical Analysis and Applications 65(3) (1978), 511 – 530, DOI: 10.1016/0022-247X(78)90160-9.

A. Spence and C. Poulton, Photonic band structure calculations using nonlinear eigenvalue techniques, Journal of Computational Physics 204 (2005), 65 – 81, DOI: 10.1016/j.jcp.2004.09.016.

H. Volkmer, Multiparameter Eigenvalue Problems and Expansion Theorems, Lecture Notes in Mathematics (LNM, Vol. 1356), Springer, Berlin (1988), DOI: 10.1007/BFb0089295.

H. Voss, Nonlinear eigenvalue problems, in: Handbook of Linear Algebra, Hamburg University of Technology, Hamburg (2013), URL: https://www.mat.tuhh.de/forschung/rep/rep174.pdf.

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Published

29-11-2022
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How to Cite

Bora, N., Chutia, B., Moran, R., & Bora, M. C. (2022). On Nonlinearization of 3-parameter Eigenvalue Problems. Communications in Mathematics and Applications, 13(3), 1061–1073. https://doi.org/10.26713/cma.v13i3.1816

Issue

Vol. 13 No. 3 (2022)

Section

Research Article

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