# zbMATH — the first resource for mathematics

Numerical methods for nonlinear two-parameter eigenvalue problems. (English) Zbl 1342.65116
This paper deals with the nonlinear two-parameter eigenvalue problem (N2EP), which appears in the study of critical delays of delay-differential equations and can be seen as a generalization of both the nonlinear eigenvalue problem (NEP) and the two-parameter eigenvalue problem (2EP). Several numerical methods for NEP, including inverse iteration, residual inverse iteration, successive linear approximation, and Jacobi-Davidson method, are generalized to N2EP. Numerical tests show that the inverse iteration is the most competitive local method and the Jacobi-Davidson method can be used as a global method.

##### MSC:
 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 15A18 Eigenvalues, singular values, and eigenvectors
MultiParEig
Full Text:
##### References:
 [1] Akinola, RO; Freitag, MA; Spence, A, The computation of Jordan blocks in parameter-dependent matrices, IMA J. Numer. Anal., 34, 955-976, (2014) · Zbl 1302.65136 [2] Andrew, AL; Eric Chu, K-W; Lancaster, P, Derivatives of eigenvalues and eigenvectors of matrix functions, SIAM J. Matrix Anal. Appl., 14, 903-926, (1993) · Zbl 0786.15011 [3] Atkinson, F.V.: Multiparameter Eigenvalue Problems. Academic Press, New York (1972) · Zbl 0555.47001 [4] Betcke, T; Voss, H, A Jacobi-Davidson-type projection method for nonlinear eigenvalue problems, Future Gener. Comput. Syst., 20, 363-372, (2004) [5] Bohte, Z, Calculation of the derivative of the determinant, Obzornik Mat. Fiz., 28, 33-50, (1981) · Zbl 0452.65023 [6] Bohte, Z.: Numerical solution of some twoparameter eigenvalue problems. Anton Kuhelj Memorial Volume, Ljubljana, pp. 17-28 (1982) [7] Cox, D.A., Little, J.B., O’Shea, D.: Using Algebraic Geometry, 2nd edn. Springer, New York (2005) · Zbl 1079.13017 [8] Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996) · Zbl 0865.65009 [9] Hochstenbach, ME; Košir, T; Plestenjak, B, A Jacobi-Davidson type method for the nonsingular two-parameter eigenvalue problem, SIAM J. Matrix Anal. Appl., 26, 477-497, (2005) · Zbl 1077.65036 [10] Hochstenbach, ME; Muhič, A; Plestenjak, B, On linearizations of the quadratic two-parameter eigenvalue problems, Linear Algebr. Appl., 436, 2725-2743, (2012) · Zbl 1245.65042 [11] Hochstenbach, M.E., Muhič, A., Plestenjak, B.: Jacobi-Davidson methods for polynomial two-parameter eigenvalue problems . J. Comput. Appl. Math. 288, 251-263 (2015) · Zbl 06448790 [12] Hochstenbach, ME; Plestenjak, B, A Jacobi-Davidson type method for a right definite two-parameter eigenvalue problem, SIAM J. Matrix Anal. Appl., 24, 392-410, (2002) · Zbl 1025.65023 [13] Hochstenbach, ME; Plestenjak, B, Harmonic Rayleigh-Ritz extraction for the multiparameter eigenvalue problem, Electron. Trans. Numer. Anal., 29, 81-96, (2008) · Zbl 1171.65378 [14] Jarlebring, E.: The spectrum of delay-differential equations: numerical methods, stability and perturbation. PhD thesis, TU Braunschweig (2008) · Zbl 1183.34001 [15] Jarlebring, E; Hochstenbach, ME, Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations, Linear Algebr. Appl., 431, 369-380, (2009) · Zbl 1170.65063 [16] Ji, X.R., Jiang, H., Lee, B.H.K: A generalized Rayleigh quotient iteration for coupled eigenvalue problems. Technical Report 92-338, Department of Computing and Information Science, Queen’s University (1992) [17] Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia (1995) · Zbl 0832.65046 [18] Lancaster, P.: Lambda-Matrices and Vibrating Systems. Pergamon Press, Oxford (1966) · Zbl 0146.32003 [19] Meerbergen, K., Plestenjak, B.: A Sylvester-Arnoldi type method for the generalized eigenvalue problem with two-by-two operator determinants. Report TW 653, Department of Computer Science, KU Leuven (2014) · Zbl 1349.65122 [20] Mehrmann, V; Voss, H, Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods, GAMM Mitt. Ges. Angew. Math. Mech., 27, 121-152, (2004) · Zbl 1071.65074 [21] Muhič, A; Plestenjak, B, On the singular two-parameter eigenvalue problem, Electron. J. Linear Algebr., 18, 420-437, (2009) · Zbl 1190.15011 [22] Muhič, A; Plestenjak, B, On the quadratic two-parameter eigenvalue problem and its linearization, Linear Algebr. Appl., 432, 2529-2542, (2010) · Zbl 1189.65070 [23] Neumaier, A, Residual inverse iteration for the nonlinear eigenvalue problem, SIAM J. Numer. Anal., 22, 914-923, (1985) · Zbl 0594.65026 [24] Plestenjak, B, A continuation method for a right definite two-parameter eigenvalue problem, SIAM J. Matrix Anal. Appl., 21, 1163-1184, (2000) · Zbl 0995.65036 [25] Plestenjak, B, Numerical methods for the tridiagonal hyperbolic quadratic eigenvalue problem, SIAM J. Matrix Anal. Appl., 28, 1157-1172, (2006) · Zbl 1130.65055 [26] Ruhe, A, Algorithms for the nonlinear eigenvalue problem, SIAM J. Numer. Anal., 10, 674-689, (1973) · Zbl 0261.65032 [27] Schreiber, K.: Nonlinear eigenvalue problems: Newton-type methods and nonlinear Rayleigh functionals. PhD thesis, Department of Mathematics, TU Berlin (2008) · Zbl 1213.65064 [28] Sleijpen, GLG; Booten, AGL; Fokkema, DR; Vorst, HA, Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT, 36, 595-633, (1996) · Zbl 0861.65035 [29] Spence, A; Poulton, C, Photonic band structure calculations using nonlinear eigenvalue techniques, J. Comput. Phys., 204, 65-81, (2005) · Zbl 1143.82336
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.