Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-05T10:27:17.526Z Has data issue: false hasContentIssue false
  • English
  • Français

The nonlinear eigenvalue problem*

Published online by Cambridge University Press:  05 May 2017

Stefan Güttel  and
Françoise Tisseur
Stefan Güttel
Affiliation:
School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK E-mail: [email protected], [email protected]
Françoise Tisseur
Affiliation:
School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK E-mail: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Nonlinear eigenvalue problems arise in a variety of science and engineering applications, and in the past ten years there have been numerous breakthroughs in the development of numerical methods. This article surveys nonlinear eigenvalue problems associated with matrix-valued functions which depend nonlinearly on a single scalar parameter, with a particular emphasis on their mathematical properties and available numerical solution techniques. Solvers based on Newton’s method, contour integration and sampling via rational interpolation are reviewed. Problems of selecting the appropriate parameters for each of the solver classes are discussed and illustrated with numerical examples. This survey also contains numerous MATLAB code snippets that can be used for interactive exploration of the discussed methods.

Type
Research Article
Information
Acta Numerica , Volume 26 , 01 May 2017 , pp. 1 - 94
Copyright
© Cambridge University Press, 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES 4

Ahlfors, L. V. (1953), Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, McGraw-Hill.Google Scholar
Al-Ammari, M. and Tisseur, F. (2012), ‘‘Hermitian matrix polynomials with real eigenvalues of definite type I: Classification’’, Linear Algebra Appl. 436, 3954–3973.CrossRefGoogle Scholar
Alam, R. and Behera, N. (2016), ‘Linearizations for rational matrix functions and Rosenbrock system polynomials’, SIAM J. Matrix Anal. Appl. 37, 354–380.Google Scholar
Amiraslani, A., Corless, R. M. and Lancaster, P. (2009), ‘Linearization of matrix polynomials expressed in polynomial bases’, IMA J. Numer. Anal. 29, 141–157.Google Scholar
Andrew, A. L., Chu, K. E. and Lancaster, P. (1993), ‘Derivatives of eigenvalues and eigenvectors of matrix functions’, SIAM J. Matrix Anal. Appl. 14, 903–926.CrossRefGoogle Scholar
Andrew, A. L., Chu, K. E. and Lancaster, P. (1995), ‘On the numerical solution of nonlinear eigenvalue problems’, Computing 55, 91–111.CrossRefGoogle Scholar
Anselone, P. M. and Rall, L. B. (1968), ‘The solution of characteristic value-vector problems by Newton’s method’, Numer. Math. 11, 38–45.Google Scholar
Arbenz, P. and Gander, W. (1986), ‘Solving nonlinear eigenvalue problems by algorithmic differentiation’, Computing 36, 205–215.CrossRefGoogle Scholar
Asakura, J., Sakurai, T., Tadano, H., Ikegami, T. and Kimura, K. (2009), ‘A numerical method for nonlinear eigenvalue problems using contour integral’, JSIAM Letters 1, 52–55.CrossRefGoogle Scholar
Austin, A. P., Kravanja, P. and Trefethen, L. N. (2014), ‘Numerical algorithms based on analytic function values at roots of unity’, SIAM J. Numer. Anal. 52, 1795–1821.Google Scholar
Bagby, T. (1967), ‘The modulus of a plane condenser’, J. Math. Mech. 17, 315–329.Google Scholar
Bagby, T. (1969), ‘On interpolation by rational functions’, Duke Math. J. 36, 95–104.Google Scholar
Bai, Z., Demmel, J. W., Dongarra, J. J., Ruhe, A. & van der Vorst, H. A. (Eds) (2000), Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, SIAM.CrossRefGoogle Scholar
Balay, S., Abhyankar, S., Adams, M. F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W. D., Kaushik, D., Knepley, M. G., McInnes, L. C., Rupp, K., Smith, B. F., Zampini, S., Zhang, H. and Zhang, H. (2016), PETSc users’ manual. Technical report ANL-95/11, revision 3.7, Argonne National Laboratory.Google Scholar
Berljafa, M. and Güttel, S. (2014), A Rational Krylov Toolbox for MATLAB. MIMS EPrint 2014.56, Manchester Institute for Mathematical Sciences, The University of Manchester, UK. http://rktoolbox.org/ Google Scholar
Berljafa, M. and Güttel, S. (2015), ‘Generalized rational Krylov decompositions with an application to rational approximation’, SIAM J. Matrix Anal. Appl. 36, 894–916.CrossRefGoogle Scholar
Berljafa, M. and Güttel, S. (2017), Parallelization of the rational Arnoldi algorithm. MIMS EPrint 2016.32, Manchester Institute for Mathematical Sciences, The University of Manchester, UK. SIAM J. Sci. Comput., to appear.Google Scholar
Betcke, T. and Voss, H. (2004), ‘A Jacobi–Davidson-type projection method for nonlinear eigenvalue problems’, Future Gener. Comput. Syst. 20, 363–372.CrossRefGoogle Scholar
Betcke, T., Higham, N. J., Mehrmann, V., Schröder, C. and Tisseur, F. (2013), ‘NLEVP: A collection of nonlinear eigenvalue problems’, ACM Trans. Math. Software 39, 7:1–7:28.CrossRefGoogle Scholar
Beyn, W.-J. (2012), ‘An integral method for solving nonlinear eigenvalue problems’, Linear Algebra Appl. 436, 3839–3863.CrossRefGoogle Scholar
Beyn, W.-J. and Thümmler, V. (2009), ‘Continuation of invariant subspaces for parameterized quadratic eigenvalue problems’, SIAM J. Matrix Anal. Appl. 31, 1361–1381.CrossRefGoogle Scholar
Beyn, W.-J., Effenberger, C. and Kressner, D. (2011), ‘Continuation of eigenvalues and invariant pairs for parameterized nonlinear eigenvalue problems’, Numer. Math. 119, 489–516.Google Scholar
Bindel, D. and Hood, A. (2013), ‘Localization theorems for nonlinear eigenvalue problems’, SIAM J. Matrix Anal. Appl. 34, 1728–1749.Google Scholar
Blatt, H.-P., Saff, E. B. and Simkani, M. (1988), ‘Jentzsch–Szegö type theorems for the zeros of best approximants’, J. London Math. Soc. 2, 307–316.Google Scholar
Botchev, M. A., Sleijpen, G. L. G. and Sopaheluwakan, A. (2009), ‘An SVD-approach to Jacobi–Davidson solution of nonlinear Helmholtz eigenvalue problems’, Linear Algebra Appl. 431, 427–440.CrossRefGoogle Scholar
Botten, L., Craig, M. and McPhedran, R. (1983), ‘Complex zeros of analytic functions’, Comput. Phys. Commun. 29, 245–259.Google Scholar
Bühler, T. and Hein, M. (2009), Spectral clustering based on the graph p-Laplacian. In 26th Annual International Conference on Machine Learning, ACM, pp. 81–88.Google Scholar
Byers, R., Mehrmann, V. and Xu, H. (2008), ‘Trimmed linearizations for structured matrix polynomials’, Linear Algebra Appl. 429, 2373–2400.CrossRefGoogle Scholar
Campos, C. and Roman, J. E. (2016a)), ‘Parallel iterative refinement in polynomial eigenvalue problems’, Numer. Linear Algebra Appl. 23, 730–745.Google Scholar
Campos, C. and Roman, J. E. (2016b), ‘Parallel Krylov solvers for the polynomial eigenvalue problem in SLEPc’, SIAM J. Sci. Comput. 38, S385–S411.CrossRefGoogle Scholar
Corless, R. M. (2004), Generalized companion matrices in the Lagrange basis. In Proceedings EACA (Gonzalez-Vega, L. and Recio, T., eds), SIAM, pp. 317–322.Google Scholar
Courant, R. (1920), ‘Über die Eigenwerte bei den Differentialgleichungen der mathematischen Physik’, Math. Z. 7, 1–57.Google Scholar
Davidson, E. R. (1975), ‘The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices’, J. Comput. Phys. 17, 87–94.Google Scholar
Davis, P. J. and Rabinowitz, P. (2007), Methods of Numerical Integration, Courier Corporation.Google Scholar
Davis, T. A. (2004), ‘Algorithm 832: UMFPACK V4.3: An unsymmetric-pattern multifrontal method’, ACM Trans. Math. Software 30, 196–199.Google Scholar
Delves, L. and Lyness, J. (1967), ‘A numerical method for locating the zeros of an analytic function’, Math. Comp. 21(100), 543–560.CrossRefGoogle Scholar
Du, N. H., Linh, V. H., Mehrmann, V. and Thuan, D. D. (2013), ‘Stability and robust stability of linear time-invariant delay differential-algebraic equations’, SIAM J. Matrix Anal. Appl. 34, 1631–1654.Google Scholar
Duffin, R. J. (1955), ‘A minimax theory for overdamped networks’, J. Rat. Mech. Anal. 4, 221–233.Google Scholar
Eaton, J. W., Bateman, D., Hauberg, S. and Wehbring, R. (2016), GNU Octave Version 4.2.0 Manual: A High-Level Interactive Language for Numerical Computations. http://www.gnu.org/software/octave/doc/interpreter Google Scholar
Effenberger, C. (2013a), Robust solution methods for nonlinear eigenvalue problems. PhD thesis, EPFL, Lausanne.Google Scholar
Effenberger, C. (2013b)), ‘Robust successive computation of eigenpairs for nonlinear eigenvalue problems’, SIAM J. Matrix Anal. Appl. 34, 1231–1256.Google Scholar
Effenberger, C. and Kressner, D. (2012), ‘Chebyshev interpolation for nonlinear eigenvalue problems’, BIT 52, 933–951.Google Scholar
Ferng, W. R., Lin, W.-W., Pierce, D. J. and Wang, C.-S. (2001), ‘Nonequivalence transformation of $\unicode[STIX]{x1D706}$ -matrix eigenproblems and model embedding approach to model tuning’, Numer. Linear Algebra Appl. 8, 53–70.3.0.CO;2-3>CrossRefGoogle Scholar
Fischer, E. (1905), ‘Über quadratische Formen mit reellen Koeffizienten’, Monatshefte für Mathematik und Physik 16, 234–249.CrossRefGoogle Scholar
Freund, R. W. and Nachtigal, N. M. (1996), ‘QMRPACK: A package of QMR algorithms’, ACM Trans. Math. Software 22, 46–77.Google Scholar
Gaaf, S. W. and Jarlebring, E. (2016), The infinite bi-Lanczos method for nonlinear eigenvalue problems. arXiv:1607.03454 Google Scholar
Gander, W., Gander, M. J. and Kwok, F. (2014), Scientific Computing: An Introduction using Maple and MATLAB, Springer.CrossRefGoogle Scholar
Garrett, C. K. and Li, R.-C. (2013), Unstructurally banded nonlinear eigenvalue solver software. http://www.csm.ornl.gov/newsite/software.html Google Scholar
Garrett, C. K., Bai, Z. and Li, R.-C. (2016), ‘A nonlinear QR algorithm for banded nonlinear eigenvalue problems’, ACM Trans. Math. Software 43, 4:1–4:19.Google Scholar
Gohberg, I. and Rodman, L. (1981), ‘Analytic matrix functions with prescribed local data’, J. Analyse Math. 40.Google Scholar
Gohberg, I., Kaashoek, M. and van Schagen, F. (1993), ‘On the local theory of regular analytic matrix functions’, Linear Algebra Appl. 182, 9–25.Google Scholar
Gohberg, I., Lancaster, P. and Rodman, L. (2009), Matrix Polynomials, SIAM. Unabridged republication of book first published by Academic Press in 1982.CrossRefGoogle Scholar
Gonchar, A. A. (1969), ‘Zolotarev problems connected with rational functions’, Math. USSR Sb. 7, 623–635.Google Scholar
Grammont, L., Higham, N. J. and Tisseur, F. (2011), ‘A framework for analyzing nonlinear eigenproblems and parametrized linear systems’, Linear Algebra Appl. 435, 623–640.Google Scholar
Griewank, A. and Walther, A. (2008), Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, SIAM.CrossRefGoogle Scholar
Gu, K., Kharitonov, V. L. and Chen, J. (2003), Stability of Time-Delay Systems, Springer Science & Business Media.Google Scholar
Güttel, S. (2013), ‘Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection’, GAMM-Mitt. 36, 8–31.Google Scholar
Güttel, S., Van Beeumen, R., Meerbergen, K. and Michiels, W. (2014), ‘NLEIGS: A class of fully rational Krylov methods for nonlinear eigenvalue problems’, SIAM J. Sci. Comput. 36, A2842–A2864.Google Scholar
Hadeler, K. P. (1967), ‘Mehrparametrige und nichtlineare Eigenwertaufgaben’, Arch. Rational Mech. Anal. 27, 306–328.Google Scholar
Hadeler, K. P. (1968), ‘Variationsprinzipien bei nichtlinearen Eigenwertaufgaben’, Arch. Rational Mech. Anal. 30, 297–307.Google Scholar
Hale, N., Higham, N. J. and Trefethen, L. N. (2008), ‘Computing $A^{\unicode[STIX]{x1D6FC}}$ , $\log (A)$ , and related matrix functions by contour integrals’, SIAM J. Numer. Anal. 46, 2505–2523.Google Scholar
Hernandez, V., Roman, J. E. and Vidal, V. (2005), ‘SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems’, ACM Trans. Math. Software 31, 351–362.Google Scholar
Hernandez, V., Roman, J. E., Tomas, A. and Vidal, V. (2009), A survey of software for sparse eigenvalue problems. SLEPc Technical Report STR-6, Universidad Politécnica de Valencia.Google Scholar
Higham, D. J. and Higham, N. J. (2017), MATLAB Guide, third edition, SIAM.Google Scholar
Higham, N. J. (2008), Functions of Matrices: Theory and Computation, SIAM.CrossRefGoogle Scholar
Higham, N. J. and Tisseur, F. (2002), ‘More on pseudospectra for polynomial eigenvalue problems and applications in control theory’, Linear Algebra Appl. 351–352, 435–453.Google Scholar
Higham, N. J., Mackey, D. S., Mackey, N. and Tisseur, F. (2006), ‘Symmetric linearizations for matrix polynomials’, SIAM J. Matrix Anal. Appl. 29, 143–159.Google Scholar
Hochstenbach, M. E. and Sleijpen, G. L. G. (2003), ‘Two-sided and alternating Jacobi–Davidson’, Linear Algebra Appl. 358, 145–172.Google Scholar
Horn, R. A. and Johnson, C. R. (1985), Matrix Analysis, Cambridge University Press.Google Scholar
Horn, R. A. and Johnson, C. R. (1991), Topics in Matrix Analysis, Cambridge University Press.CrossRefGoogle Scholar
HSL(2016), A collection of Fortran codes for large-scale scientific computation. http://www.hsl.rl.ac.uk/ Google Scholar
Huang, T.-M., Lin, W.-W. and Mehrmann, V. (2016), ‘A Newton-type method with nonequivalence deflation for nonlinear eigenvalue problems arising in photonic crystal modeling’, SIAM J. Sci. Comput. 38, B191–B218.Google Scholar
Ioakimidis, N. I. (1987), Quadrature methods for the determination of zeros of transcendental functions: A review. In Numerical Integration: Recent Developments, Software and Applications, Springer, pp. 61–82.Google Scholar
Ipsen, I. C. F. (1997), ‘Computing an eigenvector with inverse iteration’, SIAM Rev. 39, 254–291.Google Scholar
Jarlebring, E. and Güttel, S. (2014), ‘A spatially adaptive iterative method for a class of nonlinear operator eigenproblems’, Electron. Trans. Numer. Anal. 41, 21–41.Google Scholar
Jarlebring, E., Meerbergen, K. and Michiels, W. (2012a)), The infinite Arnoldi method and an application to time-delay systems with distributed delay. In Time Delay Systems: Methods, Applications and New Trends, (Sipahi, R., Vyhlídal, T., Niculescu, S.-I. and Pepe, P., eds), Vol. 423 of Lecture Notes in Control and Information Sciences, Springer, pp. 229–239.Google Scholar
Jarlebring, E., Meerbergen, K. and Michiels, W. (2012b)), ‘A linear eigenvalue algorithm for the nonlinear eigenvalue problem’, Numer. Math. 122, 169–195.Google Scholar
Jarlebring, E., Meerbergen, K. and Michiels, W. (2014), ‘Computing a partial Schur factorization of nonlinear eigenvalue problems using the infinite Arnoldi method’, SIAM J. Matrix Anal. Appl. 35, 411–436.Google Scholar
Jarlebring, E., Mele, G. and Runborg, O. (2017), ‘The waveguide eigenvalue problem and the tensor infinite Arnoldi method’, SIAM J. Sci. Comput., to appear.Google Scholar
Jentzsch, R. (1916), ‘Untersuchungen zur Theorie der Folgen analytischer Funktionen’, Acta Math. 41, 219–251.CrossRefGoogle Scholar
Kamiya, N., Andoh, E. and Nogae, K. (1993), ‘Eigenvalue analysis by the boundary element method: New developments’, Eng. Anal. Bound. Elem. 12, 151–162.Google Scholar
Karma, O. (1996a)), ‘Approximation in eigenvalue problems for holomorphic Fredholm operator functions I’, Numer. Funct. Anal. Optim. 17, 365–387.Google Scholar
Karma, O. (1996b)), ‘Approximation in eigenvalue problems for holomorphic Fredholm operator functions II: Convergence rate’, Numer. Funct. Anal. Optim. 17, 389–408.Google Scholar
Keener, J. P. (1993), ‘The Perron–Frobenius theorem and the ranking of football teams’, SIAM Rev. 35, 80–93.Google Scholar
Khazanov, V. B. and Kublanovskaya, V. N. (1988), ‘Spectral problems for matrix pencils: Methods and algorithms II’, Sov. J. Numer. Anal. Math. Modelling 3, 467–485.Google Scholar
Kimeswenger, A., Steinbach, O. and Unger, G. (2014), ‘Coupled finite and boundary element methods for fluid–solid interaction eigenvalue problems’, SIAM J. Numer. Anal. 52, 2400–2414.Google Scholar
Kozlov, V. and Maz’ja, V. G. (1999), Differential Equations with Operator Coefficients, Springer Monographs in Mathematics, Springer.Google Scholar
Krantz, S. G. (1982), Function Theory of Several Complex Variables, Wiley.Google Scholar
Kravanja, P., Sakurai, T. and Van Barel, M. (1999a)), ‘On locating clusters of zeros of analytic functions’, BIT 39, 646–682.Google Scholar
Kravanja, P., Van Barel, M. and Haegemans, A. (1999b)), On computing zeros and poles of meromorphic functions. In Third Conference on Computational Methods and Function Theory 1997, World Scientific, pp. 359–370.Google Scholar
Kravanja, P., Van Barel, M., Ragos, O., Vrahatis, M. and Zafiropoulos, F. (2000), ‘ZEAL: A mathematical software package for computing zeros of analytic functions’, Comput. Phys. Commun. 124, 212–232.CrossRefGoogle Scholar
Kressner, D. (2009), ‘A block Newton method for nonlinear eigenvalue problems’, Numer. Math. 114, 355–372.Google Scholar
Kressner, D. and Roman, J. E. (2014), ‘Memory-efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis’, Numer. Linear Algebra Appl. 21, 569–588.Google Scholar
Kublanovskaja, V. N. (1969), ‘On an application of Newton’s method to the determination of eigenvalues of $\unicode[STIX]{x1D706}$ -matrices’, Soviet Math. Dokl. 10, 1240–1241.Google Scholar
Kublanovskaya, V. N. (1970), ‘On an approach to the solution of the generalized latent value problem for $\unicode[STIX]{x1D706}$ -matrices’, SIAM J. Numer. Anal. 7, 532–537.Google Scholar
Lancaster, P. (1961), ‘A generalised Rayleigh quotient iteration for lambda-matrices’, Arch. Rational Mech. Anal. 8, 309–322.Google Scholar
Lancaster, P. (1966), Lambda-Matrices and Vibrating Systems, Pergamon Press. Reprinted by Dover, 2002.Google Scholar
Leblanc, A. and Lavie, A. (2013), ‘Solving acoustic nonlinear eigenvalue problems with a contour integral method’, Engrg Anal. Bound. Elem. 37, 162–166.Google Scholar
Lehoucq, R. B. and Meerbergen, K. (1998), ‘Using generalized Cayley transformations within an inexact rational Krylov sequence method’, SIAM J. Matrix Anal. Appl. 20, 131–148.Google Scholar
Lehoucq, R. B., Sorensen, D. C. and Yang, C. (1998), ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, SIAM.Google Scholar
Leiterer, J. (1978), ‘Local and global equivalence of meromorphic operator functions II’, Math. Nachr. 84, 145–170.Google Scholar
Levin, E. and Saff, E. B. (2006), Potential theoretic tools in polynomial and rational approximation. In Harmonic Analysis and Rational Approximation, (Fournier, J.-D. et al. , ed.), Vol. 327 of Lecture Notes in Control and Information Sciences, Springer, pp. 71–94.Google Scholar
Liao, B.-S., Bai, Z., Lee, L.-Q. and Ko, K. (2010), ‘Nonlinear Rayleigh–Ritz iterative method for solving large scale nonlinear eigenvalue problems’, Taiwan. J. Math. 14, 869–883.Google Scholar
Lu, D., Huang, X., Bai, Z. and Su, Y. (2015), ‘A Padé approximate linearization algorithm for solving the quadratic eigenvalue problem with low-rank damping’, Internat. J. Numer. Methods Engrg 103, 840–858.Google Scholar
Lu, D., Su, Y. and Bai, Z. (2016), ‘Stability analysis of the two-level orthogonal Arnoldi procedure’, SIAM J. Matrix Anal. Appl. 37, 195–214.Google Scholar
Mackey, D. S. and Perović, V. (2016), ‘Linearizations of matrix polynomials in Bernstein bases’, Linear Algebra Appl. 501, 162–197.Google Scholar
Mackey, D. S., Mackey, N. and Tisseur, F. (2015), Polynomial eigenvalue problems: Theory, computation, and structure. In Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory (Benner, P. et al. , ed.), Springer, pp. 319–348.Google Scholar
Mackey, D. S., Mackey, N., Mehl, C. and Mehrmann, V. (2006a)), ‘Structured polynomial eigenvalue problems: Good vibrations from good linearizations’, SIAM J. Matrix Anal. Appl. 28, 1029–1051.Google Scholar
Mackey, D. S., Mackey, N., Mehl, C. and Mehrmann, V. (2006b)), ‘Vector spaces of linearizations for matrix polynomials’, SIAM J. Matrix Anal. Appl. 28, 971–1004.Google Scholar
Maeda, Y., Sakurai, T. and Roman, J. (2016), Contour integral spectrum slicing method in SLEPc. SLEPc Technical Report STR-11, Universidad Politécnica de Valencia.Google Scholar
Mehrmann, V. and Voss, H. (2004), ‘Nonlinear eigenvalue problems: A challenge for modern eigenvalue methods’, GAMM-Mitt. 27, 121–152.Google Scholar
Mennicken, R. and Möller, M. (2003), Non-Self-Adjoint Boundary Eigenvalue Problems, Vol. 192 of North-Holland Mathematics Studies, Elsevier Science.Google Scholar
Michiels, W. and Guglielmi, N. (2012), ‘An iterative method for computing the pseudospectral abscissa for a class of nonlinear eigenvalue problems’, SIAM J. Sci. Comput. 34, A2366–A2393.Google Scholar
Michiels, W. and Niculescu, S.-I. (2007), Stability and Stabilization of Time-Delay Systems: An Eigenvalue-Based Approach, SIAM.Google Scholar
Michiels, W., Green, K., Wagenknecht, T. and Niculescu, S.-I. (2006), ‘Pseudospectra and stability radii for analytic matrix functions with application to time-delay systems’, Linear Algebra Appl. 418, 315–335.Google Scholar
MUMPS(2016), MUltifrontal Massively Parallel Solver: Users’ guide. http://mumps.enseeiht.fr/ Google Scholar
Nehari, Z. (1975), Conformal Mapping, Dover. Unabridged and unaltered republication of book originally published by McGraw-Hill in 1952.Google Scholar
Neumaier, A. (1985), ‘Residual inverse iteration for the nonlinear eigenvalue problem’, SIAM J. Numer. Anal. 22, 914–923.Google Scholar
Niendorf, V. and Voss, H. (2010), ‘Detecting hyperbolic and definite matrix polynomials’, Linear Algebra Appl. 432, 1017–1035.Google Scholar
Noferini, V. and Pérez, J. (2016), ‘Fiedler-comrade and Fiedler–Chebyshev pencils’, SIAM J. Matrix Anal. Appl. 37, 1600–1624.Google Scholar
Opitz, G. (1964), ‘Steigungsmatrizen’, Z. Angew. Math. Mech. 44, T52–T54.Google Scholar
Peters, G. and Wilkinson, J. H. (1979), ‘Inverse iteration, ill-conditioned equations and Newton’s method’, SIAM Rev. 21, 339–360.Google Scholar
Poincaré, H. (1890), ‘Sur les équations aux dérivées partielles de la physique mathématique’, Amer. J. Math. 211–294.Google Scholar
Rogers, E. H. (1964), ‘A minimax theory for overdamped systems’, Arch. Rational Mech. Anal. 16, 89–96.Google Scholar
Ruhe, A. (1973), ‘Algorithms for the nonlinear eigenvalue problem’, SIAM J. Numer. Anal. 10, 674–689.Google Scholar
Ruhe, A. (1998), ‘Rational Krylov: A practical algorithm for large sparse nonsymmetric matrix pencils’, SIAM J. Sci. Comput. 19, 1535–1551.Google Scholar
Saad, Y. (2011), Numerical Methods for Large Eigenvalue Problems, revised edition, SIAM.CrossRefGoogle Scholar
Saad, Y. and Schultz, M. H. (1986), ‘GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems’, SIAM J. Sci. Statist. Comput. 7, 856–869.Google Scholar
Saad, Y., Stathopoulos, A., Chelikowsky, J., Wu, K. and Öğüt, S. (1996), ‘Solution of large eigenvalue problems in electronic structure calculations’, BIT 36, 563–578.Google Scholar
Sakurai, T. and Sugiura, H. (2003), ‘A projection method for generalized eigenvalue problems using numerical integration’, J. Comput. Appl. Math. 159, 119–128.Google Scholar
Schenk, O. and Gärtner, K. (2004), ‘Solving unsymmetric sparse systems of linear equations with PARDISO’, Future Gener. Comput. Syst. 20, 475–487. http://www.pardiso-project.org/ Google Scholar
Schreiber, K. (2008), Nonlinear eigenvalue problems: Newton-type methods and nonlinear Rayleigh functionals. PhD thesis, Technische Universität Berlin. http://www.math.tu-berlin.de/∼schreibe/ Google Scholar
Schwetlick, H. and Schreiber, K. (2012), ‘Nonlinear Rayleigh functionals’, Linear Algebra Appl. 436, 3991–4016.Google Scholar
Sleijpen, G. L. G. and van der Vorst, H. A. (1996), ‘A Jacobi–Davidson iteration method for linear eigenvalue problems’, SIAM J. Matrix Anal. Appl. 17, 401–425.Google Scholar
Sleijpen, G. L. G., Booten, A. G. L., Fokkema, D. R. and van der Vorst, H. A. (1996), ‘Jacobi–Davidson type methods for generalized eigenproblems and polynomial eigenproblems’, BIT 36, 595–633.Google Scholar
Solov’ëv, S. I. (2006), ‘Preconditioned iterative methods for a class of nonlinear eigenvalue problems’, Linear Algebra Appl. 415, 210–229.Google Scholar
Spence, A. and Poulton, C. (2005), ‘Photonic band structure calculations using nonlinear eigenvalue techniques’, J. Comput. Phys. 204, 65–81.Google Scholar
Stahl, H. (1996), ‘Convergence of rational interpolants’, Bull. Belg. Math. Soc. Simon Stevin 3, 11–32.Google Scholar
Stewart, G. W. (2002), ‘A Krylov–Schur algorithm for large eigenproblems’, SIAM J. Matrix Anal. Appl. 23, 601–614.Google Scholar
Su, Y. and Bai, Z. (2011), ‘Solving rational eigenvalue problems via linearization’, SIAM J. Matrix Anal. Appl. 32, 201–216.Google Scholar
Szyld, D. B. and Xue, F. (2013a)), ‘Local convergence analysis of several inexact Newton-type algorithms for general nonlinear eigenvalue problems’, Numer. Math. 123, 333–362.Google Scholar
Szyld, D. B. and Xue, F. (2013b)), ‘Several properties of invariant pairs of nonlinear algebraic eigenvalue problems’, IMA J. Numer. Anal. 34, 921–954.Google Scholar
Szyld, D. B. and Xue, F. (2015), ‘Local convergence of Newton-like methods for degenerate eigenvalues of nonlinear eigenproblems I: Classical algorithms’, Numer. Math. 129, 353–381.Google Scholar
Szyld, D. B. and Xue, F. (2016), ‘Preconditioned eigensolvers for large-scale nonlinear Hermitian eigenproblems with variational characterizations I: Extreme eigenvalues’, Math. Comp. 85(302), 2887–2918.Google Scholar
Tisseur, F. (2000), ‘Backward error and condition of polynomial eigenvalue problems’, Linear Algebra Appl. 309, 339–361.Google Scholar
Tisseur, F. and Higham, N. J. (2001), ‘Structured pseudospectra for polynomial eigenvalue problems, with applications’, SIAM J. Matrix Anal. Appl. 23, 187–208.Google Scholar
Tisseur, F. and Meerbergen, K. (2001), ‘The quadratic eigenvalue problem’, SIAM Rev. 43, 235–286.Google Scholar
Trefethen, L. N. and Embree, M. (2005), Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton University Press.Google Scholar
Trefethen, L. N. and Weideman, J. (2014), ‘The exponentially convergent trapezoidal rule’, SIAM Rev. 56, 385–458.Google Scholar
Trofimov, V. (1968), ‘The root subspaces of operators that depend analytically on a parameter’, Mat. Issled 3, 117–125.Google Scholar
Unger, H. (1950), ‘Nichtlineare Behandlung von Eigenwertaufgaben’, Z. Angew. Math. Mech. 30, 281–282.Google Scholar
Van Barel, M. (2016), ‘Designing rational filter functions for solving eigenvalue problems by contour integration’, Linear Algebra Appl. 502, 346–365.Google Scholar
Van Barel, M. and Kravanja, P. (2016), ‘Nonlinear eigenvalue problems and contour integrals’, J. Comput. Appl. Math. 292, 526–540.Google Scholar
Van Beeumen, R. (2015), Rational Krylov methods for nonlinear eigenvalue problems. PhD thesis, KU Leuven.CrossRefGoogle Scholar
Van Beeumen, R., Jarlebring, E. and Michiels, W. (2016a)), ‘A rank-exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems’, Numer. Linear Algebra Appl. 23, 607–628.Google Scholar
Van Beeumen, R., Meerbergen, K. and Michiels, W. (2013), ‘A rational Krylov method based on Hermite interpolation for nonlinear eigenvalue problems’, SIAM J. Sci. Comput. 35, A327–A350.Google Scholar
Van Beeumen, R., Meerbergen, K. and Michiels, W. (2015a)), ‘Compact rational Krylov methods for nonlinear eigenvalue problems’, SIAM J. Matrix Anal. Appl. 36, 820–838.Google Scholar
Van Beeumen, R., Meerbergen, K. and Michiels, W. (2016b), Connections between contour integration and rational Krylov methods for eigenvalue problems. Technical report TW673, Department of Computer Science, KU Leuven.Google Scholar
Van Beeumen, R., Michiels, W. and Meerbergen, K. (2015b)), ‘Linearization of Lagrange and Hermite interpolating matrix polynomials’, IMA J. Numer. Anal. 35, 909–930.Google Scholar
van der Vorst, H. A. (1992), ‘Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems’, SIAM J. Sci. Statist. Comput. 13, 631–644.Google Scholar
Verhees, D., Van Beeumen, R., Meerbergen, K., Guglielmi, N. and Michiels, W. (2014), ‘Fast algorithms for computing the distance to instability of nonlinear eigenvalue problems, with application to time-delay systems’, Int. J. Dynam. Control 2, 133–142.Google Scholar
Voss, H. (2004a)), ‘An Arnoldi method for nonlinear eigenvalue problems’, BIT 44, 387–401.Google Scholar
Voss, H. (2004b)), Eigenvibrations of a plate with elastically attached loads. In European Congress on Computational Methods in Applied Sciences and Engineering: ECCOMAS 2004 (Neittaanmäki, P. et al. , ed.). http://www.mit.jyu.fi/eccomas2004/proceedings/proceed.html Google Scholar
Voss, H. (2007), ‘A Jacobi–Davidson method for nonlinear and nonsymmetric eigenproblems’, Comput. Struct. 85, 1284–1292.Google Scholar
Voss, H. (2009), ‘A minmax principle for nonlinear eigenproblems depending continuously on the eigenparameter’, Numer. Linear Algebra Appl. 16, 899–913.Google Scholar
Voss, H. (2014), Nonlinear eigenvalue problems. In Handbook of Linear Algebra, (Hogben, L., ed.), second edition, Chapman and Hall/CRC, 115:1–115:24.Google Scholar
Voss, H. and Werner, B. (1982), ‘A minimax principle for nonlinear eigenvalue problems with applications to nonoverdamped systems’, Math. Methods Appl. Sci. 4, 415–424.Google Scholar
Walsh, J. L. (1932), ‘On interpolation and approximation by rational functions with preassigned poles’, Trans. Amer. Math. Soc. 34, 22–74.Google Scholar
Walsh, J. L. (1935), Interpolation and Approximation by Rational Functions in the Complex Domain, Vol. 20 of American Mathematical Society Colloquium Publications, AMS.Google Scholar
Werner, B. (1970), Das Spektrum von Operatorenscharen mit verallgemeinerten Rayleighquotienten. PhD thesis, Universität Hamburg.Google Scholar
Weyl, H. (1912), ‘Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)’, Math. Ann. 71, 441–479.Google Scholar
Wieners, C. and Xin, J. (2013), ‘Boundary element approximation for Maxwell’s eigenvalue problem’, Math. Methods Appl. Sci. 36, 2524–2539.Google Scholar
Wilkinson, J. H. (1965), The Algebraic Eigenvalue Problem, Oxford University Press.Google Scholar
Wobst, R. (1987), ‘The generalized eigenvalue problem and acoustic surface wave computations’, Computing 39, 57–69.Google Scholar
Xiao, J., Zhang, C., Huang, T.-M. and Sakurai, T. (2016a)), ‘Solving large-scale nonlinear eigenvalue problems by rational interpolation and resolvent sampling based Rayleigh–Ritz method’, Internat. J. Numer. Methods Engrg. https://doi.org/10.1002/nme.5441 Google Scholar
Xiao, J., Zhou, H., Zhang, C. and Xu, C. (2016b)), ‘Solving large-scale finite element nonlinear eigenvalue problems by resolvent sampling based Rayleigh–Ritz method’, Comput. Mech. 1–18.Google Scholar
Yang, W. H. (1983), ‘A method for eigenvalues of sparse $\unicode[STIX]{x1D706}$ -matrices’, Internat. J. Numer. Methods Engrg 19, 943–948.Google Scholar
Yokota, S. and Sakurai, T. (2013), ‘A projection method for nonlinear eigenvalue problems using contour integrals’, JSIAM Letters 5, 41–44.Google Scholar