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Convergence factors of Newton methods for nonlinear eigenvalue problems. (English) Zbl 1250.65072
Author’s abstract: Consider a complex sequence $$\{\lambda_k\}_{k=0}^{\infty}$$ convergent to $$\lambda _{\ast }\in \mathbb{C}$$ with order $$p\in N$$. The convergence factor is typically defined as the fraction $$c_{k}:=(\lambda _{k+1}-\lambda _{\ast })/(\lambda _{k}-\lambda _{\ast })^{p}$$ in the limit $$k\to \infty$$. In this paper, we prove formulas characterizing $$c_{k}$$ in the limit $$k\to \infty$$ for two different Newton-type methods for nonlinear eigenvalue problems. The formulas are expressed in terms of the left and right eigenvectors.
The two treated methods are called the method of successive linear problems (MSLP) and augmented Newton and are widely used in the literature. We prove several explicit formulas for $$c_{k}$$ for both methods. Formulas for both methods are found for simple as well as double eigenvalues. In some cases, we observe in examples that the limit $$c_{k}$$ as k$$\to \infty$$ does not exist. For cases where this limit does not appear to exist, we prove other limiting expressions such that a characterization of $$c_{k}$$ in the limit is still possible.

##### MSC:
 65H17 Numerical solution of nonlinear eigenvalue and eigenvector problems
##### Software:
NLEVP; DDE-BIFTOOL
Full Text:
##### References:
  Anselone, P.; Rall, L., The solution of characteristic value-vector problems by newton’s method, Numer. math., 11, 38-45, (1968) · Zbl 0162.46602  T. Betcke, N.J. Higham, V. Mehrmann, C. Schröder, F. Tisseur, NLEVP: a collection of nonlinear eigenvalue problems, Tech. Rep., Manchester Institute for Mathematical Sciences, 2008. · Zbl 1295.65140  Betcke, T.; Voss, H., A jacobi – davidson type projection method for nonlinear eigenvalue problems, Future gener. comput. systems, 20, 3, 363-372, (2004)  Engelborghs, K.; Luzyanina, T.; Roose, D., Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM trans. math. software, 28, 1, 1-24, (2002) · Zbl 1070.65556  Freitag, M.; Spence, A., Convergence of inexact inverse iteration with application to preconditioned iterative solves, Bit, 47, 27-44, (2007) · Zbl 1121.65038  Gohberg, I.; Lancaster, P.; Rodman, L., Matrix polynomials, (1982), Academic Press  Hryniv, R.; Lancaster, P., On the perturbation of analytic matrix functions, Integral equations operator theory, 34, 3, 325-338, (1999) · Zbl 0940.47008  Ipsen, I.C.F., Computing an eigenvector with inverse iteration, SIAM rev., 39, 2, 254-291, (1997) · Zbl 0874.65029  Jain, N.K.; Singhal, K., On kublanovskaya’s approach to the solution of the generalized latent value problem for functional lambda-matrices, SIAM J. numer. anal., 20, 1062-1070, (1983) · Zbl 0528.65019  E. Jarlebring, The spectrum of delay-differential equations: numerical methods, stability and perturbation, Ph.D. Thesis, TU Braunschweig, 2008. · Zbl 1183.34001  Jarlebring, E.; Michiels, W., Invariance properties in the root sensitivity of time-delay systems with double imaginary roots, Automatica, 46, 1112-1115, (2010) · Zbl 1191.93049  Kressner, D., A block Newton method for nonlinear eigenvalue problems, Numer. math., 114, 2, 355-372, (2009) · Zbl 1191.65054  Kublanovskaya, V., On an approach to the solution of the generalized latent value problem for $$\lambda$$-matrices, SIAM J. numer. anal., 7, 532-537, (1970) · Zbl 0225.65048  Lancaster, P., A generalized Rayleigh quotient iteration for lambda-matrices, Arch. ration. mech. anal., 8, 309-322, (1961) · Zbl 0105.31705  Lancaster, P., Lambda-matrices and vibrating systems, (2002), Dover Publications Mineola, NY · Zbl 1048.34002  Li, R.-C., Compute multiply nonlinear eigenvalues, J. comput. math., 10, 1, 1-20, (1992)  Mehrmann, V.; Voss, H., Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods, GAMM mitt., 27, 121-152, (2004) · Zbl 1071.65074  Michiels, W.; Niculescu, S.-I., Stability and stabilization of time-delay systems: an eigenvalue-based approach, () · Zbl 1052.93029  Neumaier, A., Residual inverse iteration for the nonlinear eigenvalue problem, SIAM J. numer. anal., 22, 914-923, (1985) · Zbl 0594.65026  J. Ortega, W. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, SIAM, 2000, ISBN: 0-89871-461-3. · Zbl 0949.65053  Ostrowski, A., On the convergence of the Rayleigh quotient iteration for the computation of the characteristic roots and vectors. I, II, Arch. ration. mech. anal., 1, 233-241, (1959) · Zbl 0083.12002  Peters, G.; Wilkinson, J., Inverse iterations, ill-conditioned equations and newton’s method, SIAM rev., 21, 339-360, (1979) · Zbl 0424.65021  Rall, L., A note on the convergence of newton’s method, SIAM J. numer. anal., 11, 34-36, (1974) · Zbl 0239.65052  Ruhe, A., Algorithms for the nonlinear eigenvalue problem, SIAM J. numer. anal., 10, 674-689, (1973) · Zbl 0261.65032  K. Schreiber, Nonlinear eigenvalue problems: Newton-type methods and nonlinear Rayleigh functionals, Ph.D. Thesis, TU Berlin, 2008. · Zbl 1213.65064  Schröder, J., Über das newtonsche verfahren, Arch. ration. mech. anal., 1, 154-180, (1957) · Zbl 0079.13605  H. Schwetlick, K. Schreiber, A primal – dual Jacobi-Davidson-like method for nonlinear eigenvalue problems, Tech. Rep. ZIH-IR-0613, Techn. Univ. Dresden, Zentrum für Informationsdienste und Hochleistungsrechnen, 2006, pp. 1-20.  Sleijpen, G.L.; Booten, A.G.; Fokkema, D.R.; van der Vorst, H.A., Jacobi – davidson type methods for generalized eigenproblems and polynomial eigenproblems, Bit, 36, 3, 595-633, (1996) · Zbl 0861.65035  Tisseur, F.; Meerbergen, K., The quadratic eigenvalue problem, SIAM rev., 43, 2, 235-286, (2001) · Zbl 0985.65028  Unger, H., Nichtlineare behandlung von eigenwertaufgaben, Z. angew. math. mech., 30, 281-282, (1950), English Trans. Available from: · Zbl 0040.06602  H. Voss, Numerical methods for sparse nonlinear eigenvalue problems, in: Proceedings of the XVth Summer School on Software and Algorithms of Numerical Mathematics, Hejnice, Czech Republic, 2004, Report 70, Arbeitsbereich Mathematik, TU Hamburg-Harburg.  Yang, W.H., A method for eigenvalues of sparse lambda-matrices, Int. J. numer. methods eng., 19, 943-948, (1983) · Zbl 0517.65018
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