Keywords: eigenvalue, eigenvector, iterative projection method, robust expansion, Jacobi--Davidson method, nonlinear Arnoldi method, structure preservation.

A wide variety of applications require the solution of a large--scale nonlinear eigenvalue problem T(lambda)x=0 where T(lambda) is a family of matrices depending on a complex parameter lambda in D. As in the linear case lambda is called an eigenvalue of T(.) if the equation T(lambda)x=0 has a nontrivial solution x not= 0, and x is called a corresponding eigenvector.

Quadratic problems T(lambda):=lambda^2 M+lambda C+K arise in the dynamic analysis of structures, or vibrations of spinning structures yielding conservative gyroscopic systems, constrained least squares problems, and control of linear mechanical systems with a quadratic cost functional. Polynomial eigenvalue problems of higher degree than two arise when discretising a linear eigenproblem by dynamic elements or by least squares elements, and the study of corner singularities in anisotropic elastic materials. Rational eigenproblems govern free vibrations of fluid solid structures or describe the electronic states of semi-conductor hetero-structures. Finally, a more general dependence on the eigenparameter appears in vibrations of poroelastic and piezoelectric structures and in the stability analysis of vibrating systems under state delay feedback control.

Most of the examples mentioned above are large and sparse, and typically only a small number of eigenvalues in a particular complex domain are of interest. Numerical methods have to exploit the sparseness fully to be efficient in storage and computing time.

In this paper we review iterative projection methods for general (i.e. not necessarily polynomial) sparse nonlinear eigenvalue problems which generalise the Jacobi--Davidson approach for linear problems and which have proved to be very efficient. Here the eigenvalue problem is projected to a subspace V of small dimension which yields approximate eigenpairs. If an error tolerance is not met then the search space V is expanded in an iterative way with the aim that some of the eigenvalues of the reduced matrix become good approximations of some of the wanted eigenvalues of the given large matrix. Methods of this type are the nonlinear Arnoldi method [1,2] and the Jacobi-Davidson method [3-5].

[1]

H. Voss, "An Arnoldi method for nonlinear eigenvalue problems", BIT Numerical Mathematics, 44, 387-401, 2004. doi:10.1023/B:BITN.0000039424.56697.8b

[2]

K. Meerbergen, "Fast frequency response computation for Rayleigh damping", Internat. J. Numer. Meth. Engrg., 73, 96-106, 2008. doi:10.1002/nme.2058

[3]

T. Betcke, H. Voss, "A Jacobi-Davidson-type projection method for nonlinear eigenvalue problems", Future Generation Comput. Syst., 20(3), 363-372, 2004. doi:10.1016/S0167-739X(03)00179-1

[4]

G.L. Sleijpen, G.L. Booten, D.R. Fokkema, H.A. van der Vorst, "Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems", BIT, 36, 595-633, 1996. doi:10.1007/BF01731936

[5]

H. Voss, "A Jacobi-Davidson method for nonlinear and nonsymmetric eigenproblems", Computers & Structures, 85, 1284-1292, 2007. doi:10.1016/j.compstruc.2006.08.088

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