Keywords: eigenvalue, eigenvector, nonlinear eigenproblem, iterative projection method, Jacobi-Davidson method, rational eigenproblem, damped vibrations of structures.

In this paper we consider the nonlinear eigenvalue problem

where is a family of large and sparse matrices depending on a parameter . Problems of this type arise in damped vibrations of structures, free vibrations of plates with elastically attached masses, and of rotating structures, stability of linear systems with retarded argument, lateral buckling problems or vibrations of fluid-solid structures, to name just a few. As in the linear case a parameter is called an eigenvalue of if problem (65) has a nontrivial solution which is called a corresponding eigenvector.

For linear sparse eigenproblems iterative projection methods such as the Lanczos, Arnoldi, rational Krylov or Jacobi-Davidson methods are very efficient. In these approaches one determines approximations to the wanted eigenvalues and corresponding eigenvectors from projections of the large eigenproblem to low-dimensional subspaces which are expanded in the course of the algorithm. The small projected eigenproblems are solved by standard techniques.

In some sense, Ruhe [1] generalised the rational Krylov approach to sparse nonlinear eigenvalue problems by nesting the linearisation of problem (65) by Regula falsi and the solution of the resulting linear eigenproblem by Arnoldi's method, where the Regula falsi iteration and the Arnoldi recursion are knit together. Similarly as in the rational Krylov process he constructs a sequence of subspaces of , and at the same time he updates Hessenberg matrices which approximate the projection of to . Here denotes a shift and an approximation to the wanted eigenvalue of (65). Then a Ritz vector of corresponding to an eigenvalue of small modulus approximates an eigenvector of the nonlinear problem from which a (hopefully) improved eigenvalue approximation of problem (65) is obtained. Hence, in this approach the two numerical subtasks, reducing the large dimension to a much smaller one, and solving the projected nonlinear eigenproblems are attacked simultaneously.

In this paper we suggest an iterative projection method for the nonlinear eigenproblem where the two subtasks mentioned in the last paragraph are handled separately. We order the eigenvalues in some way and determine them one after another. If denotes the subspace of of small dimension constructed in the course of the algorithm, and is an orthonormal basis of , we solve the projected nonlinear eigenvalue problem of dimension by a dense solver to obtain approximations and to an eigenvalue and eigenvector, respectively. Thereafter we expand the search space to span and repeat the projection step. Similarly as in the Jacobi-Davidson method for linear eigenproblems the expansion direction of is chosen such that for some has a high approximation potential for the eigenvector we are just aiming at. The projection step and the expansion step are repeated alternately until convergence.

Inverse iteration with variable shifts converges quadratically to simple eigenvalues. Therefore, the expansion would be a reasonable choice. However, in this case we would have to solve a high dimensional linear system in every iteration step where the coefficient matrix varies. A way out is to replace this expansion of the search space by suggested by residual inverse iteration. This yields to the nonlinear Arnoldi method proposed in [2].

For linear eigenproblems it is well known that the Jacobi-Davidson method yields the same expansion of the search space as inverse iteration if the the correction equation is solved exactly, and solving it only approximately still yields fast convergence. The same holds true for nonlinear eigenproblems if the search space is expanded by the solution of the correction equation

and as in the linear case this equation has to be solved only approximately in a Jacobi-Davidson like manner. For a symmetric problem allowing a minmax characterisation of its eigenvalues this was discussed in [3]. The present paper deals with the general case.

1

A. Ruhe. "A rational Krylov algorithm for nonlinear matrix eigenvalue problems", Zapiski Nauchnyh Seminarov POMI, 268, 176 - 180, 2000.

2

H. Voss. "An Arnoldi method for nonlinear eigenvalue problems", Technical Report 56, Section of Mathematics, Hamburg University of Technology, 2002. To appear in BIT Numerical Mathematics.

3

T. Betcke, H. Voss. "A Jacobi-Davidson-type projection method for nonlinear eigenvalue problems", Future Generation Computer Systems, 20, 363 - 372, 2004. doi:10.1016/j.future.2003.07.003

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