A Jacobi–Davidson method for nonlinear and nonsymmetric eigenproblems

For the nonlinear eigenvalue problem T(λ)x = 0 we consider a Jacobi–Davidson type iterative projection method for computing a few eigenvalues close to a given parameter. We discuss the numerical solution of the projected eigenvalue problems in particular for nonsymmetric systems. We present methods how to prevent the algorithm from converging to the same eigenpair repeatedly. To verify the Jacobi–Davidson method it is applied to a rational eigenvalue problem governing damped vibrations of a structure and to a damped gyroscopic eigenvalue problem.