Zentrum für Informationsdienste und Hochleistungsrechnen (ZIH) A primal-dual Jacobi–Davidson-like method for nonlinear eigenvalue problems

We propose a method for the solution of non-Hermitian ill-conditioned nonlinear eigenvalue problems T (λ)x = 0, which is based on singularity theory of nonlinear equations. The singularity of T (λ) is characterized by a scalar condition μ(λ) = 0 where the singularity function μ is implicitly defined by a nonsingular linear system with the appropriately bordered matrix T (λ). One Newton step for μ(λ) = 0 is performed which leads to a generalized Rayleigh quotient as new eigenvalue approximation, and the bordering vectors are updated, too. Quadratic convergence is shown. Rearranging of the system leads to equivalent Jacobi–Davidson-like correction equations. An important difference to standard JD is that the correction equations do not depend on the derivative of T and that only orthogonal projectors occur. Numerical examples demonstrate a considerably better performance of the new method compared to standard Jacobi–Davidson when applied to highly nonnormal eigenproblems.