Robust Rayleigh quotient minimization and nonlinear eigenvalue problems

A Comparison of Parallel Preconditioners for the Sparse Generalized Eigenvalue Problems by Rayleigh-Quotient Minimization

Nonlinear Rank-one Modification of the Symmetric Eigenvalue Problem

Nonlinear rank-one modification of the symmetric eigenvalue problem arises from eigenvibrations of mechanical structures with elastically attached loads and calculation of the propagation modes in optical fiber. In this paper, we first study the existence and uniqueness of eigenvalues, and then investigate three numerical algorithms, namely Picard iteration, nonlinear Rayleigh quotient iteratio...

Convergence orders of iterative methods for nonlinear eigenvalue problems

The convergence analysis of iterative methods for nonlinear eigenvalue problems is in the most cases restricted either to algebraic simple eigenvalues or to polynomial eigenvalue problems. In this paper we consider two classical methods for general holomorphic eigenvalue problems, namely the nonlinear generalized Rayleigh quotient iteration (NGRQI) and the augmented Newton method. For both meth...

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...

Convergence Analysis of Restarted Krylov Subspace Eigensolvers

The A-gradient minimization of the Rayleigh quotient allows to construct robust and fastconvergent eigensolvers for the generalized eigenvalue problem for (A,M) with symmetric and positive definite matrices. The A-gradient steepest descent iteration is the simplest case of more general restarted Krylov subspace iterations for the special case that all step-wise generated Krylov subspaces are tw...