The Rational Krylov Algorithm for Nonlinear Eigenvalue Problems

Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods

“…reducing the large dimension to a much smaller one and solving a nonlinear eigenproblem are attacked simultaneously. This method was applied in [31,32] to the rational eigenvalue problem (9) governing damped vibrations of a structure.…”

“…Both these problems have real eigenvalues which can be characterized as min-max values of a Rayleigh functional [106], and in both cases one is interested in a small number of eigenvalues at the lower end of the spectrum or which are close to an excitation frequency. Another type of rational eigenproblem is obtained for the free vibrations of a structure if one uses a viscoelastic constitutive relation to describe the behavior of a material [31,32]. A finite element model takes the form…”

Solving nonlinear eigenproblems by AMLS

“…governing the damped vibrations of a structure where the behavior of the material in the equations of motion is modeled by a viscoelastic constitutive relation [3]. In particular we consider a finite element model of a three dimensional structure with 193617 degrees of freedom, and a stiffness and mass matrices which are quite populated with more than 7.6 million nonzero elements in the stiffness matrix.…”

A Jacobi–Davidson Method for Nonlinear Eigenproblems

“…Similar approaches for general nonlinear eigenproblems were studied in [3], [4], [8], [9], [12], and for symmetric problems allowing maxmin characterizations of the eigenvalues in [1] and [13].…”