Locking and Restarting Quadratic Eigenvalue Solvers

• Authors: Karl Meerbergen
• Venue: SIAM Journal on Scientific Computing
• Year: 2000

This paper studies the solution of quadratic eigenvalue problems by the quadratic residual iteration method. The focus is on applications arising from finite-element simulations in acoustics. One approach is the shift-and-invert Arnoldi method applied to the linearized problem. When more than one eigenvalue is wanted, it is advisable to use locking or deflation of converged eigenvectors (or Schur vectors). In order to avoid unlimited growth of the subspace dimension, one can restart the method by purging unwanted eigenvectors (or Schur vectors). Both locking and restarting use the partial Schur form. The disadvantage of this approach is that the dimension of the linearized problem is twice that of the quadratic problem. The quadratic residual iteration and Jacobi--Davidson methods directly solve the quadratic problem. Unfortunately, the Schur form is not defined, nor are locking and restarting. This paper shows a link between methods for solving quadratic eigenvalue problems and the linearized problem. It aims to combine the benefits of the quadratic and the linearized approaches by employing a locking and restarting scheme based on the Schur form of the linearized problem in quadratic residual iteration and Jacobi--Davidson. Numerical experiments illustrate quadratic residual iteration and Jacobi--Davidson for computing the linear Schur form. It also makes a comparison with the shift-and-invert Arnoldi method.