Computing interior eigenpairs in augmented Krylov subspace produced by Jacobi–Davidson correction equation

As we know, the Jacobi–Davidson iteration method is very efficient for computing both extreme and interior eigenvalues of standard eigenvalue problems. However, the involved Jacobi–Davidson correction equation and the harmonic Rayleigh–Ritz process are more complicated and costly for computing the interior eigenvalues than those for computing the extreme eigenvalues. Thus, in this paper we adopt the originally elegant correction equation to generate the projection subspace in a skillful way, which is used to extract the desired eigenpair by the harmonic Rayleigh–Ritz process. The projection subspace, called as augmented Krylov subspace, inherits the benefits from both the standard Krylov subspace and the Jacobi–Davidson correction equation, which results in the constructed augmented Krylov subspace method being more effective than the Jacobi–Davidson method. A few numerical experiments are executed to exhibit the convergence and the competitiveness of the method.