Criteria for Combining Inverse and Rayleigh Quotient Iteration

A method is presented to find selected eigenvalue-eigenvector pairs of the generalized problem $Ax = \lambda Bx$. Here A and B are real symmetric matrices and B is positive definite. It is further assumed that the matrices are large and sparse and that factorization of either of them is impractical. Our method finds an eigenvalue in a given interval $J = (\gamma - \eta ,\gamma + \eta )$ or determines that J is free of eigenvalues while computing an approximation to the eigenvalue closest to this interval. The corresponding eigenvector is also computed. The method consists of inverse and Rayleigh quotient iteration steps. The convergence is studied and it is shown how an inclusion theorem gives one of the criteria for switching from inverse to Rayleigh quotient iteration. The existence of an eigenvalue in the desired interval is guaranteed when this criterion is fulfilled. Some numerical experiments are reported which suggest that the method is optimal in the sense that when the mesh is refined the computational effort grows only linearly with the number of mesh points.