Analyzing the convergence factor of residual inverse iteration

A complete characterization of the convergence factor can be very useful when analyzing the asymptotic convergence of an iterative method. We will here establish a formula for the convergence factor of the method called residual inverse iteration, which is a method for nonlinear eigenvalue problems and a generalization of the well known inverse iteration. The formula for the convergence factor is explicit and only involves quantities associated with the eigenvalue the iteration converges to, in particular the eigenvalue and eigenvector. Besides deriving the explicit formula we also use the formula to characterize the convergence of the method. In particular, we derive a formula for the first order expansion when the shift is close to the eigenvalue. The residual inverse iteration allows some freedom in the choice of a vector rk. In the analysis we characterize the convergence for different choices of rk. We use the explicit formula for the first order expansion to show that the convergence factor approaches zero when the shift approaches the eigenvalue for an arbitrary choice of rk. Moreover, we show that using an approximation of the left eigenvector as rk as proposed in the literature, is natural since it results in accurate eigenvalue approximations; but it is not necessarily optimal in terms of the first order expansion of the convergence factor when the shift is close to the eigenvalue. The convergence factor also allows us to completely characterize the behavior of the method for double eigenvalues. For non-semisimple double eigenvalues it turns out that the convergence factor is one, implying slow or no convergence at all. For the case of a semisimple eigenvalue, the method behaves in a similar as for a simple eigenvalue, except that when converging, the iterates converge to the subspace of eigenvectors, not necessarily to a particular eigenvector. manuscript No. (will be inserted by the editor) Analyzing the convergence factor of residual inverse iteration Elias Jarlebring · Wim Michiels Received: date / Accepted: date Abstract A complete characterization of the convergence factor can be very useful when analyzing the asymptotic convergence of an iterative method. We will here establish a formula for the convergence factor of the method called residual inverse iteration, which is a method for nonlinear eigenvalue problems and a generalization of the well known inverse iteration. The formula for the convergence factor is explicit and only involves quantities associated with the eigenvalue the iteration converges to, in particular the eigenvalue and eigenvector. Besides deriving the explicit formula we also use the formula to characterize the convergence of the method. In particular, we derive a formula for the first order expansion when the shift is close to the eigenvalue. The residual inverse iteration allows some freedom in the choice of a vector rk. In the analysis we characterize the convergence for different choices of rk. We use the explicit formula for the first order expansion to show that the convergence factor approaches zero when the shift approaches the eigenvalue for an arbitrary choice of rk. Moreover, we show that using an approximation of the left eigenvector as rk as proposed in the literature, is natural since it results in accurate eigenvalue approximations; but it is not necessarily optimal in terms of the first order expansion of the convergence factor when the shift is close to the eigenvalue. The convergence factor also allows us to completely characterize the behavior of the method for double eigenvalues. For non-semisimple double eigenvalues it turns out that the convergence factor is one, implying slow or no convergence at all. For the case of a semisimple eigenvalue, the method behaves in a similar as for a simple eigenvalue, except that when converging, the iterates converge to the subspace of eigenvectors, not necessarily to a particular eigenvector.A complete characterization of the convergence factor can be very useful when analyzing the asymptotic convergence of an iterative method. We will here establish a formula for the convergence factor of the method called residual inverse iteration, which is a method for nonlinear eigenvalue problems and a generalization of the well known inverse iteration. The formula for the convergence factor is explicit and only involves quantities associated with the eigenvalue the iteration converges to, in particular the eigenvalue and eigenvector. Besides deriving the explicit formula we also use the formula to characterize the convergence of the method. In particular, we derive a formula for the first order expansion when the shift is close to the eigenvalue. The residual inverse iteration allows some freedom in the choice of a vector rk. In the analysis we characterize the convergence for different choices of rk. We use the explicit formula for the first order expansion to show that the convergence factor approaches zero when the shift approaches the eigenvalue for an arbitrary choice of rk. Moreover, we show that using an approximation of the left eigenvector as rk as proposed in the literature, is natural since it results in accurate eigenvalue approximations; but it is not necessarily optimal in terms of the first order expansion of the convergence factor when the shift is close to the eigenvalue. The convergence factor also allows us to completely characterize the behavior of the method for double eigenvalues. For non-semisimple double eigenvalues it turns out that the convergence factor is one, implying slow or no convergence at all. For the case of a semisimple eigenvalue, the method behaves in a similar as for a simple eigenvalue, except that when converging, the iterates converge to the subspace of eigenvectors, not necessarily to a particular eigenvector.