Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Inexact Inverse Subspace Iteration with Preconditioning Applied to Non-Hermitian Eigenvalue Problems
SIAM Journal on Matrix Analysis and Applications
A block Newton method for nonlinear eigenvalue problems
Numerische Mathematik
Convergence Analysis of Iterative Solvers in Inexact Rayleigh Quotient Iteration
SIAM Journal on Matrix Analysis and Applications
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This work is focused on the computation of the invariant subspace associated with a separated group of eigenvalues near a specified shift of a large sparse matrix. First, we consider the inverse subspace iteration with the preconditioned GMRES method. It guarantees a convergence to the desired invariant subspace but the rate of convergence is at best linear. We propose to use it as a preprocessing for a Newton scheme which necessitates, at each iteration, the solution of a Sylvester type equation for which an iterative algorithm based on the preconditioned GMRES method is specially devised. This combination results in a fast and reliable method. We discuss the implementation aspects and propose a theory of convergence. Numerical tests are given to illustrate our approach.