Adaptive Inner - Outer Inverse Iteration

Inverse iteration is a standard technique for nding selected eigenvectors associated with eigenvalues which are known approximately. At each step the solution of a linear system of equations is required, which is usually done by factorizing the system matrix. When direct factorization is impractical, iterative methods can be applied for solving the linear systems. Several results about termination criteria for the inner iteration of this inner-outer iterative scheme have been published. Our experiences indicate that they are too stringent in practice. As an alternative an adaptive algorithm is motivated and proposed. The inner tolerance is decreased only when required. A detailed discussion of implementation details of the inner-outer iterative approach is given. Results with prototype implementations have been very promising: Adaptively selecting the tolerance in the inner iteration results in signi cant reductions in the computational e ort compared to other approaches.