Block Algorithms with Augmented Rayleigh-Ritz Projections for Large-Scale Eigenpair Computation (Journal of Computational Mathematics, 2021)
Most iterative algorithms for eigenpair computation consist of two main steps: a subspace update (SU) step that generates bases for approximate eigenspaces, followed bya Rayleigh-Ritz (RR) projection step that extracts approximate eigenpairs. So far thepredominant methodology for the SU step is based on Krylov subspaces that builds orthonormal bases piece by piece in a sequential manner. In this work, we investigate blockmethods in the SU step that allow a higher level of concurrency than what is reachableby Krylov subspace methods. To achieve a competitive speed, we propose an augmentedRayleigh-Ritz (ARR) procedure. Combining this ARR procedure with a set of polynomialaccelerators, as well as utilizing a few other techniques such as continuation and deflation,we construct a block algorithm designed to reduce the number of RR steps and elevateconcurrency in the SU steps. Extensive computational experiments are conducted in $C$ ona representative set of test problems to evaluate the performance of two variants of ouralgorithm. Numerical results, obtained on a many-core computer without explicit codeparallelization, show that when computing a relatively large number of eigenpairs, the performance of our algorithms is competitive with that of several state-of-the-art eigensolvers.