Iterative Validation of Eigensolvers: A Scheme for Improving the Reliability of Hermitian Eigenvalue Solvers

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Abstract
Iterative eigenvalue solvers for large, sparse matrices may miss some of the required eigenvalues that are of high algebraic multiplicity or tightly clustered. Block methods, locking, a posteriori validation, or simply increasing the required accuracy are often used to avoid missing or to detect a missed eigenvalue, but each has its own shortcomings in robustness or performance. To resolve these shortcomings, we have developed a postprocessing algorithm, iterative validation of eigensolvers (IVE), that combines the advantages of each technique. IVE detects numerically multiple eigenvalues among the approximate eigenvalues returned by a given solver, adjusts the block size accordingly, then calls the given solver using locking to compute a new approximation in the subspace orthogonal to the current approximate eigenvectors. This process is repeated until no additional missed eigenvalues can be identified. IVE is general and can be applied as a wrapper to any Rayleigh–Ritz‐based, Hermitian eigensolver. Our experiments show that IVE is very effective in computing missed eigenvalues even with eigensolvers that lack locking or block capabilities, although such capabilities may further enhance robustness. By focusing on robustness in a postprocessing stage, IVE allows the user to decouple the notion of robustness from that of performance when choosing the block size or the convergence tolerance.