Convergence Estimates for Preconditioned Gradient Subspace Iteration Eigensolvers

Subspace iteration for computing several eigenpairs (i.e. eigenvalues and eigenvectors) of an eigenvalue problem is an alternative to the deflation technique whereby the eigenpairs are computed successively by projecting the problem onto the subspace orthogonal to the already found eigenvectors. The main advantage of the subspace iteration over the deflation is its ‘cluster robustness’: even if some of the computed eigenvalues form a cluster (i.e. are very close to each other), the convergence does not deteriorate. For standard subspace iteration eigensolvers the above fact is well-known, and it is supported by convergence estimates. This paper tackles the so-called preconditioned gradient subspace iteration eigensolvers – a relatively new class of methods designed to efficiently compute several extreme eigenpairs of large-scale eigenvalue problems. Using a new approach to the convergence analysis of subspace iterations, based on dealing with eigenvalue sums rather than individual eigenvalues, the paper presents new convergence results for a class of preconditioned gradient subspace iteration eigensolvers which are fully cluster robust, i.e. involve the distances between the eigenvalues in a cluster neither in the assumptions nor in the estimates themselves.