Nearly Optimal Preconditioned Methods for Hermitian Eigenproblems Under Limited Memory. Part II: Seeking Many Eigenvalues

Abstract

In a recent companion paper, we proposed two methods, GD+k and JDQMR, as nearly optimal methods for finding one eigenpair of a real symmetric matrix. In this paper, we seek nearly optimal methods for a large number, $nev$, of eigenpairs that work with a search space whose size is $O(1)$, independent from $nev$. The motivation is twofold: avoid the additional $O(nev N)$ storage and the $O(nev^2N)$ iteration costs. First, we provide an analysis of the oblique projectors required in the Jacobi–Davidson method and identify ways to avoid them during the inner iterations, either completely or partially. Second, we develop a comprehensive set of performance models for GD+k, Jacobi–Davidson type methods, and ARPACK. Based both on theoretical arguments and on our models we argue that any eigenmethod with $O(1)$ basis size, preconditioned or not, will be superseded asymptotically by Lanczos-type methods that use $O(nev)$ vectors in the basis. However, this may not happen until $nev > O(1000)$. Third, we perform an extensive set of experiments with our methods and against other state-of-the-art software that validates our models and confirms our GD+k and JDQMR methods as nearly optimal within the class of O(1) basis size methods.