A Projected Preconditioned Conjugate Gradient Algorithm for Computing Many Extreme Eigenpairs of a Hermitian Matrix

TL;DR

This paper introduces a block iterative algorithm that efficiently computes many of the smallest eigenvalues and eigenvectors of large Hermitian matrices, reducing computational costs and enabling high-performance implementation.

Contribution

The proposed projected preconditioned conjugate gradient algorithm reduces Rayleigh-Ritz calculations and leverages BLAS3 operations for efficient high-performance computing.

Findings

Fewer Rayleigh-Ritz calculations than existing methods

Effective for large invariant subspaces in electronic structure problems

Suitable for high concurrency implementations

Abstract

We present an iterative algorithm for computing an invariant subspace associated with the algebraically smallest eigenvalues of a large sparse or structured Hermitian matrix A. We are interested in the case in which the dimension of the invariant subspace is large (e.g., over several hundreds or thousands) even though it may still be small relative to the dimension of A. These problems arise from, for example, density functional theory based electronic structure calculations for complex materials. The key feature of our algorithm is that it performs fewer Rayleigh--Ritz calculations compared to existing algorithms such as the locally optimal precondition conjugate gradient or the Davidson algorithm. It is a block algorithm, hence can take advantage of efficient BLAS3 operations and be implemented with multiple levels of concurrency. We discuss a number of practical issues that must be addressed in order to implement the algorithm efficiently on a high performance computer.