A fast iterative algorithm for near-diagonal eigenvalue problems

TL;DR

This paper presents a new iterative eigenvalue algorithm called IPT, inspired by perturbation theory, which efficiently computes multiple eigenpairs for near-diagonal matrices, outperforming classical methods especially in parallel computing environments.

Contribution

The paper introduces IPT, a novel parallelizable eigenvalue algorithm for near-diagonal matrices, offering improved efficiency over traditional methods.

Findings

IPT achieves faster convergence on dense and sparse matrices.

IPT outperforms classical eigenvalue algorithms like LAPACK and CUSOLVER.

Demonstrated effectiveness on quantum chemistry matrices.

Abstract

We introduce a novel eigenvalue algorithm for near-diagonal matrices inspired by Rayleigh-Schr\"odinger perturbation theory and termed Iterative Perturbative Theory (IPT). Contrary to standard eigenvalue algorithms, which are either 'direct' (to compute all eigenpairs) or 'iterative' (to compute just a few), IPT computes any number of eigenpairs with the same basic iterative procedure. Thanks to this perfect parallelism, IPT proves more efficient than classical methods (LAPACK or CUSOLVER for the full-spectrum problem, preconditioned Davidson solvers for extremal eigenvalues). We give sufficient conditions for linear convergence and demonstrate performance on dense and sparse test matrices, including one from quantum chemistry.