A communication-avoiding parallel algorithm for the symmetric eigenvalue problem

TL;DR

This paper presents a new parallel algorithm for symmetric eigenvalue problems that significantly reduces communication costs by using innovative reduction techniques and a novel QR factorization, improving efficiency in large-scale computations.

Contribution

It introduces a communication-avoiding parallel algorithm with novel reduction methods and QR factorization, reducing interprocessor communication in symmetric eigenvalue computations.

Findings

Requires $ heta(\sqrt{c})$ less communication than previous algorithms.

Achieves reduction to banded matrices with fewer communication steps.

Employs new algorithms leveraging a novel QR factorization for rectangular matrices.

Abstract

Many large-scale scientific computations require eigenvalue solvers in a scaling regime where efficiency is limited by data movement. We introduce a parallel algorithm for computing the eigenvalues of a dense symmetric matrix, which performs asymptotically less communication than previously known approaches. We provide analysis in the Bulk Synchronous Parallel (BSP) model with additional consideration for communication between a local memory and cache. Given sufficient memory to store $c$ copies of the symmetric matrix, our algorithm requires $\Theta(\sqrt{c})$ less interprocessor communication than previously known algorithms, for any $c\leq p^{1/3}$ when using $p$ processors. The algorithm first reduces the dense symmetric matrix to a banded matrix with the same eigenvalues. Subsequently, the algorithm employs successive reduction to $O(\log p)$ thinner banded matrices. We employ two new parallel algorithms that achieve lower communication costs for the full-to-band and band-to-band reductions. Both of these algorithms leverage a novel QR factorization algorithm for rectangular matrices.