A Communication-Avoiding Parallel Algorithm for the Symmetric Eigenvalue Problem

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.