A Parallel Eigensolver for Dense Symmetric Matrices Based on Multiple Relatively Robust Representations

Abstract

We present a new parallel algorithm for the dense symmetric eigenvalue/eigenvector problem that is based upon the tridiagonal eigensolver, Algorithm MR 3 , recently developed by Dhillon and Parlett. Algorithm MR 3 has a complexity of O(n 2) operations for computing all eigenvalues and eigenvectors of a symmetric tridiagonal problem. Moreover the algorithm requires only O(n) extra workspace and can be adapted to compute any subset of k eigenpairs in O(nk) time. In contrast, all earlier stable parallel algorithms for the tridiagonal eigenproblem require O(n 3) operations in the worst case, while some implementations, such as divide and conquer, have an extra O(n 2) memory requirement. The proposed parallel algorithm balances the workload equally among the processors by traversing a matrix-dependent representation tree which captures the sequence of computations performed by Algorithm MR 3. The resulting implementation allows problems of very large size to be solved efficiently-the largest dense eigenproblem solved in-core on a 256 processor machine with 2 GBytes of memory per processor is for a matrix of size 128,000 × 128,000, which required about 8 hours of CPU time. We present comparisons with other eigensolvers and results on matrices that arise in the applications of computational quantum chemistry and finite element modeling of automobile bodies.

The ELPA library: scalable parallel eigenvalue solutions for electronic structure theory and computational science

“…[51,52,53,54] For the purposes of this paper, we note that ELPA relies upon an efficient implementation of the divide-and-conquer approach, the steps of which are reviewed in Ref. [22].…”

Out‐of‐core macromolecular simulations on multithreaded architectures

“…including some OOC extensions (SOLAR, POOCLAPACK). Among the conventional solvers for the symmetric eigenproblem, in [7] we introduced a practical OOC-GPU implementation of a two-stage reduction-based eigensolver which first transforms the matrix C from dense to band form, to then refine this intermediate matrix to tridiagonal form, and finally obtain the eigenvalues using the MR 3 tridiagonal eigensolver [6,8]. There, we demonstrated how, by carefully orchestrating the PCI data transfers between host and accelerator, in-core performance is maintained or even increased for the OOC solution of general large-scale eigenproblems on hybrid CPU-GPU platforms.…”

Out-of-Core Solution of Eigenproblems for Macromolecular Simulations

“…In the subsequent stage,Ĉ is further reduced to a tridiagonal matrix T ∈ R n×n as Q T 2Ĉ Q 2 = T with Q 2 ∈ R n×n orthogonal. Finally, the eigenvalues of T (which are also those of the C) and the associated eigenvectors, in Z ∈ R n×s , are computed using, e.g., the MR 3 solver [7], [4]; and the eigenvectors are recovered from Y := Q 1 Q 2 Z.…”