SCALAPACK’S MRRR ALGORITHM

Abstract. The sequential algorithm of Multiple Relatively Robust Representations, MRRR, can compute numerically orthogonal eigenvectors of an unreduced symmetric tridiagonal matrix T ∈ Rn×n with O(n2) cost. This paper describes the design of ScaLAPACK’s parallel MRRR algorithm. One emphasis is on the critical role of the representation tree in achieving both numerical accuracy and parallel scalability. A second point concerns the favorable properties of this code: subset computation, the use of static memory, and scalability. Unlike ScaLAPACK’s Divide & Conquer and QR, MRRR can compute subsets of eigenpairs at reduced cost. And in contrast to inverse iteration, it is guaranteed to produce the right answer while maintaining memory scalability. Compared to ParEig, the parallel MRRR algorithm for PLAPACK, our code avoids dynamic memory allocation at marginal additional cost. It also uses a different representation tree criterion that allows for more accurate computation of the eigenvectors but can make parallelization more difficult. AMS subject classifications. 65F15, 65Y15. Key words. Symmetric eigenproblem, multiple relatively robust representations, ScaLAPACK