Abstract. GPUs for numerical computations are becoming an attractive alternative in research. In this paper, we propose a new parallel processing environment for matrix multiplications by using both CPUs and GPUs. The execution time of matrix multiplications can be decreased to 40.1 % by our method, compared with using the fastest of either CPU only case or GPU only case. Our method performs well when matrix sizes are large. 1

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.

This paper provides some reflections on the field of mathematical software on the occasion of John Rice's 65th birthday. I describe some of the common themes of research in this field and recall some significant events in its evolution. Finally, I raise a number of issues that are of concern to future developments.

In the 90s, Dhillon and Parlett devised a new algorithm (Multiple Relatively Robust Representations, MRRR) for computing numerically orthogonal eigenvectors of a symmetric tridiagonal matrix T with ) cost. In this paper, we describe the design of pdsyevr, a ScaLAPACK implementation of the MRRR algorithm to compute the eigenpairs in parallel. It represents a substantial improvement over the symmetric eigensolver pdsyevx that is currently in ScaLAPACK and is going to be part of the next ScaLAPACK release. AMS subject classifications. 65F15, 65Y15. Key words. Multiple relatively robust representations, ScaLAPACK, symmetric eigenvalue problem, parallel computation, numerical software, design, implementation. 1.

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