Performance and Accuracy of LAPACK's Symmetric TridiagonalEigensolvers

We compare four algorithms from the latest LAPACK 3.1 release for computing eigenpairs of a symmetric tridiagonal matrix. These include QR iteration, bisection and inverse iteration (BI), the Divide-and-Conquer method (DC), and the method of Multiple Relatively Robust Representations (MR). Our evaluation considers speed and accuracy when computing all eigenpairs, and additionally subset computations. Using a variety of carefully selected test problems, our study includes a variety of today's computer architectures. Our conclusions can be summarized as follows. (1) DC and MR are generally much faster than QR and BI on large matrices. (2) MR almost always does the fewest floating point operations, but at a lower MFlop rate than all the other algorithms. (3) The exact performance of MR and DC strongly depends on the matrix at hand. (4) DC and QR are the most accurate algorithms with observed accuracy O({radical}ne). The accuracy of BI and MR is generally O(ne). (5) MR is preferable to BI for subset computations

A 2D algorithm with asymmetric workload for the UPC conjugate gradient method

This is a post-peer-review, pre-copyedit version of an article published in Journal of Supercomputing. The final authenticated version is available online at: https://doi.org/10.1007/s11227-014-1300-0[Abstract] This paper examines four different strategies, each one with its own data distribution, for implementing the parallel conjugate gradient (CG) method and how they impact communication and overall performance. Firstly, typical 1D and 2D distributions of the matrix involved in CG computations are considered. Then, a new 2D version of the CG method with asymmetric workload, based on leaving some threads idle during part of the computation to reduce communication, is proposed. The four strategies are independent of sparse storage schemes and are implemented using Unified Parallel C (UPC), a Partitioned Global Address Space (PGAS) language. The strategies are evaluated on two different platforms through a set of matrices that exhibit distinct sparse patterns, demonstrating that our asymmetric proposal outperforms the others except for one matrix on one platform.Ministerio de Economía y Competitividad; TIN2013-42148-PXunta de Galicia; GRC2013/055United States. Department of Energy; DEAC03-76SF0009

Eigensolvers and Applications in Finite Element Analyses

This article presents an overview of eigenproblems that arise in current finiteelement computations. We focus on a set of applications that have been studied at CERFACS, Centre Europ'een de Recherche et de Formation Avanc'ee en Calcul Scientifique, and describe the ideas and tools that have been developed to deal with them. The main characteristics of five different cases are given. We also discuss the trends as well as the research efforts to understand and tackle new applications. 1 INTRODUCTIO