Introduction to StarNEig -- A Task-based Library for Solving Nonsymmetric Eigenvalue Problems

In this paper, we present the StarNEig library for solving dense non-symmetric (generalized) eigenvalue problems. The library is built on top of the StarPU runtime system and targets both shared and distributed memory machines. Some components of the library support GPUs. The library is currently in an early beta state and only real arithmetic is supported. Support for complex data types is planned for a future release. This paper is aimed for potential users of the library. We describe the design choices and capabilities of the library, and contrast them to existing software such as ScaLAPACK. StarNEig implements a ScaLAPACK compatibility layer that should make it easy for a new user to transition to StarNEig. We demonstrate the performance of the library with a small set of computational experiments.Comment: 10 pages, 4 figures (10 when counting sub-figures), 2 tex-files. Submitted to PPAM 2019, 13th international conference on parallel processing and applied mathematics, September 8-11, 2019. Proceedings will be published after the conference by Springer in the LNCS series. Second author's first name is "Carl Christian" and last name "Kjelgaard Mikkelsen

Algorithm 1019 : A Task-based Multi-shift QR/QZ Algorithm with Aggressive Early Deflation

The QR algorithm is one of the three phases in the process of computing the eigenvalues and the eigenvectors of a dense nonsymmetric matrix. This paper describes a task-based QR algorithm for reducing an upper Hessenberg matrix to real Schur form. The task-based algorithm also supports generalized eigenvalue problems (QZ algorithm) but this paper concentrates on the standard case. The task-based algorithm adopts previous algorithmic improvements, such as tightly-coupled multi-shifts and Aggressive Early Deflation (AED), and also incorporates several new ideas that significantly improve the performance. This includes, but is not limited to, the elimination of several synchronization points, the dynamic merging of previously separate computational steps, the shortening and the prioritization of the critical path, and experimental GPU support. The task-based implementation is demonstrated to be multiple times faster than multi-threaded LAPACK and ScaLAPACK in both single-node and multi-node configurations on two different machines based on Intel and AMD CPUs. The implementation is built on top of the StarPU runtime system and is part of the open-source StarNEig library

A Task-Based Algorithm for Reordering the Eigenvalues of a Matrix in Real Schur Form

A task-based parallel algorithm for reordering the eigenvalues of a matrix in real Schur form is presented.The algorithm is realized on top of the StarPU runtime system.Only the aspects which are relevant for shared memory machines are discussed here, but the implementation can be configured to run on distributed memory machines as well.Various techniques to reduce the overhead and the core idle time are discussed.Computational experiments indicate that the new algorithm is between 1.5 and 6.6 times faster than a state of the art MPI-based implementation found in ScaLAPACK.With medium to large matrices, strong scaling efficiencies above 60\% up to 28 CPU cores are reported.The overhead and the core idle time are shown to be negligible with the exception of the smallest matrices and highest core counts

A parallel radix-4 block cyclic reduction algorithm

A conventional block cyclic reduction algorithm operates by halving the size of the linear system at each reduction step, that is, the algorithm is a radix-2 method. An algorithm analogous to the block cyclic reduction known as the radix-q partial solution variant of the cyclic reduction (PSCR) method allows the use of higher radix numbers and is thus more suitable for parallel architectures as it requires fever reduction steps. This paper presents an alternative and more intuitive way of deriving a radix-4 block cyclic reduction method for systems with a coefficient matrix of the form tridiag{ − I,D, − I}. This is performed by modifying an existing radix-2 block cyclic reduction method. The resulting algorithm is then parallelized by using the partial fraction technique. The parallel variant is demonstrated to be less computationally expensive when compared to the radix-2 block cyclic reduction method in the sense that the total number of emerging subproblems is reduced. The method is also shown to be numerically stable and equivalent to the radix-4 PSCR method. The numerical results archived correspond to the theoretical expectations.peerReviewe

D6.5 Evaluation of auto-tuning techniques

This work is c by the NLAFET Consortium, 2015–2018. Its duplication is allowed only for personal, educational, or research uses.NLAFE

Introduction to StarNEig : A Task-based Library for Solving Nonsymmetric Eigenvalue Problems

Abstract. In this paper, we present the StarNEig library for solvingdense nonsymmetric (generalized) eigenvalue problems. The library isbuilt on top of the StarPU runtime system and targets both shared anddistributed memory machines. Some components of the library supportGPUs. The library is currently in an early beta state and only real arith-metic is supported. Support for complex data types is planned for afuture release. This paper is aimed at potential users of the library. Wedescribe the design choices and capabilities of the library, and contrastthem to existing software such as ScaLAPACK. StarNEig implements aScaLAPACK compatibility layer that should make it easy for new usersto transition to StarNEig. We demonstrate the performance of the librarywith a small set of computational experiments.NLAFE

Fast Poisson solvers for graphics processing units

Two block cyclic reduction linear system solvers are considered and implemented using the OpenCL framework. The topics of interest include a simplified scalar cyclic reduction tridiagonal system solver and the impact of increasing the radix-number of the algorithm. Both implementations are tested for the Poisson problem in two and three dimensions, using a Nvidia GTX 580 series GPU and double precision floating-point arithmetic. The numerical results indicate up to 6-fold speed increase in the case of the two-dimensional problems and up to 3- fold speed increase in the case of the three-dimensional problems when compared to equivalent CPU implementations run on a Intel Core i7 quad-core CPU.peerReviewe

D6.5 Evaluation of auto-tuning techniques

This work is c by the NLAFET Consortium, 2015–2018. Its duplication is allowed only for personal, educational, or research uses.NLAFE

D6.5 Evaluation of auto-tuning techniques

This work is c by the NLAFET Consortium, 2015–2018. Its duplication is allowed only for personal, educational, or research uses.NLAFE