University of Tsukuba | Center for Computational Sciences

Numerical Computation
Triple and Quadruple Precision BLAS Subroutines on GPUs
Background

Floating - point operations have round - off errors. These errors may become a critical issue for someapplications. Especially in large - scale computing, an accumulation of round – off errors may become a more
serious problem. Double precision accuracy may be insufficient in some cases, and there is a demand for higher
precision operations.

Overview

We have implemented triple and quadruple precision Basic Linear Algebra Subprograms (BLAS) subroutines,
AXPY (y = αx+y), GEMV (y = αAx+βy) and GEMM (C = αAB+βC) on GPUs. For quadruple precision, we used
Double – Double (DD) type quadruple precision operations (11bits exponent & 104bits significand). On theother hand, we propose Double+Single (D+S) type triple precision floating – point format (8bites exponent & 75
bits significand) and triple precision operations using DD – operations internally.

Performance

Some level - 1 and 2 BLAS subroutines are memory - bound on the Tesla M2090, not only in single and double,
but also triple and quadruple precision: the execution time of triple and quadruple precision subroutines is close
to only 1.5x and 2.0x of that of double precision. For memory - bound operations where double precision isinsufficient but quadruple precision is not needed, triple precision operations should be used.

Auto - Tuning of SpMV for CRS format
on GPUs
Background

Performance of sparse matrix - vector multiplication(SpMV) on GPUs is highly dependent on the structure ofthe sparse matrix used in the computation, the computing
environment, and the selection of certain parameters.

Overview

We show that the performance achieved using kernel
SpMV on GPUs for the compressed row storage (CRS)
format depends greatly on optimal selection of aparameter T that is a number of threads to compute an
output vector element, and we propose an efficientalgorithm for the automatic selection of the optimal
parameter.

PerformanceDevelopment of Parallel Sparse
Eigensolver Package: z - Pares
The aim of this research project is to develop numerical software for largescale eigenvalue problems in post-petascale computing environment. Aneigensolver based on contour integral (the SS method) has been proposedby Sakurai and Sugiura [3]. This method has a hierarchical structure and issuitable for massively parallel supercomputers [2]. Moreover, the SS
method can be applicable for nonlinear eigenvalue problem [1]. Block
Krylov method [4] improve the performance of the method. We aredeveloping software on both Trilinos and PETSc. MATLAB version is available
at http://zpares.cs.tsukuba.ac.jp/

Hierarchical Parallel Structure
Hardware is grouped according
to a hierarchical structure of thealgorithm.

Numerical Example on the K Computer
Application for band calculation with real
space density functional theory (RSDFT)[2].

CRS - T* that is kernel SpMV for the CRS format using
automatic parameter selection achieves up toapproximately 26% improvement over NVIDIA’s CUSPARSE
library.
Band structure of silicon nanowire of 9,924 atoms.
(matrix size = 8,719,488, Number of cores = 6,144)*The results are tentative since they are obtained byearly access to the K computer.
Reference[1] J. Asakura, T. Sakurai, H. Tadano, T. Ikegami and K. Kimura, A numerical method for nonlinear eigenvalue problems using
contour integrals, JSIAM Letters, 1 (2009) 52-55.
[2] Y. Futamura, T. Sakurai, S. Furuya and J.-I. Iwata, Efficient algorithm for linear systems arising in solutions ofeigenproblems and its application to electronic-structure calculations, Proc. 10th International Meeting on High-Performance
Computing for Computational Science (VECPAR 2012) (accepted).
[3] T. Sakurai and H. Sugiura, A projection method for generalized eigenvalue problems, J. Comput. Appl. Math., 159 (2003)
119-128.
[4] H. Tadano, T. Sakurai and Y. Kuramashi, Block BiCGGR: A new block Krylov subspace method for computing high accuracy
solutions, JSIAM Letters, 1 (2009) 44-47.
Acknowledgment
This research was partially supported by the Core Research for Evolutional Science and Technology (CREST) Project of JST and
the Grant-in-Aid for Scientific Re- search of Ministry of Education, Culture, Sports, Science and Technology, Japan, Grant
number: 23105702.

http://www.ccs.tsukuba.ac.jp/