To utilize the full computing potential of the K computer and post-K computer, and to produce simulations with higher efficiency and accuracy, it is necessary to take advantage of numerical software libraries that have been tuned to the needs of computer science and applied mathematics. We are researching and developing a large-scale, high-performance numerical library called KMATHLIB. Typical library algorithms commonly employed in simulation programs are used to handle systems of linear equations, eigenvalue calculations, three-dimensional Fast Fourier transforms, and long-period random number generators. We maintain corresponding library algorithms such as EigenExa, KMATH_FFT3D, and KMATH_RANDOM. We are modifying the existing KMATHLIB algorithms for use as a component of KMATHLIB2, which works in a scalable manner on the post-K computer.

Furthermore, we are promoting R&D of innovative algorithms that can deal with the unprecedented challenges raised by the K computer, such as the eigenvalue problem of a nonsymmetric matrix, and higher order tensor calculations. In addition to the modification and development of such algorithms, we are extending a high-precision calculation framework developed for the K computer to use on the post-K computer. We have also developed a method to reduce accumulated errors, and which guarantees the reproducibility of calculations by controlling the number of effective digits and removing non-determinism hidden in parallel calculations and when repeating these calculations. Collaboration is another important issue. We are collaborating with researchers and companies in Japan and overseas with the aim of establishing fundamental technologies for numerical libraries that can continue to be used well into the future.

World Largest Dense Eigenvalue Computation
The solution of real symmetric dense Eigen value problems is one of the fundamental matrix computations. To date, several new high-performance Eigen solvers have been developed for peta and postpeta scale systems. One of these, the Eigen Exa Eigen solver, has been developed in Japan. Eigen Exa provides two routines: eigens, which is based on traditional tridiagonalization, and eigensx, which employs a new method via a pentadiagonal matrix. Recently, we conducted a detailed performance evaluation of Eigen Exa by using 4,800 nodes of the Oak leaf-FX supercomputer system. In this paper, we report the results of our evaluation, which is mainly focused on investigating the differences between the two routines.

The results clearly indicate both the advantages and disadvantages of eigensx over eigens, which will contribute to further performance improvement of Eigen Exa. The obtained results are also expected to be useful for other parallel dense matrix computations, in addition to Eigen value problems. We have successfully solved a world largest-scale dense eigenvalue problem (one million dimension) by EigenExa taking advantage of the overall nodes (82,944 processors) of K computer in 3,464 seconds. Our EigenExa achieves 1.7 PFLOPS (16% of the K computer’s peak performance). It is the world highest performance for solving an eigenvalue problem of a dense matrix.

CCSE JAEA
CREST project led by Prof. Tetsuya Sakurai at University of Tsukuba