Accelerating 128-bit Floating-Point Matrix Multiplication on FPGAs

General Matrix Multiplication (GEMM) is a fundamental operation widely used in scientific computations. Its performance and accuracy significantly impact the performance and accuracy of applications that depend on it. One such application is semidefinite programming (SDP), and it often requires binary128 or higher precision arithmetic to solve problems involving SDP stably. However, only some processors support binary128 arithmetic, which makes SDP solvers generally slow. In this study, we focused on accelerating GEMM with binary128 arithmetic on field-programmable gate arrays (FPGAs) to enable the flexible design of accelerators for the desired computations. Our binary128 GEMM designs on a recent high-performance FPGA achieved approximately 90GFlops, 147x faster than the computation executed on a recent CPU with 20 threads for large matrices. Using our binary128 GEMM design on the FPGA, we successfully accelerated two numerical applications: LU decomposition and SDP problems, for the first time.Comment: 12 pages, 8 figure

Numerical Simulations of Globular Cluster Formation

We examine various physical processes associated with the formation of globular clusters by using the three-dimensional Smoothed Particle Hydrodynamics (SPH) code. Our code includes radiative cooling of gases, star formation, energy feedback from stars including stellar winds and supernovae, and chemical enrichment by stars. We assume that, in the collapsing galaxy, isothermal cold clouds form through thermal condensations and become proto-globular clouds. We calculate the size of proto-globular clouds by solving the linearized equations for perturbation. We compute the evolution of the inner region of the proto-cloud with our SPH code for various initial radius and initial composition of gases. When the initial gases contain no heavy elements, the evolution of proto-clouds sensitively depends on the initial radius. For a smaller initial radius, the initial star burst is so intense that the subsequent star formation occurs in the central regions to form a dense star cluster as massive as the globular cluster. When the initial gases contain some heavy elements, the metallicity of gases affects the evolution and the final stellar mass. If the initial radius of the proto-globular clouds was relatively large, the formation of a star cluster as massive as the globular clusters requires the initial metallicity as high as [Fe/H] $\geq -2$. The self-enrichment of heavy elements in the star cluster does not occur in all cases.Comment: Accpeted for publication in the ApJ. Correctiong errors in Table