  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.
