This paper addresses emulation algorithms for matrix multiplication. General Matrix-Matrix Multiplication (GEMM), a fundamental operation in the Basic Linear Algebra Subprograms (BLAS), is typically optimized for specific hardware architectures. The Ozaki scheme is a well-established GEMM-based emulation method for matrix multiplication, wherein input matrices are decomposed into several low-precision components to ensure that the resulting matrix product is computed exactly through numerical operations. This study proposes a novel GEMM-based emulation method for matrix multiplication that leverages the Chinese Remainder Theorem. The proposed method inherits the computational efficiency of highly optimized GEMM routines and further enables control over the number of matrix multiplications, which can enhance computational accuracy. We present numerical experiments featuring INT8 Tensor Core operations on GPUs and FP64 arithmetic on CPUs as case studies. The results demonstrate that FP64 emulation using the proposed method achieves performance levels of up to 7.4 to 9.8 TFLOPS on the NVIDIA RTX 4090 and 56.6 to 80.2 TFLOPS on the NVIDIA GH200, exceeding the measured performance of native FP64 arithmetic. Furthermore, for FP64 computations on CPUs, the proposed method achieved up to a 2.3x speedup in emulating quadruple-precision arithmetic compared to the conventional Ozaki scheme.

We propose a method to rapidly generate matrices for real-symmetric eigenproblems. The proposed method produces a reproducible matrix with explicit eigenpairs, where the distribution of the eigenvalues can be controlled by a user. All elements of the generated matrix are rigorous floating-point numbers and can be represented in simple expressions involving the exact eigenvalues. The exact eigenpairs of the generated matrix are known in advance; thus, the proposed method contributes to the validation of errors in approximate eigenpairs. Several constraints on the matrix generated by the proposed method, were produced theoretically.

This study was aimed at simultaneously achieving sufficient accuracy and high performance for general matrix multiplications. Recent architectures, such as NVIDIA GPUs, feature high-performance units designed for low-precision matrix multiplications in machine learning models, and next-generation architectures are expected to follow the same design principle. The key to achieving superior performance is to fully leverage such architectures. The Ozaki scheme, a highly accurate matrix multiplication algorithm using error-free transformations, enables higher-precision matrix multiplication to be performed through multiple lower-precision matrix multiplications and higher-precision matrix additions. Ootomo et al. implemented the Ozaki scheme on high-performance matrix multiplication units with the aim of achieving both sufficient accuracy and high performance. This paper proposes alternative approaches to improving performance by reducing the numbers of lower-precision matrix multiplications and higher-precision matrix additions. Numerical experiments demonstrate the accuracy of the results and conduct performance benchmarks of the proposed approaches. These approaches are expected to yield more efficient results in next-generation architectures.

This study proposes a high-performance and reliable eigensolver via mixed-precision arithmetic between ordinary and highly-accurate precisions. Eigenvalue decomposition is ubiquitous in simulations. Various eigensolvers for computing approximations have been developed thus far. If eigenvalues are narrowly clustered, the computation of eigenvectors may be ill-posed. Thus, the computed eigenpairs may not be sufficiently accurate and lack reliability. In this study, we introduce mixed-precision iterative refinement methods to improve the accuracy of eigenvectors obtained using numerical methods. This approach contributes to obtaining sufficiently accurate results without arbitrary precision eigensolvers. We construct a high-performance and reliable eigensolver by combining the iterative refinement methods and EigenExa, a modern high-performance solver for large-scale and highly parallel computations. Numerical experiment results demonstrate the accuracy of the results and performance benchmark of the proposed approach.