The Strassen algorithm and Winograd's variant accelerate matrix multiplication by using fewer arithmetic operations than standard matrix multiplication. Although many papers have been published to accelerate single-as well as double-precision matrix multiplication by using these algorithms, no research to date has been undertaken to accelerate multiple precision matrix multiplication. In this paper, we propose a multiple precision matrix multiplication program for matrices of any size and test its performance. We also reveal special properties of our program through its application to LU decomposition.
It is well known that Strassen and Winograd algorithms can reduce the computational costs associated with dense matrix multiplication. We have already shown that they are also very effective for software-based multiple precision floating-point arithmetic environments such as the MPFR/GMP library. In this paper, we show that we can obtain the same effectiveness for double-double (DD) and quadrupledouble (QD) environments supported by the QD library, and that parallelization can increase the speed of these multiple precision matrix multiplications. Finally, we demonstrate that our implemented parallelized Strassen and Winograd algorithms can increase the speed of parallelized LU decomposition.
The Strassen matrix multiplication can be categorized into divide-and-conquer algorithms, and they are known as the most efficient algorithms. We previously implemented them supporting multiple precision floating-point arithmetic using MPFR and Bailey’s QD libraries and have shown their effectiveness in our papers and open-source codes. In preparation for a future release, we have introduced an optimized triple-word floating-point arithmetic proposed by Fabiano et al., and we found its utility in our implementation of multiple precision matrix multiplication. In this paper, we demonstrate the effectiveness of the Strassen triple-double precision matrix multiplication through performance evaluation compared to those based on QD and MPFR libraries.
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