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Error-free transformations of matrix multiplication by using fast routines of matrix multiplication and its applications

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Abstract

This paper is concerned with accurate matrix multiplication in floating-point arithmetic. Recently, an accurate summation algorithm was developed by Rump et al. (SIAM J Sci Comput 31(1):189–224, 2008). The key technique of their method is a fast error-free splitting of floating-point numbers. Using this technique, we first develop an error-free transformation of a product of two floating-point matrices into a sum of floating-point matrices. Next, we partially apply this error-free transformation and develop an algorithm which aims to output an accurate approximation of the matrix product. In addition, an a priori error estimate is given. It is a characteristic of the proposed method that in terms of computation as well as in terms of memory consumption, the dominant part of our algorithm is constituted by ordinary floating-point matrix multiplications. The routine for matrix multiplication is highly optimized using BLAS, so that our algorithms show a good computational performance. Although our algorithms require a significant amount of working memory, they are significantly faster than ‘gemmx’ in XBLAS when all sizes of matrices are large enough to realize nearly peak performance of ‘gemm’. Numerical examples illustrate the efficiency of the proposed method.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Shibaura Institute of Technology, 307 Fukasaku, Minuma-ku, Saitama-shi, Saitama, 337-8570, Japan

    Katsuhisa Ozaki

  2. Japan Science and Technology Agency (JST)/CREST, Tokyo, Japan

    Katsuhisa Ozaki, Takeshi Ogita & Shin’ichi Oishi

  3. Division of Mathematical Sciences, Tokyo Woman’s Christian University, 2-6-1 Zempukuji, Suginami-ku, Tokyo, 167-8585, Japan

    Takeshi Ogita

  4. Faculty and Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjyuku-ku, Tokyo, 169-0072, Japan

    Shin’ichi Oishi & Siegfried M. Rump

  5. Institute for Reliable Computing, Hamburg University of Technology, Schwarzenbergstr. 95, 21071, Hamburg, Germany

    Siegfried M. Rump

Authors
  1. Katsuhisa Ozaki
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  2. Takeshi Ogita
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  3. Shin’ichi Oishi
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  4. Siegfried M. Rump
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Correspondence to Katsuhisa Ozaki.

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Ozaki, K., Ogita, T., Oishi, S. et al. Error-free transformations of matrix multiplication by using fast routines of matrix multiplication and its applications. Numer Algor 59, 95–118 (2012). https://doi.org/10.1007/s11075-011-9478-1

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  • DOI: https://doi.org/10.1007/s11075-011-9478-1

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