### 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.

Original language | English |
---|---|

Pages (from-to) | 95-118 |

Number of pages | 24 |

Journal | Numerical Algorithms |

Volume | 59 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2012 Jan |

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### Keywords

- Accurate computations
- Error-free transformation
- Floating-point arithmetic
- Matrix multiplication

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*Numerical Algorithms*,

*59*(1), 95-118. https://doi.org/10.1007/s11075-011-9478-1

**Error-free transformations of matrix multiplication by using fast routines of matrix multiplication and its applications.** / Ozaki, Katsuhisa; Ogita, Takeshi; Oishi, Shin'ichi; Rump, Siegfried M.

Research output: Contribution to journal › Article

*Numerical Algorithms*, vol. 59, no. 1, pp. 95-118. https://doi.org/10.1007/s11075-011-9478-1

}

TY - JOUR

T1 - Error-free transformations of matrix multiplication by using fast routines of matrix multiplication and its applications

AU - Ozaki, Katsuhisa

AU - Ogita, Takeshi

AU - Oishi, Shin'ichi

AU - Rump, Siegfried M.

PY - 2012/1

Y1 - 2012/1

N2 - 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.

AB - 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.

KW - Accurate computations

KW - Error-free transformation

KW - Floating-point arithmetic

KW - Matrix multiplication

UR - http://www.scopus.com/inward/record.url?scp=82255175133&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=82255175133&partnerID=8YFLogxK

U2 - 10.1007/s11075-011-9478-1

DO - 10.1007/s11075-011-9478-1

M3 - Article

VL - 59

SP - 95

EP - 118

JO - Numerical Algorithms

JF - Numerical Algorithms

SN - 1017-1398

IS - 1

ER -