### Abstract

In this study, we examine the accurate matrix multiplication in floating-point arithmetic. We demonstrate the error-free transformations of matrix multiplication using high performance basic linear algebra subprograms. These transformations can be applied to accurately compute the product of two matrices using floating-point entries. A key technique for this calculation is error-free splitting in floating-point matrices. In this study, we improve upon our previous method by a posteriori validation using floating-point exception. In the method, we utilize the presence of overflow in a positive manner for detecting whether rounding error occurs. If overflow occurs, the result contains some exceptional values such as ±∞ and NaN, that is, the method fails by necessity. Otherwise, we can confirm that no rounding error occurs in the process. Therefore, reducing the possibility of overflow is important. The numerical results suggest that the proposed algorithm provides more accurate results compared with the original algorithm. Moreover, for the product of n × n matrices, when n ≥5000, the new algorithm reduces the computing time for error-free transformation by an average of 20 % and up to 30 % compared with the original algorithm. Furthermore, the new algorithm can be used when matrix multiplication is performed using divide-and-conquer methods.

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

Pages (from-to) | 931-946 |

Number of pages | 16 |

Journal | Numerical Linear Algebra with Applications |

Volume | 23 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2016 Oct 1 |

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### Keywords

- accurate computing
- floating-point arithmetic
- interval arithmetic
- matrix multiplication

### ASJC Scopus subject areas

- Algebra and Number Theory
- Applied Mathematics

### Cite this

*Numerical Linear Algebra with Applications*,

*23*(5), 931-946. https://doi.org/10.1002/nla.2061

**Error-free transformation of matrix multiplication with a posteriori validation.** / Ozaki, Katsuhisa; Ogita, Takeshi; Oishi, Shinichi.

Research output: Contribution to journal › Article

*Numerical Linear Algebra with Applications*, vol. 23, no. 5, pp. 931-946. https://doi.org/10.1002/nla.2061

}

TY - JOUR

T1 - Error-free transformation of matrix multiplication with a posteriori validation

AU - Ozaki, Katsuhisa

AU - Ogita, Takeshi

AU - Oishi, Shinichi

PY - 2016/10/1

Y1 - 2016/10/1

N2 - In this study, we examine the accurate matrix multiplication in floating-point arithmetic. We demonstrate the error-free transformations of matrix multiplication using high performance basic linear algebra subprograms. These transformations can be applied to accurately compute the product of two matrices using floating-point entries. A key technique for this calculation is error-free splitting in floating-point matrices. In this study, we improve upon our previous method by a posteriori validation using floating-point exception. In the method, we utilize the presence of overflow in a positive manner for detecting whether rounding error occurs. If overflow occurs, the result contains some exceptional values such as ±∞ and NaN, that is, the method fails by necessity. Otherwise, we can confirm that no rounding error occurs in the process. Therefore, reducing the possibility of overflow is important. The numerical results suggest that the proposed algorithm provides more accurate results compared with the original algorithm. Moreover, for the product of n × n matrices, when n ≥5000, the new algorithm reduces the computing time for error-free transformation by an average of 20 % and up to 30 % compared with the original algorithm. Furthermore, the new algorithm can be used when matrix multiplication is performed using divide-and-conquer methods.

AB - In this study, we examine the accurate matrix multiplication in floating-point arithmetic. We demonstrate the error-free transformations of matrix multiplication using high performance basic linear algebra subprograms. These transformations can be applied to accurately compute the product of two matrices using floating-point entries. A key technique for this calculation is error-free splitting in floating-point matrices. In this study, we improve upon our previous method by a posteriori validation using floating-point exception. In the method, we utilize the presence of overflow in a positive manner for detecting whether rounding error occurs. If overflow occurs, the result contains some exceptional values such as ±∞ and NaN, that is, the method fails by necessity. Otherwise, we can confirm that no rounding error occurs in the process. Therefore, reducing the possibility of overflow is important. The numerical results suggest that the proposed algorithm provides more accurate results compared with the original algorithm. Moreover, for the product of n × n matrices, when n ≥5000, the new algorithm reduces the computing time for error-free transformation by an average of 20 % and up to 30 % compared with the original algorithm. Furthermore, the new algorithm can be used when matrix multiplication is performed using divide-and-conquer methods.

KW - accurate computing

KW - floating-point arithmetic

KW - interval arithmetic

KW - matrix multiplication

UR - http://www.scopus.com/inward/record.url?scp=84985930305&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84985930305&partnerID=8YFLogxK

U2 - 10.1002/nla.2061

DO - 10.1002/nla.2061

M3 - Article

AN - SCOPUS:84985930305

VL - 23

SP - 931

EP - 946

JO - Numerical Linear Algebra with Applications

JF - Numerical Linear Algebra with Applications

SN - 1070-5325

IS - 5

ER -