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**Matrix multiplication via arithmetic progressions.**
*(English)*
Zbl 0702.65046

A new method is presented for accelerating matrix multiplication asymptotically by a basic trilinear form which is not a matrix product. The method is based on Schönhage’s \(\tau\)-theorem, Strassen’s construction, and the Salem-Spencer theorem. The first variant of construction gives a matrix exponent \(\omega =2.404\). An improvement to of 2.388 is obtained by considering more terms and indices. Finally, some more complicated techniques offer a better estimate of 2.376. A combinatorial construction (whose realization is not guaranteed) is proposed which would yield \(\omega =2\).

Reviewer: O.Brudaru

### MSC:

65F30 | Other matrix algorithms (MSC2010) |

65Y20 | Complexity and performance of numerical algorithms |

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\textit{D. Coppersmith} and \textit{S. Winograd}, J. Symb. Comput. 9, No. 3, 251--280 (1990; Zbl 0702.65046)

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### References:

[1] | Behrend, F. A., On sets of integers which contain no three terms in arithmetical progression, Proc. Nat. Acad. Sci. USA, 32, 331-332 (1946) · Zbl 0060.10302 |

[2] | Coppersmith, D.; Winograd, S., On the Asymptotic Complexity of Matrix Multiplication, SIAM Journal on Computing, Vol. 11, No. 3, 472-492 (1982) · Zbl 0486.68030 |

[3] | Coppersmith, D.; Winograd, S., Matrix Multiplication via Behrend’s Theorem, (Research Report RC 12104 (1986), IBM T. J. Watson Research Center: IBM T. J. Watson Research Center Yorktown Heights, N.Y. 10598), August 29, 1986 |

[4] | Coppersmith, D.; Winograd, S., Matrix Multiplication via Arithmetic Progressions, (Proc. 19th Ann. ACM Symp. on theory of Computing (1987)), 1-6 |

[5] | Pan, V. Ya., Strassen Algorithm Is Not Optimal. Trilinear Technique of Aggregating Uniting and Canceling for Constructing Fast Algorithms for Matrix Multiplication, (Proc. 19th Ann. IEEE Symp. on Foundations of Computer Science (1978)), 166-176 |

[6] | Pan, V. Ya., How to Multiply Matrices Faster, Springer Lecture Notes in Computer Science, vol. 179 (1984) · Zbl 0548.65022 |

[7] | Schönhage, A., Partial and Total Matrix Multiplication, SIAM J. on Computing, 10, 3, 434-456 (1981) · Zbl 0462.68018 |

[8] | Salem, R.; Spencer, D. C., On sets of integers which contain no three terms in arithmetical progression, Proc. Nat. Acad. Sci. USA, 28, 561-563 (1942) · Zbl 0060.10301 |

[9] | Strassen, V., The Asymptotic Spectrum of Tensors and the Exponent of Matrix Multiplication, (Proc. 27th Ann. IEEE Symp. on Foundations of Computer Science (1986)), 49-54 |

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