×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
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, (), August 29, 1986
[4] Coppersmith, D.; Winograd, S., Matrix multiplication via arithmetic progressions, (), 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, (), 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, (), 49-54
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.