The matrix capacity of a~tensor
Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 2, pp. 107-166.

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In 1990, D. Coppersmith and S. Winograd published an estimate of the amount of arithmetic operations necessary for the multiplication of square matrices $n\times n$, which equals $O(n^{2.3755})$. In this article, we make a systematization of the theoretical instruments that were used by D. Coppersmith and S. Winograd for their estimate. The improved estimate $O(n^{2.373})$ is one of the results of this systematization.
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D. V. Zhdanovich. The matrix capacity of a~tensor. Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 2, pp. 107-166. http://geodesic.mathdoc.fr/item/FPM_2012_17_2_a4/

[1] Dukhin A. A., Teoriya informatsii, Gelios ARV, M., 2007

[2] Behrend F., “On sets of integers which contain no three in arithmetic progression”, Proc. Nat. Acad. Sci. U.S.A., 32 (1946), 331–332 | DOI | MR | Zbl

[3] Bini D., “Relations between exact and approximate bilinear algorithms”, Calcolo, 17 (1980), 87–97 | DOI | MR | Zbl

[4] Bini D., Capovani M., Lotti G., Romani F., “$O(n^{2.7799})$ complexity for matrix multiplication. Strassen algorithm is not optimal”, Inform. Process. Lett., 8:5 (1979), 234–235 | DOI | MR | Zbl

[5] Coppersmith D., Winograd S., “Matrix multiplication via arithmetic progressions”, J. Symbol. Comput., 9 (1990), 251–280 | DOI | MR | Zbl

[6] Erdős P., Turan P., “On some sequences of integer”, J. London Math. Soc., 11 (1936), 261–264 | DOI | MR | Zbl

[7] Pan V., “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 Foundation of Computer Science, Ann Arbor, 1978, 166–176 | MR

[8] Salem R., Spencer D., “On sets of integers which contain no three in arithmetic progression”, Proc. Nat. Acad. Sci. U.S.A., 28 (1942), 561–563 | DOI | MR | Zbl

[9] Schönhage A., “Partial and total matrix multiplication”, SIAM J. Comput., 10:3 (1981), 434–456 | DOI | MR

[10] Strassen V., “Gaussian elimination is not optimal”, Numer. Math., 13 (1969), 354–356 | DOI | MR | Zbl

[11] Strassen V., “Relative bilinear complexity and matrix multiplication”, J. Reine Angew. Math., 375–376 (1987), 406–443 | MR | Zbl