On the complexity of the multiplication of matrices of small formats

We prove a lower bound of 2mn + 2n - m - 2 for the bilinear complexity of the multiplication of n × m-matrices with m × n-matrices using the substitution method (m ≥ n ≥ 3). In particular, we obtain the improved lower bound of 19 for the bilinear complexity of 3 × 3-matrix multiplication.

[1]  Volker Strassen,et al.  On the Algorithmic Complexity of Associative Algebras , 1981, Theor. Comput. Sci..

[2]  Michael Clausen,et al.  Algebraic complexity theory , 1997, Grundlehren der mathematischen Wissenschaften.

[3]  V. Strassen.  Gaussian elimination is not optimal , 1969 .

[4]  V. Strassen.  Relative bilinear complexity and matrix multiplication. , 1987 .

[5]  Julian D. Laderman,et al.  A noncommutative algorithm for multiplying $3 \times 3$ matrices using 23 multiplications , 1976 .

[6]  Abraham Waksman.  On Winograd's Algorithm for Inner Products , 1970, IEEE Transactions on Computers.

[7]  David P. Dobkin,et al.  On the optimal evaluation of a set of bilinear forms , 1973, SWAT.

[8]  Hans F. de Groote.  Lectures on the Complexity of Bilinear Problems , 1987, Lecture Notes in Computer Science.

[9]  Rodney W. Johnson,et al.  Noncommutative Bilinear Algorithms for 3 x 3 Matrix Multiplication , 1986, SIAM J. Comput..

[10]  V. Pan.  METHODS OF COMPUTING VALUES OF POLYNOMIALS , 1966 .

[11]  Markus Bläser.  Lower bounds for the bilinear complexity of associative algebras , 2000, computational complexity.

[12]  Arnold Schönhage,et al.  Partial and Total Matrix Multiplication , 1981, SIAM J. Comput..

[13]  Hans F. de Groote.  On Varieties of Optimal Algorithms for the Computation of Bilinear Mappings. II. Optimal Algorithms for 2x2-Matrix Multiplication , 1978, Theor. Comput. Sci..

[14]  L. R. Kerr,et al.  On Minimizing the Number of Multiplications Necessary for Matrix Multiplication , 1969 .

[15]  M. Bläser.  A - Lower Bound for the Rank of - Matrix Multiplication over Arbitrary Fields , 1999, FOCS 1999.

[16]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[17]  Hans F. de Groote.  On Varieties of Optimal Algorithms for the Computation of Bilinear Mappings I. The Isotropy Group of a Bilinear Mapping , 1978, Theor. Comput. Sci..

[18]  Shmuel Winograd,et al.  On multiplication of 2 × 2 matrices , 1971 .

[19]  Grazia Lotti,et al.  O(n2.7799) Complexity for n*n Approximate Matrix Multiplication , 1979, Inf. Process. Lett..

[20]  Markus Bläser.  Lower bounds for the multiplicative complexity of matrix multiplication , 1999, computational complexity.