Description of fast matrix multiplication algorithm: ⟨3×7×7:115⟩

Algorithm type

16X3Y3Z2+24X3Y3Z+X3Y2Z2+2X2Y3Z2+15X2Y2Z2+XY4Z+4XY3Z+4X2YZ+14XY2Z+34XYZ16X3Y3Z224X3Y3ZX3Y2Z22X2Y3Z215X2Y2Z2XY4Z4XY3Z4X2YZ14XY2Z34XYZ16*X^3*Y^3*Z^2+24*X^3*Y^3*Z+X^3*Y^2*Z^2+2*X^2*Y^3*Z^2+15*X^2*Y^2*Z^2+X*Y^4*Z+4*X*Y^3*Z+4*X^2*Y*Z+14*X*Y^2*Z+34*X*Y*Z

Algorithm definition

The algorithm ⟨3×7×7:115⟩ could be constructed using the following decomposition:

⟨3×7×7:115⟩ = ⟨3×7×3:49⟩ + ⟨3×7×4:66⟩.

This decomposition is defined by the following equality:

TraceMulA_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_1_7A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_2_7A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_3_7B_1_1B_1_2B_1_3B_1_4B_1_5B_1_6B_1_7B_2_1B_2_2B_2_3B_2_4B_2_5B_2_6B_2_7B_3_1B_3_2B_3_3B_3_4B_3_5B_3_6B_3_7B_4_1B_4_2B_4_3B_4_4B_4_5B_4_6B_4_7B_5_1B_5_2B_5_3B_5_4B_5_5B_5_6B_5_7B_6_1B_6_2B_6_3B_6_4B_6_5B_6_6B_6_7B_7_1B_7_2B_7_3B_7_4B_7_5B_7_6B_7_7C_1_1C_1_2C_1_3C_2_1C_2_2C_2_3C_3_1C_3_2C_3_3C_4_1C_4_2C_4_3C_5_1C_5_2C_5_3C_6_1C_6_2C_6_3C_7_1C_7_2C_7_3=TraceMulA_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_1_7A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_2_7A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_3_7B_1_1B_1_2B_1_3B_2_1B_2_2B_2_3B_3_1B_3_2B_3_3B_4_1B_4_2B_4_3B_5_1B_5_2B_5_3B_6_1B_6_2B_6_3B_7_1B_7_2B_7_3C_1_1C_1_2C_1_3C_2_1C_2_2C_2_3C_3_1C_3_2C_3_3+TraceMulA_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_1_7A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_2_7A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_3_7B_1_4B_1_5B_1_6B_1_7B_2_4B_2_5B_2_6B_2_7B_3_4B_3_5B_3_6B_3_7B_4_4B_4_5B_4_6B_4_7B_5_4B_5_5B_5_6B_5_7B_6_4B_6_5B_6_6B_6_7B_7_4B_7_5B_7_6B_7_7C_4_1C_4_2C_4_3C_5_1C_5_2C_5_3C_6_1C_6_2C_6_3C_7_1C_7_2C_7_3TraceMulA_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_1_7A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_2_7A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_3_7B_1_1B_1_2B_1_3B_1_4B_1_5B_1_6B_1_7B_2_1B_2_2B_2_3B_2_4B_2_5B_2_6B_2_7B_3_1B_3_2B_3_3B_3_4B_3_5B_3_6B_3_7B_4_1B_4_2B_4_3B_4_4B_4_5B_4_6B_4_7B_5_1B_5_2B_5_3B_5_4B_5_5B_5_6B_5_7B_6_1B_6_2B_6_3B_6_4B_6_5B_6_6B_6_7B_7_1B_7_2B_7_3B_7_4B_7_5B_7_6B_7_7C_1_1C_1_2C_1_3C_2_1C_2_2C_2_3C_3_1C_3_2C_3_3C_4_1C_4_2C_4_3C_5_1C_5_2C_5_3C_6_1C_6_2C_6_3C_7_1C_7_2C_7_3TraceMulA_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_1_7A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_2_7A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_3_7B_1_1B_1_2B_1_3B_2_1B_2_2B_2_3B_3_1B_3_2B_3_3B_4_1B_4_2B_4_3B_5_1B_5_2B_5_3B_6_1B_6_2B_6_3B_7_1B_7_2B_7_3C_1_1C_1_2C_1_3C_2_1C_2_2C_2_3C_3_1C_3_2C_3_3TraceMulA_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_1_7A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_2_7A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_3_7B_1_4B_1_5B_1_6B_1_7B_2_4B_2_5B_2_6B_2_7B_3_4B_3_5B_3_6B_3_7B_4_4B_4_5B_4_6B_4_7B_5_4B_5_5B_5_6B_5_7B_6_4B_6_5B_6_6B_6_7B_7_4B_7_5B_7_6B_7_7C_4_1C_4_2C_4_3C_5_1C_5_2C_5_3C_6_1C_6_2C_6_3C_7_1C_7_2C_7_3Trace(Mul(Matrix(3, 7, [[A_1_1,A_1_2,A_1_3,A_1_4,A_1_5,A_1_6,A_1_7],[A_2_1,A_2_2,A_2_3,A_2_4,A_2_5,A_2_6,A_2_7],[A_3_1,A_3_2,A_3_3,A_3_4,A_3_5,A_3_6,A_3_7]]),Matrix(7, 7, [[B_1_1,B_1_2,B_1_3,B_1_4,B_1_5,B_1_6,B_1_7],[B_2_1,B_2_2,B_2_3,B_2_4,B_2_5,B_2_6,B_2_7],[B_3_1,B_3_2,B_3_3,B_3_4,B_3_5,B_3_6,B_3_7],[B_4_1,B_4_2,B_4_3,B_4_4,B_4_5,B_4_6,B_4_7],[B_5_1,B_5_2,B_5_3,B_5_4,B_5_5,B_5_6,B_5_7],[B_6_1,B_6_2,B_6_3,B_6_4,B_6_5,B_6_6,B_6_7],[B_7_1,B_7_2,B_7_3,B_7_4,B_7_5,B_7_6,B_7_7]]),Matrix(7, 3, [[C_1_1,C_1_2,C_1_3],[C_2_1,C_2_2,C_2_3],[C_3_1,C_3_2,C_3_3],[C_4_1,C_4_2,C_4_3],[C_5_1,C_5_2,C_5_3],[C_6_1,C_6_2,C_6_3],[C_7_1,C_7_2,C_7_3]]))) = Trace(Mul(Matrix(3, 7, [[A_1_1,A_1_2,A_1_3,A_1_4,A_1_5,A_1_6,A_1_7],[A_2_1,A_2_2,A_2_3,A_2_4,A_2_5,A_2_6,A_2_7],[A_3_1,A_3_2,A_3_3,A_3_4,A_3_5,A_3_6,A_3_7]]),Matrix(7, 3, [[B_1_1,B_1_2,B_1_3],[B_2_1,B_2_2,B_2_3],[B_3_1,B_3_2,B_3_3],[B_4_1,B_4_2,B_4_3],[B_5_1,B_5_2,B_5_3],[B_6_1,B_6_2,B_6_3],[B_7_1,B_7_2,B_7_3]]),Matrix(3, 3, [[C_1_1,C_1_2,C_1_3],[C_2_1,C_2_2,C_2_3],[C_3_1,C_3_2,C_3_3]])))+Trace(Mul(Matrix(3, 7, [[A_1_1,A_1_2,A_1_3,A_1_4,A_1_5,A_1_6,A_1_7],[A_2_1,A_2_2,A_2_3,A_2_4,A_2_5,A_2_6,A_2_7],[A_3_1,A_3_2,A_3_3,A_3_4,A_3_5,A_3_6,A_3_7]]),Matrix(7, 4, [[B_1_4,B_1_5,B_1_6,B_1_7],[B_2_4,B_2_5,B_2_6,B_2_7],[B_3_4,B_3_5,B_3_6,B_3_7],[B_4_4,B_4_5,B_4_6,B_4_7],[B_5_4,B_5_5,B_5_6,B_5_7],[B_6_4,B_6_5,B_6_6,B_6_7],[B_7_4,B_7_5,B_7_6,B_7_7]]),Matrix(4, 3, [[C_4_1,C_4_2,C_4_3],[C_5_1,C_5_2,C_5_3],[C_6_1,C_6_2,C_6_3],[C_7_1,C_7_2,C_7_3]])))

N.B.: for any matrices A, B and C such that the expression Tr(Mul(A,B,C)) is defined, one can construct several trilinear homogeneous polynomials P(A,B,C) such that P(A,B,C)=Tr(Mul(A,B,C)) (P(A,B,C) variables are A,B and C's coefficients). Each trilinear P expression encodes a matrix multiplication algorithm: the coefficient in C_i_j of P(A,B,C) is the (i,j)-th entry of the matrix product Mul(A,B)=Transpose(C).

Algorithm description

These encodings are given in compressed text format using the maple computer algebra system. In each cases, the last line could be understood as a description of the encoding with respect to classical matrix multiplication algorithm. As these outputs are structured, one can construct easily a parser to its favorite format using the maple documentation without this software.


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