Description of fast matrix multiplication algorithm: ⟨6×6×7:185⟩

Algorithm type

3⁢X4⁢Y4⁢Z4+6⁢X4⁢Y2⁢Z2+32⁢X3⁢Y2⁢Z3+48⁢X3⁢Y⁢Z3+24⁢X2⁢Y2⁢Z2+36⁢X2⁢Y⁢Z+36⁢X⁢Y⁢Z3X4Y4Z46X4Y2Z232X3Y2Z348X3YZ324X2Y2Z236X2YZ36XYZ3*X^4*Y^4*Z^4+6*X^4*Y^2*Z^2+32*X^3*Y^2*Z^3+48*X^3*Y*Z^3+24*X^2*Y^2*Z^2+36*X^2*Y*Z+36*X*Y*Z

Algorithm definition

The algorithm ⟨6×6×7:185⟩ could be constructed using the following decomposition:

⟨6×6×7:185⟩ = ⟨6×6×4:105⟩ + ⟨6×6×3:80⟩.

This decomposition is defined by the following equality:

Trace⁡Mul⁡A_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_4_1A_4_2A_4_3A_4_4A_4_5A_4_6A_5_1A_5_2A_5_3A_5_4A_5_5A_5_6A_6_1A_6_2A_6_3A_6_4A_6_5A_6_6B_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_7C_1_1C_1_2C_1_3C_1_4C_1_5C_1_6C_2_1C_2_2C_2_3C_2_4C_2_5C_2_6C_3_1C_3_2C_3_3C_3_4C_3_5C_3_6C_4_1C_4_2C_4_3C_4_4C_4_5C_4_6C_5_1C_5_2C_5_3C_5_4C_5_5C_5_6C_6_1C_6_2C_6_3C_6_4C_6_5C_6_6C_7_1C_7_2C_7_3C_7_4C_7_5C_7_6=Trace⁡Mul⁡A_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_4_1A_4_2A_4_3A_4_4A_4_5A_4_6A_5_1A_5_2A_5_3A_5_4A_5_5A_5_6A_6_1A_6_2A_6_3A_6_4A_6_5A_6_6B_1_1B_1_2B_1_3B_1_4B_2_1B_2_2B_2_3B_2_4B_3_1B_3_2B_3_3B_3_4B_4_1B_4_2B_4_3B_4_4B_5_1B_5_2B_5_3B_5_4B_6_1B_6_2B_6_3B_6_4C_1_1C_1_2C_1_3C_1_4C_1_5C_1_6C_2_1C_2_2C_2_3C_2_4C_2_5C_2_6C_3_1C_3_2C_3_3C_3_4C_3_5C_3_6C_4_1C_4_2C_4_3C_4_4C_4_5C_4_6+Trace⁡Mul⁡A_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_4_1A_4_2A_4_3A_4_4A_4_5A_4_6A_5_1A_5_2A_5_3A_5_4A_5_5A_5_6A_6_1A_6_2A_6_3A_6_4A_6_5A_6_6B_1_5B_1_6B_1_7B_2_5B_2_6B_2_7B_3_5B_3_6B_3_7B_4_5B_4_6B_4_7B_5_5B_5_6B_5_7B_6_5B_6_6B_6_7C_5_1C_5_2C_5_3C_5_4C_5_5C_5_6C_6_1C_6_2C_6_3C_6_4C_6_5C_6_6C_7_1C_7_2C_7_3C_7_4C_7_5C_7_6TraceMulA_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_4_1A_4_2A_4_3A_4_4A_4_5A_4_6A_5_1A_5_2A_5_3A_5_4A_5_5A_5_6A_6_1A_6_2A_6_3A_6_4A_6_5A_6_6B_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_7C_1_1C_1_2C_1_3C_1_4C_1_5C_1_6C_2_1C_2_2C_2_3C_2_4C_2_5C_2_6C_3_1C_3_2C_3_3C_3_4C_3_5C_3_6C_4_1C_4_2C_4_3C_4_4C_4_5C_4_6C_5_1C_5_2C_5_3C_5_4C_5_5C_5_6C_6_1C_6_2C_6_3C_6_4C_6_5C_6_6C_7_1C_7_2C_7_3C_7_4C_7_5C_7_6TraceMulA_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_4_1A_4_2A_4_3A_4_4A_4_5A_4_6A_5_1A_5_2A_5_3A_5_4A_5_5A_5_6A_6_1A_6_2A_6_3A_6_4A_6_5A_6_6B_1_1B_1_2B_1_3B_1_4B_2_1B_2_2B_2_3B_2_4B_3_1B_3_2B_3_3B_3_4B_4_1B_4_2B_4_3B_4_4B_5_1B_5_2B_5_3B_5_4B_6_1B_6_2B_6_3B_6_4C_1_1C_1_2C_1_3C_1_4C_1_5C_1_6C_2_1C_2_2C_2_3C_2_4C_2_5C_2_6C_3_1C_3_2C_3_3C_3_4C_3_5C_3_6C_4_1C_4_2C_4_3C_4_4C_4_5C_4_6TraceMulA_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_4_1A_4_2A_4_3A_4_4A_4_5A_4_6A_5_1A_5_2A_5_3A_5_4A_5_5A_5_6A_6_1A_6_2A_6_3A_6_4A_6_5A_6_6B_1_5B_1_6B_1_7B_2_5B_2_6B_2_7B_3_5B_3_6B_3_7B_4_5B_4_6B_4_7B_5_5B_5_6B_5_7B_6_5B_6_6B_6_7C_5_1C_5_2C_5_3C_5_4C_5_5C_5_6C_6_1C_6_2C_6_3C_6_4C_6_5C_6_6C_7_1C_7_2C_7_3C_7_4C_7_5C_7_6Trace(Mul(Matrix(6, 6, [[A_1_1,A_1_2,A_1_3,A_1_4,A_1_5,A_1_6],[A_2_1,A_2_2,A_2_3,A_2_4,A_2_5,A_2_6],[A_3_1,A_3_2,A_3_3,A_3_4,A_3_5,A_3_6],[A_4_1,A_4_2,A_4_3,A_4_4,A_4_5,A_4_6],[A_5_1,A_5_2,A_5_3,A_5_4,A_5_5,A_5_6],[A_6_1,A_6_2,A_6_3,A_6_4,A_6_5,A_6_6]]),Matrix(6, 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]]),Matrix(7, 6, [[C_1_1,C_1_2,C_1_3,C_1_4,C_1_5,C_1_6],[C_2_1,C_2_2,C_2_3,C_2_4,C_2_5,C_2_6],[C_3_1,C_3_2,C_3_3,C_3_4,C_3_5,C_3_6],[C_4_1,C_4_2,C_4_3,C_4_4,C_4_5,C_4_6],[C_5_1,C_5_2,C_5_3,C_5_4,C_5_5,C_5_6],[C_6_1,C_6_2,C_6_3,C_6_4,C_6_5,C_6_6],[C_7_1,C_7_2,C_7_3,C_7_4,C_7_5,C_7_6]]))) = Trace(Mul(Matrix(6, 6, [[A_1_1,A_1_2,A_1_3,A_1_4,A_1_5,A_1_6],[A_2_1,A_2_2,A_2_3,A_2_4,A_2_5,A_2_6],[A_3_1,A_3_2,A_3_3,A_3_4,A_3_5,A_3_6],[A_4_1,A_4_2,A_4_3,A_4_4,A_4_5,A_4_6],[A_5_1,A_5_2,A_5_3,A_5_4,A_5_5,A_5_6],[A_6_1,A_6_2,A_6_3,A_6_4,A_6_5,A_6_6]]),Matrix(6, 4, [[B_1_1,B_1_2,B_1_3,B_1_4],[B_2_1,B_2_2,B_2_3,B_2_4],[B_3_1,B_3_2,B_3_3,B_3_4],[B_4_1,B_4_2,B_4_3,B_4_4],[B_5_1,B_5_2,B_5_3,B_5_4],[B_6_1,B_6_2,B_6_3,B_6_4]]),Matrix(4, 6, [[C_1_1,C_1_2,C_1_3,C_1_4,C_1_5,C_1_6],[C_2_1,C_2_2,C_2_3,C_2_4,C_2_5,C_2_6],[C_3_1,C_3_2,C_3_3,C_3_4,C_3_5,C_3_6],[C_4_1,C_4_2,C_4_3,C_4_4,C_4_5,C_4_6]])))+Trace(Mul(Matrix(6, 6, [[A_1_1,A_1_2,A_1_3,A_1_4,A_1_5,A_1_6],[A_2_1,A_2_2,A_2_3,A_2_4,A_2_5,A_2_6],[A_3_1,A_3_2,A_3_3,A_3_4,A_3_5,A_3_6],[A_4_1,A_4_2,A_4_3,A_4_4,A_4_5,A_4_6],[A_5_1,A_5_2,A_5_3,A_5_4,A_5_5,A_5_6],[A_6_1,A_6_2,A_6_3,A_6_4,A_6_5,A_6_6]]),Matrix(6, 3, [[B_1_5,B_1_6,B_1_7],[B_2_5,B_2_6,B_2_7],[B_3_5,B_3_6,B_3_7],[B_4_5,B_4_6,B_4_7],[B_5_5,B_5_6,B_5_7],[B_6_5,B_6_6,B_6_7]]),Matrix(3, 6, [[C_5_1,C_5_2,C_5_3,C_5_4,C_5_5,C_5_6],[C_6_1,C_6_2,C_6_3,C_6_4,C_6_5,C_6_6],[C_7_1,C_7_2,C_7_3,C_7_4,C_7_5,C_7_6]])))

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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