Description of fast matrix multiplication algorithm: ⟨2 × 4 × 7:46⟩

Algorithm type

[[1, 1, 1]$26,[1, 2, 1]$6,[1, 3, 1]$4,[2, 2, 2]$10]

Algorithm definition

The algorithm ⟨2 × 4 × 7:46⟩ could be constructed using the following decomposition:

⟨2 × 4 × 7:46⟩ = ⟨2 × 4 × 3:20⟩ + ⟨2 × 4 × 4:26⟩.

This decomposition is defined by the following equality:

Trace⁡Mul⁡A_1_1A_1_2A_1_3A_1_4A_2_1A_2_2A_2_3A_2_4B_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_7C_1_1C_1_2C_2_1C_2_2C_3_1C_3_2C_4_1C_4_2C_5_1C_5_2C_6_1C_6_2C_7_1C_7_2=Trace⁡Mul⁡A_1_1A_1_2A_1_3A_1_4A_2_1A_2_2A_2_3A_2_4B_1_1B_1_2B_1_3B_2_1B_2_2B_2_3B_3_1B_3_2B_3_3B_4_1B_4_2B_4_3C_1_1C_1_2C_2_1C_2_2C_3_1C_3_2+Trace⁡Mul⁡A_1_1A_1_2A_1_3A_1_4A_2_1A_2_2A_2_3A_2_4B_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_7C_4_1C_4_2C_5_1C_5_2C_6_1C_6_2C_7_1C_7_2TraceMulA_1_1A_1_2A_1_3A_1_4A_2_1A_2_2A_2_3A_2_4B_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_7C_1_1C_1_2C_2_1C_2_2C_3_1C_3_2C_4_1C_4_2C_5_1C_5_2C_6_1C_6_2C_7_1C_7_2TraceMulA_1_1A_1_2A_1_3A_1_4A_2_1A_2_2A_2_3A_2_4B_1_1B_1_2B_1_3B_2_1B_2_2B_2_3B_3_1B_3_2B_3_3B_4_1B_4_2B_4_3C_1_1C_1_2C_2_1C_2_2C_3_1C_3_2TraceMulA_1_1A_1_2A_1_3A_1_4A_2_1A_2_2A_2_3A_2_4B_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_7C_4_1C_4_2C_5_1C_5_2C_6_1C_6_2C_7_1C_7_2Trace(Mul(Matrix(2, 4, [[A_1_1,A_1_2,A_1_3,A_1_4],[A_2_1,A_2_2,A_2_3,A_2_4]]),Matrix(4, 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]]),Matrix(7, 2, [[C_1_1,C_1_2],[C_2_1,C_2_2],[C_3_1,C_3_2],[C_4_1,C_4_2],[C_5_1,C_5_2],[C_6_1,C_6_2],[C_7_1,C_7_2]]))) = Trace(Mul(Matrix(2, 4, [[A_1_1,A_1_2,A_1_3,A_1_4],[A_2_1,A_2_2,A_2_3,A_2_4]]),Matrix(4, 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]]),Matrix(3, 2, [[C_1_1,C_1_2],[C_2_1,C_2_2],[C_3_1,C_3_2]])))+Trace(Mul(Matrix(2, 4, [[A_1_1,A_1_2,A_1_3,A_1_4],[A_2_1,A_2_2,A_2_3,A_2_4]]),Matrix(4, 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]]),Matrix(4, 2, [[C_4_1,C_4_2],[C_5_1,C_5_2],[C_6_1,C_6_2],[C_7_1,C_7_2]])))

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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