Description of fast matrix multiplication algorithm: ⟨9×12×13:900⟩

Algorithm type

48⁢X6⁢Y6⁢Z4+72⁢X6⁢Y6⁢Z2+12⁢X4⁢Y4⁢Z4+12⁢X6⁢Y2⁢Z2+96⁢X3⁢Y3⁢Z4+24⁢X6⁢Y⁢Z+24⁢X4⁢Y2⁢Z2+240⁢X3⁢Y3⁢Z2+144⁢X3⁢Y3⁢Z+60⁢X2⁢Y2⁢Z2+24⁢X3⁢Y⁢Z+72⁢X2⁢Y⁢Z+72⁢X⁢Y⁢Z48X6Y6Z472X6Y6Z212X4Y4Z412X6Y2Z296X3Y3Z424X6YZ24X4Y2Z2240X3Y3Z2144X3Y3Z60X2Y2Z224X3YZ72X2YZ72XYZ48*X^6*Y^6*Z^4+72*X^6*Y^6*Z^2+12*X^4*Y^4*Z^4+12*X^6*Y^2*Z^2+96*X^3*Y^3*Z^4+24*X^6*Y*Z+24*X^4*Y^2*Z^2+240*X^3*Y^3*Z^2+144*X^3*Y^3*Z+60*X^2*Y^2*Z^2+24*X^3*Y*Z+72*X^2*Y*Z+72*X*Y*Z

Algorithm definition

The algorithm ⟨9×12×13:900⟩ could be constructed using the following decomposition:

⟨9×12×13:900⟩ = ⟨9×12×4:300⟩ + ⟨9×12×9:600⟩.

This decomposition is defined by the following equality:

Trace⁡Mul⁡A_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_1_7A_1_8A_1_9A_1_10A_1_11A_1_12A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_2_7A_2_8A_2_9A_2_10A_2_11A_2_12A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_3_7A_3_8A_3_9A_3_10A_3_11A_3_12A_4_1A_4_2A_4_3A_4_4A_4_5A_4_6A_4_7A_4_8A_4_9A_4_10A_4_11A_4_12A_5_1A_5_2A_5_3A_5_4A_5_5A_5_6A_5_7A_5_8A_5_9A_5_10A_5_11A_5_12A_6_1A_6_2A_6_3A_6_4A_6_5A_6_6A_6_7A_6_8A_6_9A_6_10A_6_11A_6_12A_7_1A_7_2A_7_3A_7_4A_7_5A_7_6A_7_7A_7_8A_7_9A_7_10A_7_11A_7_12A_8_1A_8_2A_8_3A_8_4A_8_5A_8_6A_8_7A_8_8A_8_9A_8_10A_8_11A_8_12A_9_1A_9_2A_9_3A_9_4A_9_5A_9_6A_9_7A_9_8A_9_9A_9_10A_9_11A_9_12B_1_1B_1_2B_1_3B_1_4B_1_5B_1_6B_1_7B_1_8B_1_9B_1_10B_1_11B_1_12B_1_13B_2_1B_2_2B_2_3B_2_4B_2_5B_2_6B_2_7B_2_8B_2_9B_2_10B_2_11B_2_12B_2_13B_3_1B_3_2B_3_3B_3_4B_3_5B_3_6B_3_7B_3_8B_3_9B_3_10B_3_11B_3_12B_3_13B_4_1B_4_2B_4_3B_4_4B_4_5B_4_6B_4_7B_4_8B_4_9B_4_10B_4_11B_4_12B_4_13B_5_1B_5_2B_5_3B_5_4B_5_5B_5_6B_5_7B_5_8B_5_9B_5_10B_5_11B_5_12B_5_13B_6_1B_6_2B_6_3B_6_4B_6_5B_6_6B_6_7B_6_8B_6_9B_6_10B_6_11B_6_12B_6_13B_7_1B_7_2B_7_3B_7_4B_7_5B_7_6B_7_7B_7_8B_7_9B_7_10B_7_11B_7_12B_7_13B_8_1B_8_2B_8_3B_8_4B_8_5B_8_6B_8_7B_8_8B_8_9B_8_10B_8_11B_8_12B_8_13B_9_1B_9_2B_9_3B_9_4B_9_5B_9_6B_9_7B_9_8B_9_9B_9_10B_9_11B_9_12B_9_13B_10_1B_10_2B_10_3B_10_4B_10_5B_10_6B_10_7B_10_8B_10_9B_10_10B_10_11B_10_12B_10_13B_11_1B_11_2B_11_3B_11_4B_11_5B_11_6B_11_7B_11_8B_11_9B_11_10B_11_11B_11_12B_11_13B_12_1B_12_2B_12_3B_12_4B_12_5B_12_6B_12_7B_12_8B_12_9B_12_10B_12_11B_12_12B_12_13C_1_1C_1_2C_1_3C_1_4C_1_5C_1_6C_1_7C_1_8C_1_9C_2_1C_2_2C_2_3C_2_4C_2_5C_2_6C_2_7C_2_8C_2_9C_3_1C_3_2C_3_3C_3_4C_3_5C_3_6C_3_7C_3_8C_3_9C_4_1C_4_2C_4_3C_4_4C_4_5C_4_6C_4_7C_4_8C_4_9C_5_1C_5_2C_5_3C_5_4C_5_5C_5_6C_5_7C_5_8C_5_9C_6_1C_6_2C_6_3C_6_4C_6_5C_6_6C_6_7C_6_8C_6_9C_7_1C_7_2C_7_3C_7_4C_7_5C_7_6C_7_7C_7_8C_7_9C_8_1C_8_2C_8_3C_8_4C_8_5C_8_6C_8_7C_8_8C_8_9C_9_1C_9_2C_9_3C_9_4C_9_5C_9_6C_9_7C_9_8C_9_9C_10_1C_10_2C_10_3C_10_4C_10_5C_10_6C_10_7C_10_8C_10_9C_11_1C_11_2C_11_3C_11_4C_11_5C_11_6C_11_7C_11_8C_11_9C_12_1C_12_2C_12_3C_12_4C_12_5C_12_6C_12_7C_12_8C_12_9C_13_1C_13_2C_13_3C_13_4C_13_5C_13_6C_13_7C_13_8C_13_9=Trace⁡Mul⁡A_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_1_7A_1_8A_1_9A_1_10A_1_11A_1_12A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_2_7A_2_8A_2_9A_2_10A_2_11A_2_12A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_3_7A_3_8A_3_9A_3_10A_3_11A_3_12A_4_1A_4_2A_4_3A_4_4A_4_5A_4_6A_4_7A_4_8A_4_9A_4_10A_4_11A_4_12A_5_1A_5_2A_5_3A_5_4A_5_5A_5_6A_5_7A_5_8A_5_9A_5_10A_5_11A_5_12A_6_1A_6_2A_6_3A_6_4A_6_5A_6_6A_6_7A_6_8A_6_9A_6_10A_6_11A_6_12A_7_1A_7_2A_7_3A_7_4A_7_5A_7_6A_7_7A_7_8A_7_9A_7_10A_7_11A_7_12A_8_1A_8_2A_8_3A_8_4A_8_5A_8_6A_8_7A_8_8A_8_9A_8_10A_8_11A_8_12A_9_1A_9_2A_9_3A_9_4A_9_5A_9_6A_9_7A_9_8A_9_9A_9_10A_9_11A_9_12B_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_4B_7_1B_7_2B_7_3B_7_4B_8_1B_8_2B_8_3B_8_4B_9_1B_9_2B_9_3B_9_4B_10_1B_10_2B_10_3B_10_4B_11_1B_11_2B_11_3B_11_4B_12_1B_12_2B_12_3B_12_4C_1_1C_1_2C_1_3C_1_4C_1_5C_1_6C_1_7C_1_8C_1_9C_2_1C_2_2C_2_3C_2_4C_2_5C_2_6C_2_7C_2_8C_2_9C_3_1C_3_2C_3_3C_3_4C_3_5C_3_6C_3_7C_3_8C_3_9C_4_1C_4_2C_4_3C_4_4C_4_5C_4_6C_4_7C_4_8C_4_9+Trace⁡Mul⁡A_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_1_7A_1_8A_1_9A_1_10A_1_11A_1_12A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_2_7A_2_8A_2_9A_2_10A_2_11A_2_12A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_3_7A_3_8A_3_9A_3_10A_3_11A_3_12A_4_1A_4_2A_4_3A_4_4A_4_5A_4_6A_4_7A_4_8A_4_9A_4_10A_4_11A_4_12A_5_1A_5_2A_5_3A_5_4A_5_5A_5_6A_5_7A_5_8A_5_9A_5_10A_5_11A_5_12A_6_1A_6_2A_6_3A_6_4A_6_5A_6_6A_6_7A_6_8A_6_9A_6_10A_6_11A_6_12A_7_1A_7_2A_7_3A_7_4A_7_5A_7_6A_7_7A_7_8A_7_9A_7_10A_7_11A_7_12A_8_1A_8_2A_8_3A_8_4A_8_5A_8_6A_8_7A_8_8A_8_9A_8_10A_8_11A_8_12A_9_1A_9_2A_9_3A_9_4A_9_5A_9_6A_9_7A_9_8A_9_9A_9_10A_9_11A_9_12B_1_5B_1_6B_1_7B_1_8B_1_9B_1_10B_1_11B_1_12B_1_13B_2_5B_2_6B_2_7B_2_8B_2_9B_2_10B_2_11B_2_12B_2_13B_3_5B_3_6B_3_7B_3_8B_3_9B_3_10B_3_11B_3_12B_3_13B_4_5B_4_6B_4_7B_4_8B_4_9B_4_10B_4_11B_4_12B_4_13B_5_5B_5_6B_5_7B_5_8B_5_9B_5_10B_5_11B_5_12B_5_13B_6_5B_6_6B_6_7B_6_8B_6_9B_6_10B_6_11B_6_12B_6_13B_7_5B_7_6B_7_7B_7_8B_7_9B_7_10B_7_11B_7_12B_7_13B_8_5B_8_6B_8_7B_8_8B_8_9B_8_10B_8_11B_8_12B_8_13B_9_5B_9_6B_9_7B_9_8B_9_9B_9_10B_9_11B_9_12B_9_13B_10_5B_10_6B_10_7B_10_8B_10_9B_10_10B_10_11B_10_12B_10_13B_11_5B_11_6B_11_7B_11_8B_11_9B_11_10B_11_11B_11_12B_11_13B_12_5B_12_6B_12_7B_12_8B_12_9B_12_10B_12_11B_12_12B_12_13C_5_1C_5_2C_5_3C_5_4C_5_5C_5_6C_5_7C_5_8C_5_9C_6_1C_6_2C_6_3C_6_4C_6_5C_6_6C_6_7C_6_8C_6_9C_7_1C_7_2C_7_3C_7_4C_7_5C_7_6C_7_7C_7_8C_7_9C_8_1C_8_2C_8_3C_8_4C_8_5C_8_6C_8_7C_8_8C_8_9C_9_1C_9_2C_9_3C_9_4C_9_5C_9_6C_9_7C_9_8C_9_9C_10_1C_10_2C_10_3C_10_4C_10_5C_10_6C_10_7C_10_8C_10_9C_11_1C_11_2C_11_3C_11_4C_11_5C_11_6C_11_7C_11_8C_11_9C_12_1C_12_2C_12_3C_12_4C_12_5C_12_6C_12_7C_12_8C_12_9C_13_1C_13_2C_13_3C_13_4C_13_5C_13_6C_13_7C_13_8C_13_9

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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