Description of fast matrix multiplication algorithm: ⟨9×12×26:1688⟩

Algorithm type

160⁢X4⁢Y6⁢Z6+32⁢X2⁢Y9⁢Z3+272⁢X2⁢Y6⁢Z6+48⁢X2⁢Y3⁢Z9+32⁢X4⁢Y6⁢Z3+48⁢X⁢Y9⁢Z3+48⁢X⁢Y6⁢Z6+72⁢X⁢Y3⁢Z9+48⁢X6⁢Y3⁢Z3+2⁢X9⁢Y⁢Z+224⁢X2⁢Y6⁢Z3+16⁢X2⁢Y3⁢Z6+X8⁢Y⁢Z+2⁢X6⁢Y2⁢Z2+16⁢X4⁢Y3⁢Z3+264⁢X⁢Y6⁢Z3+24⁢X⁢Y3⁢Z6+12⁢X7⁢Y⁢Z+9⁢X5⁢Y2⁢Z2+72⁢X3⁢Y3⁢Z3+9⁢X6⁢Y⁢Z+7⁢X4⁢Y2⁢Z2+72⁢X2⁢Y3⁢Z3+8⁢X5⁢Y⁢Z+15⁢X3⁢Y2⁢Z2+72⁢X⁢Y3⁢Z3+10⁢X4⁢Y⁢Z+15⁢X2⁢Y2⁢Z2+10⁢X3⁢Y⁢Z+17⁢X2⁢Y⁢Z+51⁢X⁢Y⁢Z160X4Y6Z632X2Y9Z3272X2Y6Z648X2Y3Z932X4Y6Z348XY9Z348XY6Z672XY3Z948X6Y3Z32X9YZ224X2Y6Z316X2Y3Z6X8YZ2X6Y2Z216X4Y3Z3264XY6Z324XY3Z612X7YZ9X5Y2Z272X3Y3Z39X6YZ7X4Y2Z272X2Y3Z38X5YZ15X3Y2Z272XY3Z310X4YZ15X2Y2Z210X3YZ17X2YZ51XYZ160*X^4*Y^6*Z^6+32*X^2*Y^9*Z^3+272*X^2*Y^6*Z^6+48*X^2*Y^3*Z^9+32*X^4*Y^6*Z^3+48*X*Y^9*Z^3+48*X*Y^6*Z^6+72*X*Y^3*Z^9+48*X^6*Y^3*Z^3+2*X^9*Y*Z+224*X^2*Y^6*Z^3+16*X^2*Y^3*Z^6+X^8*Y*Z+2*X^6*Y^2*Z^2+16*X^4*Y^3*Z^3+264*X*Y^6*Z^3+24*X*Y^3*Z^6+12*X^7*Y*Z+9*X^5*Y^2*Z^2+72*X^3*Y^3*Z^3+9*X^6*Y*Z+7*X^4*Y^2*Z^2+72*X^2*Y^3*Z^3+8*X^5*Y*Z+15*X^3*Y^2*Z^2+72*X*Y^3*Z^3+10*X^4*Y*Z+15*X^2*Y^2*Z^2+10*X^3*Y*Z+17*X^2*Y*Z+51*X*Y*Z

Algorithm definition

The algorithm ⟨9×12×26:1688⟩ could be constructed using the following decomposition:

⟨9×12×26:1688⟩ = ⟨9×12×2:168⟩ + ⟨9×12×24:1520⟩.

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_1_14B_1_15B_1_16B_1_17B_1_18B_1_19B_1_20B_1_21B_1_22B_1_23B_1_24B_1_25B_1_26B_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_2_14B_2_15B_2_16B_2_17B_2_18B_2_19B_2_20B_2_21B_2_22B_2_23B_2_24B_2_25B_2_26B_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_3_14B_3_15B_3_16B_3_17B_3_18B_3_19B_3_20B_3_21B_3_22B_3_23B_3_24B_3_25B_3_26B_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_4_14B_4_15B_4_16B_4_17B_4_18B_4_19B_4_20B_4_21B_4_22B_4_23B_4_24B_4_25B_4_26B_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_5_14B_5_15B_5_16B_5_17B_5_18B_5_19B_5_20B_5_21B_5_22B_5_23B_5_24B_5_25B_5_26B_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_6_14B_6_15B_6_16B_6_17B_6_18B_6_19B_6_20B_6_21B_6_22B_6_23B_6_24B_6_25B_6_26B_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_7_14B_7_15B_7_16B_7_17B_7_18B_7_19B_7_20B_7_21B_7_22B_7_23B_7_24B_7_25B_7_26B_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_8_14B_8_15B_8_16B_8_17B_8_18B_8_19B_8_20B_8_21B_8_22B_8_23B_8_24B_8_25B_8_26B_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_9_14B_9_15B_9_16B_9_17B_9_18B_9_19B_9_20B_9_21B_9_22B_9_23B_9_24B_9_25B_9_26B_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_10_14B_10_15B_10_16B_10_17B_10_18B_10_19B_10_20B_10_21B_10_22B_10_23B_10_24B_10_25B_10_26B_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_11_14B_11_15B_11_16B_11_17B_11_18B_11_19B_11_20B_11_21B_11_22B_11_23B_11_24B_11_25B_11_26B_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_13B_12_14B_12_15B_12_16B_12_17B_12_18B_12_19B_12_20B_12_21B_12_22B_12_23B_12_24B_12_25B_12_26C_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_9C_14_1C_14_2C_14_3C_14_4C_14_5C_14_6C_14_7C_14_8C_14_9C_15_1C_15_2C_15_3C_15_4C_15_5C_15_6C_15_7C_15_8C_15_9C_16_1C_16_2C_16_3C_16_4C_16_5C_16_6C_16_7C_16_8C_16_9C_17_1C_17_2C_17_3C_17_4C_17_5C_17_6C_17_7C_17_8C_17_9C_18_1C_18_2C_18_3C_18_4C_18_5C_18_6C_18_7C_18_8C_18_9C_19_1C_19_2C_19_3C_19_4C_19_5C_19_6C_19_7C_19_8C_19_9C_20_1C_20_2C_20_3C_20_4C_20_5C_20_6C_20_7C_20_8C_20_9C_21_1C_21_2C_21_3C_21_4C_21_5C_21_6C_21_7C_21_8C_21_9C_22_1C_22_2C_22_3C_22_4C_22_5C_22_6C_22_7C_22_8C_22_9C_23_1C_23_2C_23_3C_23_4C_23_5C_23_6C_23_7C_23_8C_23_9C_24_1C_24_2C_24_3C_24_4C_24_5C_24_6C_24_7C_24_8C_24_9C_25_1C_25_2C_25_3C_25_4C_25_5C_25_6C_25_7C_25_8C_25_9C_26_1C_26_2C_26_3C_26_4C_26_5C_26_6C_26_7C_26_8C_26_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_2_1B_2_2B_3_1B_3_2B_4_1B_4_2B_5_1B_5_2B_6_1B_6_2B_7_1B_7_2B_8_1B_8_2B_9_1B_9_2B_10_1B_10_2B_11_1B_11_2B_12_1B_12_2C_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_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_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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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