Description of fast matrix multiplication algorithm: ⟨3×4×22:200⟩

Algorithm type

12⁢X2⁢Y3⁢Z2+52⁢X2⁢Y2⁢Z2+4⁢X⁢Y4⁢Z+4⁢X3⁢Y⁢Z+12⁢X2⁢Y⁢Z2+24⁢X⁢Y3⁢Z+42⁢X⁢Y2⁢Z+18⁢X⁢Y⁢Z2+32⁢X⁢Y⁢Z12X2Y3Z252X2Y2Z24XY4Z4X3YZ12X2YZ224XY3Z42XY2Z18XYZ232XYZ12*X^2*Y^3*Z^2+52*X^2*Y^2*Z^2+4*X*Y^4*Z+4*X^3*Y*Z+12*X^2*Y*Z^2+24*X*Y^3*Z+42*X*Y^2*Z+18*X*Y*Z^2+32*X*Y*Z

Algorithm definition

The algorithm ⟨3×4×22:200⟩ could be constructed using the following decomposition:

⟨3×4×22:200⟩ = ⟨3×4×2:20⟩ + ⟨3×4×20:180⟩.

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_4A_3_1A_3_2A_3_3A_3_4B_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_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_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_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_22C_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_3C_8_1C_8_2C_8_3C_9_1C_9_2C_9_3C_10_1C_10_2C_10_3C_11_1C_11_2C_11_3C_12_1C_12_2C_12_3C_13_1C_13_2C_13_3C_14_1C_14_2C_14_3C_15_1C_15_2C_15_3C_16_1C_16_2C_16_3C_17_1C_17_2C_17_3C_18_1C_18_2C_18_3C_19_1C_19_2C_19_3C_20_1C_20_2C_20_3C_21_1C_21_2C_21_3C_22_1C_22_2C_22_3=Trace⁡Mul⁡A_1_1A_1_2A_1_3A_1_4A_2_1A_2_2A_2_3A_2_4A_3_1A_3_2A_3_3A_3_4B_1_1B_1_2B_2_1B_2_2B_3_1B_3_2B_4_1B_4_2C_1_1C_1_2C_1_3C_2_1C_2_2C_2_3+Trace⁡Mul⁡A_1_1A_1_2A_1_3A_1_4A_2_1A_2_2A_2_3A_2_4A_3_1A_3_2A_3_3A_3_4B_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_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_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_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_22C_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_3C_8_1C_8_2C_8_3C_9_1C_9_2C_9_3C_10_1C_10_2C_10_3C_11_1C_11_2C_11_3C_12_1C_12_2C_12_3C_13_1C_13_2C_13_3C_14_1C_14_2C_14_3C_15_1C_15_2C_15_3C_16_1C_16_2C_16_3C_17_1C_17_2C_17_3C_18_1C_18_2C_18_3C_19_1C_19_2C_19_3C_20_1C_20_2C_20_3C_21_1C_21_2C_21_3C_22_1C_22_2C_22_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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