Description of fast matrix multiplication algorithm: ⟨4×5×17:255⟩

Algorithm type

85⁢X2⁢Y2⁢Z2+2⁢X3⁢Y⁢Z+2⁢X2⁢Y2⁢Z+X2⁢Y⁢Z2+4⁢X⁢Y3⁢Z+4⁢X⁢Y2⁢Z2+3⁢X⁢Y⁢Z3+24⁢X2⁢Y⁢Z+46⁢X⁢Y2⁢Z+35⁢X⁢Y⁢Z2+49⁢X⁢Y⁢Z85X2Y2Z22X3YZ2X2Y2ZX2YZ24XY3Z4XY2Z23XYZ324X2YZ46XY2Z35XYZ249XYZ85*X^2*Y^2*Z^2+2*X^3*Y*Z+2*X^2*Y^2*Z+X^2*Y*Z^2+4*X*Y^3*Z+4*X*Y^2*Z^2+3*X*Y*Z^3+24*X^2*Y*Z+46*X*Y^2*Z+35*X*Y*Z^2+49*X*Y*Z

Algorithm definition

The algorithm ⟨4×5×17:255⟩ could be constructed using the following decomposition:

⟨4×5×17:255⟩ = ⟨4×5×3:47⟩ + ⟨4×5×14:208⟩.

This decomposition is defined by the following equality:

Trace⁡Mul⁡A_1_1A_1_2A_1_3A_1_4A_1_5A_2_1A_2_2A_2_3A_2_4A_2_5A_3_1A_3_2A_3_3A_3_4A_3_5A_4_1A_4_2A_4_3A_4_4A_4_5B_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_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_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_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_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_17C_1_1C_1_2C_1_3C_1_4C_2_1C_2_2C_2_3C_2_4C_3_1C_3_2C_3_3C_3_4C_4_1C_4_2C_4_3C_4_4C_5_1C_5_2C_5_3C_5_4C_6_1C_6_2C_6_3C_6_4C_7_1C_7_2C_7_3C_7_4C_8_1C_8_2C_8_3C_8_4C_9_1C_9_2C_9_3C_9_4C_10_1C_10_2C_10_3C_10_4C_11_1C_11_2C_11_3C_11_4C_12_1C_12_2C_12_3C_12_4C_13_1C_13_2C_13_3C_13_4C_14_1C_14_2C_14_3C_14_4C_15_1C_15_2C_15_3C_15_4C_16_1C_16_2C_16_3C_16_4C_17_1C_17_2C_17_3C_17_4=Trace⁡Mul⁡A_1_1A_1_2A_1_3A_1_4A_1_5A_2_1A_2_2A_2_3A_2_4A_2_5A_3_1A_3_2A_3_3A_3_4A_3_5A_4_1A_4_2A_4_3A_4_4A_4_5B_1_1B_1_2B_1_3B_2_1B_2_2B_2_3B_3_1B_3_2B_3_3B_4_1B_4_2B_4_3B_5_1B_5_2B_5_3C_1_1C_1_2C_1_3C_1_4C_2_1C_2_2C_2_3C_2_4C_3_1C_3_2C_3_3C_3_4+Trace⁡Mul⁡A_1_1A_1_2A_1_3A_1_4A_1_5A_2_1A_2_2A_2_3A_2_4A_2_5A_3_1A_3_2A_3_3A_3_4A_3_5A_4_1A_4_2A_4_3A_4_4A_4_5B_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_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_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_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_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_17C_4_1C_4_2C_4_3C_4_4C_5_1C_5_2C_5_3C_5_4C_6_1C_6_2C_6_3C_6_4C_7_1C_7_2C_7_3C_7_4C_8_1C_8_2C_8_3C_8_4C_9_1C_9_2C_9_3C_9_4C_10_1C_10_2C_10_3C_10_4C_11_1C_11_2C_11_3C_11_4C_12_1C_12_2C_12_3C_12_4C_13_1C_13_2C_13_3C_13_4C_14_1C_14_2C_14_3C_14_4C_15_1C_15_2C_15_3C_15_4C_16_1C_16_2C_16_3C_16_4C_17_1C_17_2C_17_3C_17_4

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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