Description of fast matrix multiplication algorithm: ⟨4×5×11:165⟩

Algorithm type

X4⁢Y2⁢Z+X3⁢Y2⁢Z2+6⁢X2⁢Y2⁢Z3+X3⁢Y2⁢Z+2⁢X3⁢Y⁢Z2+45⁢X2⁢Y2⁢Z2+2⁢X⁢Y2⁢Z3+4⁢X2⁢Y2⁢Z+4⁢X2⁢Y⁢Z2+2⁢X⁢Y3⁢Z+2⁢X⁢Y2⁢Z2+8⁢X⁢Y⁢Z3+8⁢X2⁢Y⁢Z+22⁢X⁢Y2⁢Z+26⁢X⁢Y⁢Z2+31⁢X⁢Y⁢ZX4Y2ZX3Y2Z26X2Y2Z3X3Y2Z2X3YZ245X2Y2Z22XY2Z34X2Y2Z4X2YZ22XY3Z2XY2Z28XYZ38X2YZ22XY2Z26XYZ231XYZX^4*Y^2*Z+X^3*Y^2*Z^2+6*X^2*Y^2*Z^3+X^3*Y^2*Z+2*X^3*Y*Z^2+45*X^2*Y^2*Z^2+2*X*Y^2*Z^3+4*X^2*Y^2*Z+4*X^2*Y*Z^2+2*X*Y^3*Z+2*X*Y^2*Z^2+8*X*Y*Z^3+8*X^2*Y*Z+22*X*Y^2*Z+26*X*Y*Z^2+31*X*Y*Z

Algorithm definition

The algorithm ⟨4×5×11:165⟩ could be constructed using the following decomposition:

⟨4×5×11:165⟩ = ⟨4×5×4:61⟩ + ⟨4×5×7:104⟩.

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_2_1B_2_2B_2_3B_2_4B_2_5B_2_6B_2_7B_2_8B_2_9B_2_10B_2_11B_3_1B_3_2B_3_3B_3_4B_3_5B_3_6B_3_7B_3_8B_3_9B_3_10B_3_11B_4_1B_4_2B_4_3B_4_4B_4_5B_4_6B_4_7B_4_8B_4_9B_4_10B_4_11B_5_1B_5_2B_5_3B_5_4B_5_5B_5_6B_5_7B_5_8B_5_9B_5_10B_5_11C_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_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_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_4C_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_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_5B_1_6B_1_7B_1_8B_1_9B_1_10B_1_11B_2_5B_2_6B_2_7B_2_8B_2_9B_2_10B_2_11B_3_5B_3_6B_3_7B_3_8B_3_9B_3_10B_3_11B_4_5B_4_6B_4_7B_4_8B_4_9B_4_10B_4_11B_5_5B_5_6B_5_7B_5_8B_5_9B_5_10B_5_11C_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_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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