Description of fast matrix multiplication algorithm: ⟨5×10×15:528⟩

Algorithm type

8⁢X4⁢Y4⁢Z4+2⁢X3⁢Y6⁢Z3+X2⁢Y8⁢Z2+2⁢X2⁢Y4⁢Z5+X⁢Y8⁢Z2+2⁢X5⁢Y2⁢Z3+2⁢X4⁢Y4⁢Z2+10⁢X3⁢Y4⁢Z3+2⁢X3⁢Y2⁢Z5+5⁢X2⁢Y6⁢Z2+X⁢Y4⁢Z5+4⁢X5⁢Y2⁢Z2+X3⁢Y4⁢Z2+2⁢X2⁢Y2⁢Z5+2⁢X5⁢Y2⁢Z+2⁢X4⁢Y2⁢Z2+X3⁢Y4⁢Z+11⁢X2⁢Y4⁢Z2+2⁢X⁢Y2⁢Z5+5⁢X3⁢Y2⁢Z2+6⁢X2⁢Y3⁢Z2+X⁢Y4⁢Z2+2⁢X4⁢Y⁢Z+3⁢X3⁢Y2⁢Z+6⁢X3⁢Y⁢Z2+3⁢X2⁢Y3⁢Z+120⁢X2⁢Y2⁢Z2+6⁢X2⁢Y⁢Z3+20⁢X⁢Y4⁢Z+2⁢X⁢Y3⁢Z2+11⁢X⁢Y2⁢Z3+25⁢X3⁢Y⁢Z+13⁢X2⁢Y2⁢Z+6⁢X2⁢Y⁢Z2+46⁢X⁢Y3⁢Z+2⁢X⁢Y2⁢Z2+22⁢X⁢Y⁢Z3+40⁢X2⁢Y⁢Z+39⁢X⁢Y2⁢Z+14⁢X⁢Y⁢Z2+75⁢X⁢Y⁢Z8X4Y4Z42X3Y6Z3X2Y8Z22X2Y4Z5XY8Z22X5Y2Z32X4Y4Z210X3Y4Z32X3Y2Z55X2Y6Z2XY4Z54X5Y2Z2X3Y4Z22X2Y2Z52X5Y2Z2X4Y2Z2X3Y4Z11X2Y4Z22XY2Z55X3Y2Z26X2Y3Z2XY4Z22X4YZ3X3Y2Z6X3YZ23X2Y3Z120X2Y2Z26X2YZ320XY4Z2XY3Z211XY2Z325X3YZ13X2Y2Z6X2YZ246XY3Z2XY2Z222XYZ340X2YZ39XY2Z14XYZ275XYZ8*X^4*Y^4*Z^4+2*X^3*Y^6*Z^3+X^2*Y^8*Z^2+2*X^2*Y^4*Z^5+X*Y^8*Z^2+2*X^5*Y^2*Z^3+2*X^4*Y^4*Z^2+10*X^3*Y^4*Z^3+2*X^3*Y^2*Z^5+5*X^2*Y^6*Z^2+X*Y^4*Z^5+4*X^5*Y^2*Z^2+X^3*Y^4*Z^2+2*X^2*Y^2*Z^5+2*X^5*Y^2*Z+2*X^4*Y^2*Z^2+X^3*Y^4*Z+11*X^2*Y^4*Z^2+2*X*Y^2*Z^5+5*X^3*Y^2*Z^2+6*X^2*Y^3*Z^2+X*Y^4*Z^2+2*X^4*Y*Z+3*X^3*Y^2*Z+6*X^3*Y*Z^2+3*X^2*Y^3*Z+120*X^2*Y^2*Z^2+6*X^2*Y*Z^3+20*X*Y^4*Z+2*X*Y^3*Z^2+11*X*Y^2*Z^3+25*X^3*Y*Z+13*X^2*Y^2*Z+6*X^2*Y*Z^2+46*X*Y^3*Z+2*X*Y^2*Z^2+22*X*Y*Z^3+40*X^2*Y*Z+39*X*Y^2*Z+14*X*Y*Z^2+75*X*Y*Z

Algorithm definition

The algorithm ⟨5×10×15:528⟩ could be constructed using the following decomposition:

⟨5×10×15:528⟩ = ⟨5×10×12:413⟩ + ⟨5×10×3:115⟩.

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_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_2_7A_2_8A_2_9A_2_10A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_3_7A_3_8A_3_9A_3_10A_4_1A_4_2A_4_3A_4_4A_4_5A_4_6A_4_7A_4_8A_4_9A_4_10A_5_1A_5_2A_5_3A_5_4A_5_5A_5_6A_5_7A_5_8A_5_9A_5_10B_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_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_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_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_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_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_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_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_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_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_15C_1_1C_1_2C_1_3C_1_4C_1_5C_2_1C_2_2C_2_3C_2_4C_2_5C_3_1C_3_2C_3_3C_3_4C_3_5C_4_1C_4_2C_4_3C_4_4C_4_5C_5_1C_5_2C_5_3C_5_4C_5_5C_6_1C_6_2C_6_3C_6_4C_6_5C_7_1C_7_2C_7_3C_7_4C_7_5C_8_1C_8_2C_8_3C_8_4C_8_5C_9_1C_9_2C_9_3C_9_4C_9_5C_10_1C_10_2C_10_3C_10_4C_10_5C_11_1C_11_2C_11_3C_11_4C_11_5C_12_1C_12_2C_12_3C_12_4C_12_5C_13_1C_13_2C_13_3C_13_4C_13_5C_14_1C_14_2C_14_3C_14_4C_14_5C_15_1C_15_2C_15_3C_15_4C_15_5=Trace⁡Mul⁡A_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_1_7A_1_8A_1_9A_1_10A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_2_7A_2_8A_2_9A_2_10A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_3_7A_3_8A_3_9A_3_10A_4_1A_4_2A_4_3A_4_4A_4_5A_4_6A_4_7A_4_8A_4_9A_4_10A_5_1A_5_2A_5_3A_5_4A_5_5A_5_6A_5_7A_5_8A_5_9A_5_10B_1_1B_1_2B_1_3B_1_4B_1_5B_1_6B_1_7B_1_8B_1_9B_1_10B_1_11B_1_12B_2_1B_2_2B_2_3B_2_4B_2_5B_2_6B_2_7B_2_8B_2_9B_2_10B_2_11B_2_12B_3_1B_3_2B_3_3B_3_4B_3_5B_3_6B_3_7B_3_8B_3_9B_3_10B_3_11B_3_12B_4_1B_4_2B_4_3B_4_4B_4_5B_4_6B_4_7B_4_8B_4_9B_4_10B_4_11B_4_12B_5_1B_5_2B_5_3B_5_4B_5_5B_5_6B_5_7B_5_8B_5_9B_5_10B_5_11B_5_12B_6_1B_6_2B_6_3B_6_4B_6_5B_6_6B_6_7B_6_8B_6_9B_6_10B_6_11B_6_12B_7_1B_7_2B_7_3B_7_4B_7_5B_7_6B_7_7B_7_8B_7_9B_7_10B_7_11B_7_12B_8_1B_8_2B_8_3B_8_4B_8_5B_8_6B_8_7B_8_8B_8_9B_8_10B_8_11B_8_12B_9_1B_9_2B_9_3B_9_4B_9_5B_9_6B_9_7B_9_8B_9_9B_9_10B_9_11B_9_12B_10_1B_10_2B_10_3B_10_4B_10_5B_10_6B_10_7B_10_8B_10_9B_10_10B_10_11B_10_12C_1_1C_1_2C_1_3C_1_4C_1_5C_2_1C_2_2C_2_3C_2_4C_2_5C_3_1C_3_2C_3_3C_3_4C_3_5C_4_1C_4_2C_4_3C_4_4C_4_5C_5_1C_5_2C_5_3C_5_4C_5_5C_6_1C_6_2C_6_3C_6_4C_6_5C_7_1C_7_2C_7_3C_7_4C_7_5C_8_1C_8_2C_8_3C_8_4C_8_5C_9_1C_9_2C_9_3C_9_4C_9_5C_10_1C_10_2C_10_3C_10_4C_10_5C_11_1C_11_2C_11_3C_11_4C_11_5C_12_1C_12_2C_12_3C_12_4C_12_5+Trace⁡Mul⁡A_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_1_7A_1_8A_1_9A_1_10A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_2_7A_2_8A_2_9A_2_10A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_3_7A_3_8A_3_9A_3_10A_4_1A_4_2A_4_3A_4_4A_4_5A_4_6A_4_7A_4_8A_4_9A_4_10A_5_1A_5_2A_5_3A_5_4A_5_5A_5_6A_5_7A_5_8A_5_9A_5_10B_1_13B_1_14B_1_15B_2_13B_2_14B_2_15B_3_13B_3_14B_3_15B_4_13B_4_14B_4_15B_5_13B_5_14B_5_15B_6_13B_6_14B_6_15B_7_13B_7_14B_7_15B_8_13B_8_14B_8_15B_9_13B_9_14B_9_15B_10_13B_10_14B_10_15C_13_1C_13_2C_13_3C_13_4C_13_5C_14_1C_14_2C_14_3C_14_4C_14_5C_15_1C_15_2C_15_3C_15_4C_15_5

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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