Description of fast matrix multiplication algorithm: ⟨6×6×10:247⟩

Algorithm type

16X4Y6Z6+24X2Y6Z6+48X2Y3Z3+72XY3Z3+21X2Y2Z2+3X3YZ+6XY3Z+6XY2Z2+6XYZ3+12XY2Z+12XYZ2+21XYZ16X4Y6Z624X2Y6Z648X2Y3Z372XY3Z321X2Y2Z23X3YZ6XY3Z6XY2Z26XYZ312XY2Z12XYZ221XYZ16*X^4*Y^6*Z^6+24*X^2*Y^6*Z^6+48*X^2*Y^3*Z^3+72*X*Y^3*Z^3+21*X^2*Y^2*Z^2+3*X^3*Y*Z+6*X*Y^3*Z+6*X*Y^2*Z^2+6*X*Y*Z^3+12*X*Y^2*Z+12*X*Y*Z^2+21*X*Y*Z

Algorithm definition

The algorithm ⟨6×6×10:247⟩ could be constructed using the following decomposition:

⟨6×6×10:247⟩ = ⟨3×3×6:40⟩ + ⟨3×3×4:29⟩ + ⟨3×3×6:40⟩ + ⟨3×3×6:40⟩ + ⟨3×3×6:40⟩ + ⟨3×3×4:29⟩ + ⟨3×3×4:29⟩.

This decomposition is defined by the following equality:

TraceMulA_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_4_1A_4_2A_4_3A_4_4A_4_5A_4_6A_5_1A_5_2A_5_3A_5_4A_5_5A_5_6A_6_1A_6_2A_6_3A_6_4A_6_5A_6_6B_1_1B_1_2B_1_3B_1_4B_1_5B_1_6B_1_7B_1_8B_1_9B_1_10B_2_1B_2_2B_2_3B_2_4B_2_5B_2_6B_2_7B_2_8B_2_9B_2_10B_3_1B_3_2B_3_3B_3_4B_3_5B_3_6B_3_7B_3_8B_3_9B_3_10B_4_1B_4_2B_4_3B_4_4B_4_5B_4_6B_4_7B_4_8B_4_9B_4_10B_5_1B_5_2B_5_3B_5_4B_5_5B_5_6B_5_7B_5_8B_5_9B_5_10B_6_1B_6_2B_6_3B_6_4B_6_5B_6_6B_6_7B_6_8B_6_9B_6_10C_1_1C_1_2C_1_3C_1_4C_1_5C_1_6C_2_1C_2_2C_2_3C_2_4C_2_5C_2_6C_3_1C_3_2C_3_3C_3_4C_3_5C_3_6C_4_1C_4_2C_4_3C_4_4C_4_5C_4_6C_5_1C_5_2C_5_3C_5_4C_5_5C_5_6C_6_1C_6_2C_6_3C_6_4C_6_5C_6_6C_7_1C_7_2C_7_3C_7_4C_7_5C_7_6C_8_1C_8_2C_8_3C_8_4C_8_5C_8_6C_9_1C_9_2C_9_3C_9_4C_9_5C_9_6C_10_1C_10_2C_10_3C_10_4C_10_5C_10_6=TraceMulA_1_1+A_4_4A_1_2+A_4_5A_1_3+A_4_6A_2_1+A_5_4A_2_2+A_5_5A_2_3+A_5_6A_3_1+A_6_4A_3_2+A_6_5A_3_3+A_6_6B_4_2B_4_3B_1_1+B_4_4B_1_5+B_4_8B_1_6+B_4_9B_1_7+B_4_10B_5_2B_5_3B_2_1+B_5_4B_2_5+B_5_8B_2_6+B_5_9B_2_7+B_5_10B_6_2B_6_3B_3_1+B_6_4B_3_5+B_6_8B_3_6+B_6_9B_3_7+B_6_10C_2_4C_2_5C_2_6C_3_4C_3_5C_3_6C_1_1+C_4_4C_1_2+C_4_5C_1_3+C_4_6C_5_1+C_8_4C_5_2+C_8_5C_5_3+C_8_6C_6_1+C_9_4C_6_2+C_9_5C_6_3+C_9_6C_7_1+C_10_4C_7_2+C_10_5C_7_3+C_10_6+TraceMulA_1_4-A_4_4A_1_5-A_4_5A_1_6-A_4_6A_2_4-A_5_4A_2_5-A_5_5A_2_6-A_5_6A_3_4-A_6_4A_3_5-A_6_5A_3_6-A_6_6B_4_1+B_4_4B_4_5+B_4_8B_4_6+B_4_9B_4_7+B_4_10B_5_1+B_5_4B_5_5+B_5_8B_5_6+B_5_9B_5_7+B_5_10B_6_1+B_6_4B_6_5+B_6_8B_6_6+B_6_9B_6_7+B_6_10C_1_1C_1_2C_1_3C_5_1C_5_2C_5_3C_6_1C_6_2C_6_3C_7_1C_7_2C_7_3+TraceMul-A_1_1+A_4_1-A_1_2+A_4_2-A_1_3+A_4_3-A_2_1+A_5_1-A_2_2+A_5_2-A_2_3+A_5_3-A_3_1+A_6_1-A_3_2+A_6_2-A_3_3+A_6_3B_1_2B_1_3B_1_1+B_1_4B_1_5+B_1_8B_1_6+B_1_9B_1_7+B_1_10B_2_2B_2_3B_2_1+B_2_4B_2_5+B_2_8B_2_6+B_2_9B_2_7+B_2_10B_3_2B_3_3B_3_1+B_3_4B_3_5+B_3_8B_3_6+B_3_9B_3_7+B_3_10C_2_4C_2_5C_2_6C_3_4C_3_5C_3_6C_4_4C_4_5C_4_6C_8_4C_8_5C_8_6C_9_4C_9_5C_9_6C_10_4C_10_5C_10_6+TraceMulA_1_1+A_1_4A_1_2+A_1_5A_1_3+A_1_6A_2_1+A_2_4A_2_2+A_2_5A_2_3+A_2_6A_3_1+A_3_4A_3_2+A_3_5A_3_3+A_3_6B_4_2B_4_3B_4_4B_4_8B_4_9B_4_10B_5_2B_5_3B_5_4B_5_8B_5_9B_5_10B_6_2B_6_3B_6_4B_6_8B_6_9B_6_10C_2_1C_2_2C_2_3C_3_1C_3_2C_3_3-C_1_1+C_4_1C_4_2-C_1_2-C_1_3+C_4_3-C_5_1+C_8_1-C_5_2+C_8_2-C_5_3+C_8_3-C_6_1+C_9_1-C_6_2+C_9_2-C_6_3+C_9_3-C_7_1+C_10_1-C_7_2+C_10_2-C_7_3+C_10_3+TraceMulA_1_1A_1_2A_1_3A_2_1A_2_2A_2_3A_3_1A_3_2A_3_3B_1_2-B_4_2B_1_3-B_4_3B_1_4-B_4_4B_1_8-B_4_8B_1_9-B_4_9B_1_10-B_4_10B_2_2-B_5_2B_2_3-B_5_3B_2_4-B_5_4B_2_8-B_5_8B_2_9-B_5_9B_2_10-B_5_10B_3_2-B_6_2B_3_3-B_6_3B_3_4-B_6_4B_3_8-B_6_8B_3_9-B_6_9B_3_10-B_6_10C_2_1+C_2_4C_2_2+C_2_5C_2_3+C_2_6C_3_1+C_3_4C_3_2+C_3_5C_3_3+C_3_6C_4_1+C_4_4C_4_2+C_4_5C_4_3+C_4_6C_8_1+C_8_4C_8_2+C_8_5C_8_3+C_8_6C_9_1+C_9_4C_9_2+C_9_5C_9_3+C_9_6C_10_1+C_10_4C_10_2+C_10_5C_10_3+C_10_6+TraceMulA_4_4A_4_5A_4_6A_5_4A_5_5A_5_6A_6_4A_6_5A_6_6-B_1_1+B_4_1-B_1_5+B_4_5-B_1_6+B_4_6-B_1_7+B_4_7-B_2_1+B_5_1-B_2_5+B_5_5-B_2_6+B_5_6-B_2_7+B_5_7-B_3_1+B_6_1-B_3_5+B_6_5-B_3_6+B_6_6-B_3_7+B_6_7C_1_1+C_1_4C_1_2+C_1_5C_1_3+C_1_6C_5_1+C_5_4C_5_2+C_5_5C_5_3+C_5_6C_6_1+C_6_4C_6_2+C_6_5C_6_3+C_6_6C_7_1+C_7_4C_7_2+C_7_5C_7_3+C_7_6+TraceMulA_4_1+A_4_4A_4_2+A_4_5A_4_3+A_4_6A_5_1+A_5_4A_5_2+A_5_5A_5_3+A_5_6A_6_1+A_6_4A_6_2+A_6_5A_6_3+A_6_6B_1_1B_1_5B_1_6B_1_7B_2_1B_2_5B_2_6B_2_7B_3_1B_3_5B_3_6B_3_7C_1_4-C_4_4C_1_5-C_4_5C_1_6-C_4_6C_5_4-C_8_4C_5_5-C_8_5C_5_6-C_8_6C_6_4-C_9_4C_6_5-C_9_5C_6_6-C_9_6C_7_4-C_10_4C_7_5-C_10_5C_7_6-C_10_6

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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