Description of fast matrix multiplication algorithm: ⟨3×3×9:63⟩

Algorithm type

16X2Y3Z3+24XY3Z3+4X2Y2Z2+2X3YZ+2XY3Z+2XYZ3+13XYZ16X2Y3Z324XY3Z34X2Y2Z22X3YZ2XY3Z2XYZ313XYZ16*X^2*Y^3*Z^3+24*X*Y^3*Z^3+4*X^2*Y^2*Z^2+2*X^3*Y*Z+2*X*Y^3*Z+2*X*Y*Z^3+13*X*Y*Z

Algorithm definition

The algorithm ⟨3×3×9:63⟩ could be constructed using the following decomposition:

⟨3×3×9:63⟩ = ⟨3×3×3:23⟩ + ⟨3×3×6:40⟩.

This decomposition is defined by the following equality:

TraceMulA_1_1A_1_2A_1_3A_2_1A_2_2A_2_3A_3_1A_3_2A_3_3B_1_1B_1_2B_1_3B_1_4B_1_5B_1_6B_1_7B_1_8B_1_9B_2_1B_2_2B_2_3B_2_4B_2_5B_2_6B_2_7B_2_8B_2_9B_3_1B_3_2B_3_3B_3_4B_3_5B_3_6B_3_7B_3_8B_3_9C_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_3=TraceMulA_1_1A_1_2A_1_3A_2_1A_2_2A_2_3A_3_1A_3_2A_3_3B_1_1B_1_2B_1_3B_2_1B_2_2B_2_3B_3_1B_3_2B_3_3C_1_1C_1_2C_1_3C_2_1C_2_2C_2_3C_3_1C_3_2C_3_3+TraceMulA_1_1A_1_2A_1_3A_2_1A_2_2A_2_3A_3_1A_3_2A_3_3B_1_4B_1_5B_1_6B_1_7B_1_8B_1_9B_2_4B_2_5B_2_6B_2_7B_2_8B_2_9B_3_4B_3_5B_3_6B_3_7B_3_8B_3_9C_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_3TraceMulA_1_1A_1_2A_1_3A_2_1A_2_2A_2_3A_3_1A_3_2A_3_3B_1_1B_1_2B_1_3B_1_4B_1_5B_1_6B_1_7B_1_8B_1_9B_2_1B_2_2B_2_3B_2_4B_2_5B_2_6B_2_7B_2_8B_2_9B_3_1B_3_2B_3_3B_3_4B_3_5B_3_6B_3_7B_3_8B_3_9C_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_3TraceMulA_1_1A_1_2A_1_3A_2_1A_2_2A_2_3A_3_1A_3_2A_3_3B_1_1B_1_2B_1_3B_2_1B_2_2B_2_3B_3_1B_3_2B_3_3C_1_1C_1_2C_1_3C_2_1C_2_2C_2_3C_3_1C_3_2C_3_3TraceMulA_1_1A_1_2A_1_3A_2_1A_2_2A_2_3A_3_1A_3_2A_3_3B_1_4B_1_5B_1_6B_1_7B_1_8B_1_9B_2_4B_2_5B_2_6B_2_7B_2_8B_2_9B_3_4B_3_5B_3_6B_3_7B_3_8B_3_9C_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_3Trace(Mul(Matrix(3, 3, [[A_1_1,A_1_2,A_1_3],[A_2_1,A_2_2,A_2_3],[A_3_1,A_3_2,A_3_3]]),Matrix(3, 9, [[B_1_1,B_1_2,B_1_3,B_1_4,B_1_5,B_1_6,B_1_7,B_1_8,B_1_9],[B_2_1,B_2_2,B_2_3,B_2_4,B_2_5,B_2_6,B_2_7,B_2_8,B_2_9],[B_3_1,B_3_2,B_3_3,B_3_4,B_3_5,B_3_6,B_3_7,B_3_8,B_3_9]]),Matrix(9, 3, [[C_1_1,C_1_2,C_1_3],[C_2_1,C_2_2,C_2_3],[C_3_1,C_3_2,C_3_3],[C_4_1,C_4_2,C_4_3],[C_5_1,C_5_2,C_5_3],[C_6_1,C_6_2,C_6_3],[C_7_1,C_7_2,C_7_3],[C_8_1,C_8_2,C_8_3],[C_9_1,C_9_2,C_9_3]]))) = Trace(Mul(Matrix(3, 3, [[A_1_1,A_1_2,A_1_3],[A_2_1,A_2_2,A_2_3],[A_3_1,A_3_2,A_3_3]]),Matrix(3, 3, [[B_1_1,B_1_2,B_1_3],[B_2_1,B_2_2,B_2_3],[B_3_1,B_3_2,B_3_3]]),Matrix(3, 3, [[C_1_1,C_1_2,C_1_3],[C_2_1,C_2_2,C_2_3],[C_3_1,C_3_2,C_3_3]])))+Trace(Mul(Matrix(3, 3, [[A_1_1,A_1_2,A_1_3],[A_2_1,A_2_2,A_2_3],[A_3_1,A_3_2,A_3_3]]),Matrix(3, 6, [[B_1_4,B_1_5,B_1_6,B_1_7,B_1_8,B_1_9],[B_2_4,B_2_5,B_2_6,B_2_7,B_2_8,B_2_9],[B_3_4,B_3_5,B_3_6,B_3_7,B_3_8,B_3_9]]),Matrix(6, 3, [[C_4_1,C_4_2,C_4_3],[C_5_1,C_5_2,C_5_3],[C_6_1,C_6_2,C_6_3],[C_7_1,C_7_2,C_7_3],[C_8_1,C_8_2,C_8_3],[C_9_1,C_9_2,C_9_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.


Back to main table