Description of fast matrix multiplication algorithm: ⟨5×5×9:173⟩

Algorithm type

X3⁢Y4⁢Z4+4⁢X3⁢Y2⁢Z2+X2⁢Y3⁢Z2+2⁢X2⁢Y2⁢Z3+43⁢X2⁢Y2⁢Z2+4⁢X⁢Y4⁢Z+4⁢X⁢Y⁢Z4+14⁢X3⁢Y⁢Z+2⁢X2⁢Y2⁢Z+9⁢X2⁢Y⁢Z2+X⁢Y3⁢Z+X⁢Y⁢Z3+11⁢X2⁢Y⁢Z+X⁢Y2⁢Z+16⁢X⁢Y⁢Z2+59⁢X⁢Y⁢ZX3Y4Z44X3Y2Z2X2Y3Z22X2Y2Z343X2Y2Z24XY4Z4XYZ414X3YZ2X2Y2Z9X2YZ2XY3ZXYZ311X2YZXY2Z16XYZ259XYZX^3*Y^4*Z^4+4*X^3*Y^2*Z^2+X^2*Y^3*Z^2+2*X^2*Y^2*Z^3+43*X^2*Y^2*Z^2+4*X*Y^4*Z+4*X*Y*Z^4+14*X^3*Y*Z+2*X^2*Y^2*Z+9*X^2*Y*Z^2+X*Y^3*Z+X*Y*Z^3+11*X^2*Y*Z+X*Y^2*Z+16*X*Y*Z^2+59*X*Y*Z

Algorithm definition

The algorithm ⟨5×5×9:173⟩ could be constructed using the following decomposition:

⟨5×5×9:173⟩ = ⟨5×5×5:97⟩ + ⟨5×5×4:76⟩.

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_5A_5_1A_5_2A_5_3A_5_4A_5_5B_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_9B_4_1B_4_2B_4_3B_4_4B_4_5B_4_6B_4_7B_4_8B_4_9B_5_1B_5_2B_5_3B_5_4B_5_5B_5_6B_5_7B_5_8B_5_9C_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_5=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_5A_5_1A_5_2A_5_3A_5_4A_5_5B_1_1B_1_2B_1_3B_1_4B_1_5B_2_1B_2_2B_2_3B_2_4B_2_5B_3_1B_3_2B_3_3B_3_4B_3_5B_4_1B_4_2B_4_3B_4_4B_4_5B_5_1B_5_2B_5_3B_5_4B_5_5C_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_5+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_5A_5_1A_5_2A_5_3A_5_4A_5_5B_1_6B_1_7B_1_8B_1_9B_2_6B_2_7B_2_8B_2_9B_3_6B_3_7B_3_8B_3_9B_4_6B_4_7B_4_8B_4_9B_5_6B_5_7B_5_8B_5_9C_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_5TraceMulA_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_5A_5_1A_5_2A_5_3A_5_4A_5_5B_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_9B_4_1B_4_2B_4_3B_4_4B_4_5B_4_6B_4_7B_4_8B_4_9B_5_1B_5_2B_5_3B_5_4B_5_5B_5_6B_5_7B_5_8B_5_9C_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_5TraceMulA_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_5A_5_1A_5_2A_5_3A_5_4A_5_5B_1_1B_1_2B_1_3B_1_4B_1_5B_2_1B_2_2B_2_3B_2_4B_2_5B_3_1B_3_2B_3_3B_3_4B_3_5B_4_1B_4_2B_4_3B_4_4B_4_5B_5_1B_5_2B_5_3B_5_4B_5_5C_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_5TraceMulA_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_5A_5_1A_5_2A_5_3A_5_4A_5_5B_1_6B_1_7B_1_8B_1_9B_2_6B_2_7B_2_8B_2_9B_3_6B_3_7B_3_8B_3_9B_4_6B_4_7B_4_8B_4_9B_5_6B_5_7B_5_8B_5_9C_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_5Trace(Mul(Matrix(5, 5, [[A_1_1,A_1_2,A_1_3,A_1_4,A_1_5],[A_2_1,A_2_2,A_2_3,A_2_4,A_2_5],[A_3_1,A_3_2,A_3_3,A_3_4,A_3_5],[A_4_1,A_4_2,A_4_3,A_4_4,A_4_5],[A_5_1,A_5_2,A_5_3,A_5_4,A_5_5]]),Matrix(5, 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],[B_4_1,B_4_2,B_4_3,B_4_4,B_4_5,B_4_6,B_4_7,B_4_8,B_4_9],[B_5_1,B_5_2,B_5_3,B_5_4,B_5_5,B_5_6,B_5_7,B_5_8,B_5_9]]),Matrix(9, 5, [[C_1_1,C_1_2,C_1_3,C_1_4,C_1_5],[C_2_1,C_2_2,C_2_3,C_2_4,C_2_5],[C_3_1,C_3_2,C_3_3,C_3_4,C_3_5],[C_4_1,C_4_2,C_4_3,C_4_4,C_4_5],[C_5_1,C_5_2,C_5_3,C_5_4,C_5_5],[C_6_1,C_6_2,C_6_3,C_6_4,C_6_5],[C_7_1,C_7_2,C_7_3,C_7_4,C_7_5],[C_8_1,C_8_2,C_8_3,C_8_4,C_8_5],[C_9_1,C_9_2,C_9_3,C_9_4,C_9_5]]))) = Trace(Mul(Matrix(5, 5, [[A_1_1,A_1_2,A_1_3,A_1_4,A_1_5],[A_2_1,A_2_2,A_2_3,A_2_4,A_2_5],[A_3_1,A_3_2,A_3_3,A_3_4,A_3_5],[A_4_1,A_4_2,A_4_3,A_4_4,A_4_5],[A_5_1,A_5_2,A_5_3,A_5_4,A_5_5]]),Matrix(5, 5, [[B_1_1,B_1_2,B_1_3,B_1_4,B_1_5],[B_2_1,B_2_2,B_2_3,B_2_4,B_2_5],[B_3_1,B_3_2,B_3_3,B_3_4,B_3_5],[B_4_1,B_4_2,B_4_3,B_4_4,B_4_5],[B_5_1,B_5_2,B_5_3,B_5_4,B_5_5]]),Matrix(5, 5, [[C_1_1,C_1_2,C_1_3,C_1_4,C_1_5],[C_2_1,C_2_2,C_2_3,C_2_4,C_2_5],[C_3_1,C_3_2,C_3_3,C_3_4,C_3_5],[C_4_1,C_4_2,C_4_3,C_4_4,C_4_5],[C_5_1,C_5_2,C_5_3,C_5_4,C_5_5]])))+Trace(Mul(Matrix(5, 5, [[A_1_1,A_1_2,A_1_3,A_1_4,A_1_5],[A_2_1,A_2_2,A_2_3,A_2_4,A_2_5],[A_3_1,A_3_2,A_3_3,A_3_4,A_3_5],[A_4_1,A_4_2,A_4_3,A_4_4,A_4_5],[A_5_1,A_5_2,A_5_3,A_5_4,A_5_5]]),Matrix(5, 4, [[B_1_6,B_1_7,B_1_8,B_1_9],[B_2_6,B_2_7,B_2_8,B_2_9],[B_3_6,B_3_7,B_3_8,B_3_9],[B_4_6,B_4_7,B_4_8,B_4_9],[B_5_6,B_5_7,B_5_8,B_5_9]]),Matrix(4, 5, [[C_6_1,C_6_2,C_6_3,C_6_4,C_6_5],[C_7_1,C_7_2,C_7_3,C_7_4,C_7_5],[C_8_1,C_8_2,C_8_3,C_8_4,C_8_5],[C_9_1,C_9_2,C_9_3,C_9_4,C_9_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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