Description of fast matrix multiplication algorithm: ⟨5×10×13:455⟩

Algorithm type

X4Y8Z4+X4Y8Z3+X4Y5Z3+2X4Y4Z4+3X3Y6Z3+X2Y8Z2+2X4Y4Z3+X4Y3Z4+X2Y7Z2+X2Y6Z3+14X3Y4Z3+X2Y7Z+X2Y6Z2+X3Y3Z3+2X2Y6Z+3X2Y5Z2+X2Y4Z3+X2Y2Z5+11X2Y4Z2+4X2Y2Z4+XY6Z+3XY4Z3+XY2Z5+X2Y4Z+16X2Y3Z2+4X2Y2Z3+4X3YZ2+99X2Y2Z2+4X2YZ3+22XY4Z+XY3Z2+7XY2Z3+12X3YZ+11X2Y2Z+47XY3Z+6XY2Z2+10XYZ3+16X2YZ+45XY2Z+9XYZ2+83XYZX4Y8Z4X4Y8Z3X4Y5Z32X4Y4Z43X3Y6Z3X2Y8Z22X4Y4Z3X4Y3Z4X2Y7Z2X2Y6Z314X3Y4Z3X2Y7ZX2Y6Z2X3Y3Z32X2Y6Z3X2Y5Z2X2Y4Z3X2Y2Z511X2Y4Z24X2Y2Z4XY6Z3XY4Z3XY2Z5X2Y4Z16X2Y3Z24X2Y2Z34X3YZ299X2Y2Z24X2YZ322XY4ZXY3Z27XY2Z312X3YZ11X2Y2Z47XY3Z6XY2Z210XYZ316X2YZ45XY2Z9XYZ283XYZX^4*Y^8*Z^4+X^4*Y^8*Z^3+X^4*Y^5*Z^3+2*X^4*Y^4*Z^4+3*X^3*Y^6*Z^3+X^2*Y^8*Z^2+2*X^4*Y^4*Z^3+X^4*Y^3*Z^4+X^2*Y^7*Z^2+X^2*Y^6*Z^3+14*X^3*Y^4*Z^3+X^2*Y^7*Z+X^2*Y^6*Z^2+X^3*Y^3*Z^3+2*X^2*Y^6*Z+3*X^2*Y^5*Z^2+X^2*Y^4*Z^3+X^2*Y^2*Z^5+11*X^2*Y^4*Z^2+4*X^2*Y^2*Z^4+X*Y^6*Z+3*X*Y^4*Z^3+X*Y^2*Z^5+X^2*Y^4*Z+16*X^2*Y^3*Z^2+4*X^2*Y^2*Z^3+4*X^3*Y*Z^2+99*X^2*Y^2*Z^2+4*X^2*Y*Z^3+22*X*Y^4*Z+X*Y^3*Z^2+7*X*Y^2*Z^3+12*X^3*Y*Z+11*X^2*Y^2*Z+47*X*Y^3*Z+6*X*Y^2*Z^2+10*X*Y*Z^3+16*X^2*Y*Z+45*X*Y^2*Z+9*X*Y*Z^2+83*X*Y*Z

Algorithm definition

The algorithm ⟨5×10×13:455⟩ could be constructed using the following decomposition:

⟨5×10×13:455⟩ = ⟨2×5×7:56⟩ + ⟨2×5×7:56⟩ + ⟨3×5×6:68⟩ + ⟨2×5×6:47⟩ + ⟨3×5×7:80⟩ + ⟨3×5×6:68⟩ + ⟨3×5×7:80⟩.

This decomposition is defined by the following equality:

TraceMulA_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_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_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_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_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_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_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_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_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_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_13C_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_5=TraceMul-A_1_6+A_1_8-A_1_7+A_1_9A_1_4-A_1_1A_1_5-A_1_2-A_1_3+A_1_10-A_2_6+A_2_8-A_2_7+A_2_9A_2_4-A_2_1A_2_5-A_2_2-A_2_3+A_2_10B_8_9-B_8_11B_8_10-B_8_12B_8_2B_8_3-B_8_1B_8_4-B_8_7B_8_5-B_8_8B_8_6-B_8_13B_9_9-B_9_11B_9_10-B_9_12B_9_2B_9_3-B_9_1B_9_4-B_9_7B_9_5-B_9_8B_9_6-B_9_13B_4_9-B_4_11B_4_10-B_4_12B_4_2B_4_3-B_4_1B_4_4-B_4_7B_4_5-B_4_8B_4_6-B_4_13B_5_9-B_5_11B_5_10-B_5_12B_5_2B_5_3-B_5_1B_5_4-B_5_7B_5_5-B_5_8B_5_6-B_5_13B_10_9-B_10_11B_10_10-B_10_12B_10_2B_10_3-B_10_1B_10_4-B_10_7B_10_5-B_10_8B_10_6-B_10_13-C_9_3+C_9_1C_9_2-C_9_4C_10_1-C_10_3C_10_2-C_10_4-C_2_3+C_2_1C_2_2-C_2_4-C_3_3+C_3_1C_3_2-C_3_4C_4_1-C_4_3C_4_2-C_4_4C_5_1-C_5_3C_5_2-C_5_4C_6_1-C_6_3C_6_2-C_6_4+TraceMulA_1_8+A_3_8A_1_9+A_3_9A_1_4+A_3_4A_3_5+A_1_5A_1_10+A_3_10A_2_8+A_4_8A_2_9+A_4_9A_2_4+A_4_4A_2_5+A_4_5A_2_10+A_4_10B_6_9+B_8_9B_6_10+B_8_10B_6_2+B_8_2B_6_3+B_8_3B_6_4+B_8_4B_6_5+B_8_5B_6_6+B_8_6B_7_9+B_9_9B_7_10+B_9_10B_7_2+B_9_2B_7_3+B_9_3B_7_4+B_9_4B_7_5+B_9_5B_7_6+B_9_6B_1_9+B_4_9B_1_10+B_4_10B_1_2+B_4_2B_1_3+B_4_3B_1_4+B_4_4B_1_5+B_4_5B_1_6+B_4_6B_2_9+B_5_9B_2_10+B_5_10B_2_2+B_5_2B_2_3+B_5_3B_2_4+B_5_4B_2_5+B_5_5B_2_6+B_5_6B_3_9+B_10_9B_3_10+B_10_10B_3_2+B_10_2B_3_3+B_10_3B_3_4+B_10_4B_3_5+B_10_5B_3_6+B_10_6C_9_1+C_11_1C_9_2+C_11_2C_10_1+C_12_1C_10_2+C_12_2C_2_1C_2_2C_3_1+C_1_1C_3_2+C_1_2C_4_1+C_7_1C_4_2+C_7_2C_5_1+C_8_1C_5_2+C_8_2C_6_1+C_13_1C_6_2+C_13_2+TraceMulA_3_6A_3_7A_3_1A_3_2A_3_3A_4_6A_4_7A_4_1A_4_2A_4_3A_5_6A_5_7A_5_1A_5_2A_5_3B_6_11B_6_12B_6_1B_6_7B_6_8B_6_13B_7_11B_7_12B_7_1B_7_7B_7_8B_7_13B_1_11B_1_12B_1_1B_1_7B_1_8B_1_13B_2_11B_2_12B_2_1B_2_7B_2_8B_2_13B_3_11B_3_12B_3_1B_3_7B_3_8B_3_13C_11_3C_11_4C_11_5C_12_3C_12_4C_12_5C_1_3C_1_4C_1_5C_7_3C_7_4C_7_5C_8_3C_8_4C_8_5C_13_3C_13_4C_13_5+TraceMulA_1_6A_1_7A_1_1A_1_2A_1_3A_2_6A_2_7A_2_1A_2_2A_2_3-B_6_9-B_8_9+B_6_11+B_8_11-B_6_10-B_8_10+B_6_12+B_8_12-B_6_3-B_8_3+B_6_1+B_8_1-B_6_4-B_8_4+B_6_7+B_8_7-B_6_5-B_8_5+B_6_8+B_8_8-B_6_6-B_8_6+B_6_13+B_8_13-B_7_9-B_9_9+B_7_11+B_9_11-B_7_10-B_9_10+B_7_12+B_9_12-B_7_3-B_9_3+B_7_1+B_9_1-B_7_4-B_9_4+B_7_7+B_9_7-B_7_5-B_9_5+B_7_8+B_9_8-B_7_6-B_9_6+B_7_13+B_9_13-B_1_9-B_4_9+B_1_11+B_4_11-B_1_10-B_4_10+B_1_12+B_4_12-B_1_3-B_4_3+B_1_1+B_4_1-B_1_4-B_4_4+B_1_7+B_4_7-B_1_5-B_4_5+B_1_8+B_4_8-B_1_6-B_4_6+B_1_13+B_4_13-B_2_9-B_5_9+B_2_11+B_5_11-B_2_10-B_5_10+B_2_12+B_5_12-B_2_3-B_5_3+B_2_1+B_5_1-B_2_4-B_5_4+B_2_7+B_5_7-B_2_5-B_5_5+B_2_8+B_5_8-B_2_6-B_5_6+B_2_13+B_5_13-B_3_9-B_10_9+B_3_11+B_10_11-B_3_10-B_10_10+B_3_12+B_10_12-B_3_3-B_10_3+B_3_1+B_10_1-B_3_4-B_10_4+B_3_7+B_10_7-B_3_5-B_10_5+B_3_8+B_10_8-B_3_6-B_10_6+B_3_13+B_10_13C_11_1C_11_2C_12_1C_12_2C_1_1C_1_2C_7_1C_7_2C_8_1C_8_2C_13_1C_13_2+TraceMulA_1_6+A_3_6-A_1_8-A_3_8A_1_7+A_3_7-A_1_9-A_3_9A_1_1+A_3_1-A_1_4-A_3_4A_1_2+A_3_2-A_1_5-A_3_5A_1_3+A_3_3-A_1_10-A_3_10A_2_6+A_4_6-A_2_8-A_4_8A_2_7+A_4_7-A_2_9-A_4_9A_2_1+A_4_1-A_2_4-A_4_4A_2_2+A_4_2-A_2_5-A_4_5A_2_3+A_4_3-A_2_10-A_4_10A_5_6-A_5_8A_5_7-A_5_9A_5_1-A_5_4A_5_2-A_5_5A_5_3-A_5_10B_6_9B_6_10B_6_2B_6_3B_6_4B_6_5B_6_6B_7_9B_7_10B_7_2B_7_3B_7_4B_7_5B_7_6B_1_9B_1_10B_1_2B_1_3B_1_4B_1_5B_1_6B_2_9B_2_10B_2_2B_2_3B_2_4B_2_5B_2_6B_3_9B_3_10B_3_2B_3_3B_3_4B_3_5B_3_6C_9_3C_9_4C_9_5C_10_3C_10_4C_10_5C_2_3C_2_4C_2_5C_3_3C_3_4C_3_5C_4_3C_4_4C_4_5C_5_3C_5_4C_5_5C_6_3C_6_4C_6_5+TraceMulA_3_8A_3_9A_3_4A_3_5A_3_10A_4_8A_4_9A_4_4A_4_5A_4_10A_5_8A_5_9A_5_4A_5_5A_5_10B_8_11B_8_12B_8_1B_8_7B_8_8B_8_13B_9_11B_9_12B_9_1B_9_7B_9_8B_9_13B_4_11B_4_12B_4_1B_4_7B_4_8B_4_13B_5_11B_5_12B_5_1B_5_7B_5_8B_5_13B_10_11B_10_12B_10_1B_10_7B_10_8B_10_13-C_9_1-C_11_1+C_9_3+C_11_3-C_9_2-C_11_2+C_9_4+C_11_4C_9_5+C_11_5-C_10_1-C_12_1+C_10_3+C_12_3-C_10_2-C_12_2+C_10_4+C_12_4C_10_5+C_12_5-C_3_1-C_1_1+C_3_3+C_1_3-C_3_2-C_1_2+C_3_4+C_1_4C_1_5+C_3_5-C_4_1-C_7_1+C_4_3+C_7_3-C_4_2-C_7_2+C_4_4+C_7_4C_4_5+C_7_5-C_5_1-C_8_1+C_5_3+C_8_3-C_5_2-C_8_2+C_5_4+C_8_4C_5_5+C_8_5-C_6_1-C_13_1+C_6_3+C_13_3-C_6_2-C_13_2+C_6_4+C_13_4C_6_5+C_13_5+TraceMulA_1_6-A_1_8-A_3_8A_1_7-A_1_9-A_3_9A_1_1-A_1_4-A_3_4A_1_2-A_1_5-A_3_5A_1_3-A_1_10-A_3_10A_2_6-A_2_8-A_4_8A_2_7-A_2_9-A_4_9A_2_1-A_2_4-A_4_4A_2_2-A_2_5-A_4_5A_2_3-A_2_10-A_4_10-A_5_8-A_5_9-A_5_4-A_5_5-A_5_10B_6_9+B_8_9-B_8_11B_6_10+B_8_10-B_8_12B_6_2+B_8_2B_6_3+B_8_3-B_8_1B_6_4+B_8_4-B_8_7B_6_5+B_8_5-B_8_8B_6_6+B_8_6-B_8_13B_7_9+B_9_9-B_9_11B_7_10+B_9_10-B_9_12B_7_2+B_9_2B_7_3+B_9_3-B_9_1B_7_4+B_9_4-B_9_7B_7_5+B_9_5-B_9_8B_7_6+B_9_6-B_9_13B_1_9+B_4_9-B_4_11B_1_10+B_4_10-B_4_12B_1_2+B_4_2B_1_3+B_4_3-B_4_1B_1_4+B_4_4-B_4_7B_1_5+B_4_5-B_4_8B_1_6+B_4_6-B_4_13B_2_9+B_5_9-B_5_11B_2_10+B_5_10-B_5_12B_2_2+B_5_2B_2_3+B_5_3-B_5_1B_2_4+B_5_4-B_5_7B_2_5+B_5_5-B_5_8B_2_6+B_5_6-B_5_13B_3_9+B_10_9-B_10_11B_3_10+B_10_10-B_10_12B_3_2+B_10_2B_3_3+B_10_3-B_10_1B_3_4+B_10_4-B_10_7B_3_5+B_10_5-B_10_8B_3_6+B_10_6-B_10_13C_9_1+C_11_1-C_9_3C_9_2+C_11_2-C_9_4-C_9_5C_10_1+C_12_1-C_10_3C_10_2+C_12_2-C_10_4-C_10_5-C_2_3+C_2_1C_2_2-C_2_4-C_2_5C_3_1+C_1_1-C_3_3C_3_2+C_1_2-C_3_4-C_3_5C_4_1+C_7_1-C_4_3C_4_2+C_7_2-C_4_4-C_4_5C_5_1+C_8_1-C_5_3C_5_2+C_8_2-C_5_4-C_5_5C_6_1+C_13_1-C_6_3C_6_2+C_13_2-C_6_4-C_6_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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