Description of fast matrix multiplication algorithm: ⟨5×12×13:543⟩

Algorithm type

16X5Y5Z4+X4Y6Z4+24X5Y5Z2+4X4Y4Z4+X3Y6Z3+10X3Y4Z3+5X4Y2Z2+16X3Y3Z2+3X2Y4Z2+64X2Y3Z3+24X3Y3Z+5X3Y2Z2+8X2Y3Z2+2XY5Z+3XY4Z2+96XY3Z3+4X3Y2Z+71X2Y2Z2+14XY4Z+X2Y2Z+18XY3Z+2XY2Z2+14X2YZ+39XY2Z+98XYZ16X5Y5Z4X4Y6Z424X5Y5Z24X4Y4Z4X3Y6Z310X3Y4Z35X4Y2Z216X3Y3Z23X2Y4Z264X2Y3Z324X3Y3Z5X3Y2Z28X2Y3Z22XY5Z3XY4Z296XY3Z34X3Y2Z71X2Y2Z214XY4ZX2Y2Z18XY3Z2XY2Z214X2YZ39XY2Z98XYZ16*X^5*Y^5*Z^4+X^4*Y^6*Z^4+24*X^5*Y^5*Z^2+4*X^4*Y^4*Z^4+X^3*Y^6*Z^3+10*X^3*Y^4*Z^3+5*X^4*Y^2*Z^2+16*X^3*Y^3*Z^2+3*X^2*Y^4*Z^2+64*X^2*Y^3*Z^3+24*X^3*Y^3*Z+5*X^3*Y^2*Z^2+8*X^2*Y^3*Z^2+2*X*Y^5*Z+3*X*Y^4*Z^2+96*X*Y^3*Z^3+4*X^3*Y^2*Z+71*X^2*Y^2*Z^2+14*X*Y^4*Z+X^2*Y^2*Z+18*X*Y^3*Z+2*X*Y^2*Z^2+14*X^2*Y*Z+39*X*Y^2*Z+98*X*Y*Z

Algorithm definition

The algorithm ⟨5×12×13:543⟩ could be constructed using the following decomposition:

⟨5×12×13:543⟩ = ⟨3×6×7:96⟩ + ⟨2×6×6:57⟩ + ⟨3×6×7:96⟩ + ⟨2×6×7:67⟩ + ⟨2×6×7:67⟩ + ⟨3×6×6:80⟩ + ⟨3×6×6:80⟩.

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_1_11A_1_12A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_2_7A_2_8A_2_9A_2_10A_2_11A_2_12A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_3_7A_3_8A_3_9A_3_10A_3_11A_3_12A_4_1A_4_2A_4_3A_4_4A_4_5A_4_6A_4_7A_4_8A_4_9A_4_10A_4_11A_4_12A_5_1A_5_2A_5_3A_5_4A_5_5A_5_6A_5_7A_5_8A_5_9A_5_10A_5_11A_5_12B_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_13B_11_1B_11_2B_11_3B_11_4B_11_5B_11_6B_11_7B_11_8B_11_9B_11_10B_11_11B_11_12B_11_13B_12_1B_12_2B_12_3B_12_4B_12_5B_12_6B_12_7B_12_8B_12_9B_12_10B_12_11B_12_12B_12_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=TraceMulA_3_7A_3_8A_3_9A_3_10A_3_11A_3_12A_1_6+A_4_7A_1_1+A_4_8A_1_2+A_4_9A_1_3+A_4_10A_1_4+A_4_11A_1_5+A_4_12A_2_6+A_5_7A_2_1+A_5_8A_2_2+A_5_9A_2_3+A_5_10A_2_4+A_5_11A_2_5+A_5_12B_7_7B_6_1+B_7_8B_6_2+B_7_9B_6_3+B_7_10B_6_4+B_7_11B_6_5+B_7_12B_6_6+B_7_13B_8_7B_1_1+B_8_8B_1_2+B_8_9B_1_3+B_8_10B_1_4+B_8_11B_1_5+B_8_12B_1_6+B_8_13B_9_7B_2_1+B_9_8B_2_2+B_9_9B_2_3+B_9_10B_2_4+B_9_11B_2_5+B_9_12B_2_6+B_9_13B_10_7B_3_1+B_10_8B_3_2+B_10_9B_3_3+B_10_10B_3_4+B_10_11B_3_5+B_10_12B_3_6+B_10_13B_11_7B_4_1+B_11_8B_4_2+B_11_9B_4_3+B_11_10B_4_4+B_11_11B_4_5+B_11_12B_4_6+B_11_13B_12_7B_5_1+B_12_8B_5_2+B_12_9B_5_3+B_12_10B_5_4+B_12_11B_5_5+B_12_12B_5_6+B_12_13C_7_3C_7_4C_7_5C_8_3C_1_1+C_8_4C_1_2+C_8_5C_9_3C_2_1+C_9_4C_2_2+C_9_5C_10_3C_3_1+C_10_4C_3_2+C_10_5C_11_3C_4_1+C_11_4C_4_2+C_11_5C_12_3C_5_1+C_12_4C_5_2+C_12_5C_13_3C_6_1+C_13_4C_6_2+C_13_5+TraceMulA_1_7-A_4_7A_1_8-A_4_8A_1_9-A_4_9A_1_10-A_4_10A_1_11-A_4_11A_1_12-A_4_12A_2_7-A_5_7A_2_8-A_5_8A_2_9-A_5_9A_2_10-A_5_10A_2_11-A_5_11A_2_12-A_5_12B_7_1+B_7_8B_7_2+B_7_9B_7_3+B_7_10B_7_4+B_7_11B_7_5+B_7_12B_7_6+B_7_13B_8_1+B_8_8B_8_2+B_8_9B_8_3+B_8_10B_8_4+B_8_11B_8_5+B_8_12B_8_6+B_8_13B_9_1+B_9_8B_9_2+B_9_9B_9_3+B_9_10B_9_4+B_9_11B_9_5+B_9_12B_9_6+B_9_13B_10_1+B_10_8B_10_2+B_10_9B_10_3+B_10_10B_10_4+B_10_11B_10_5+B_10_12B_10_6+B_10_13B_11_1+B_11_8B_11_2+B_11_9B_11_3+B_11_10B_11_4+B_11_11B_11_5+B_11_12B_11_6+B_11_13B_12_1+B_12_8B_12_2+B_12_9B_12_3+B_12_10B_12_4+B_12_11B_12_5+B_12_12B_12_6+B_12_13C_1_1C_1_2C_2_1C_2_2C_3_1C_3_2C_4_1C_4_2C_5_1C_5_2C_6_1C_6_2+TraceMulA_3_6A_3_1A_3_2A_3_3A_3_4A_3_5-A_1_6+A_4_6-A_1_1+A_4_1-A_1_2+A_4_2-A_1_3+A_4_3A_4_4-A_1_4-A_1_5+A_4_5-A_2_6+A_5_6-A_2_1+A_5_1-A_2_2+A_5_2-A_2_3+A_5_3-A_2_4+A_5_4-A_2_5+A_5_5B_6_7B_6_1+B_6_8B_6_2+B_6_9B_6_3+B_6_10B_6_4+B_6_11B_6_5+B_6_12B_6_6+B_6_13B_1_7B_1_1+B_1_8B_1_2+B_1_9B_1_3+B_1_10B_1_4+B_1_11B_1_5+B_1_12B_1_6+B_1_13B_2_7B_2_1+B_2_8B_2_2+B_2_9B_2_3+B_2_10B_2_4+B_2_11B_2_5+B_2_12B_2_6+B_2_13B_3_7B_3_1+B_3_8B_3_2+B_3_9B_3_3+B_3_10B_3_4+B_3_11B_3_5+B_3_12B_3_6+B_3_13B_4_7B_4_1+B_4_8B_4_2+B_4_9B_4_3+B_4_10B_4_4+B_4_11B_4_5+B_4_12B_4_6+B_4_13B_5_7B_5_1+B_5_8B_5_2+B_5_9B_5_3+B_5_10B_5_4+B_5_11B_5_5+B_5_12B_5_6+B_5_13C_7_3C_7_4C_7_5C_8_3C_8_4C_8_5C_9_3C_9_4C_9_5C_10_3C_10_4C_10_5C_11_3C_11_4C_11_5C_12_3C_12_4C_12_5C_13_3C_13_4C_13_5+TraceMulA_1_6+A_1_7A_1_1+A_1_8A_1_2+A_1_9A_1_3+A_1_10A_1_4+A_1_11A_1_5+A_1_12A_2_6+A_2_7A_2_1+A_2_8A_2_2+A_2_9A_2_3+A_2_10A_2_4+A_2_11A_2_5+A_2_12B_7_7B_7_8B_7_9B_7_10B_7_11B_7_12B_7_13B_8_7B_8_8B_8_9B_8_10B_8_11B_8_12B_8_13B_9_7B_9_8B_9_9B_9_10B_9_11B_9_12B_9_13B_10_7B_10_8B_10_9B_10_10B_10_11B_10_12B_10_13B_11_7B_11_8B_11_9B_11_10B_11_11B_11_12B_11_13B_12_7B_12_8B_12_9B_12_10B_12_11B_12_12B_12_13C_7_1C_7_2-C_1_1+C_8_1-C_1_2+C_8_2-C_2_1+C_9_1-C_2_2+C_9_2-C_3_1+C_10_1-C_3_2+C_10_2-C_4_1+C_11_1-C_4_2+C_11_2-C_5_1+C_12_1-C_5_2+C_12_2-C_6_1+C_13_1-C_6_2+C_13_2+TraceMulA_1_6A_1_1A_1_2A_1_3A_1_4A_1_5A_2_6A_2_1A_2_2A_2_3A_2_4A_2_5B_6_7-B_7_7B_6_8-B_7_8B_6_9-B_7_9B_6_10-B_7_10B_6_11-B_7_11B_6_12-B_7_12B_6_13-B_7_13B_1_7-B_8_7B_1_8-B_8_8B_1_9-B_8_9B_1_10-B_8_10B_1_11-B_8_11B_1_12-B_8_12B_1_13-B_8_13B_2_7-B_9_7B_2_8-B_9_8B_2_9-B_9_9B_2_10-B_9_10B_2_11-B_9_11B_2_12-B_9_12B_2_13-B_9_13B_3_7-B_10_7B_3_8-B_10_8B_3_9-B_10_9B_3_10-B_10_10B_3_11-B_10_11B_3_12-B_10_12B_3_13-B_10_13B_4_7-B_11_7B_4_8-B_11_8B_4_9-B_11_9B_4_10-B_11_10B_4_11-B_11_11B_4_12-B_11_12B_4_13-B_11_13B_5_7-B_12_7B_5_8-B_12_8B_5_9-B_12_9B_5_10-B_12_10B_5_11-B_12_11B_5_12-B_12_12B_5_13-B_12_13C_7_1+C_7_4C_7_2+C_7_5C_8_1+C_8_4C_8_2+C_8_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C_9_5C_10_3C_10_4C_10_5C_11_3C_11_4C_11_5C_12_3C_12_4C_12_5C_13_3C_13_4C_13_5TraceMulA_1_6A_1_7A_1_1A_1_8A_1_2A_1_9A_1_3A_1_10A_1_4A_1_11A_1_5A_1_12A_2_6A_2_7A_2_1A_2_8A_2_2A_2_9A_2_3A_2_10A_2_4A_2_11A_2_5A_2_12B_7_7B_7_8B_7_9B_7_10B_7_11B_7_12B_7_13B_8_7B_8_8B_8_9B_8_10B_8_11B_8_12B_8_13B_9_7B_9_8B_9_9B_9_10B_9_11B_9_12B_9_13B_10_7B_10_8B_10_9B_10_10B_10_11B_10_12B_10_13B_11_7B_11_8B_11_9B_11_10B_11_11B_11_12B_11_13B_12_7B_12_8B_12_9B_12_10B_12_11B_12_12B_12_13C_7_1C_7_2C_1_1C_8_1C_1_2C_8_2C_2_1C_9_1C_2_2C_9_2C_3_1C_10_1C_3_2C_10_2C_4_1C_11_1C_4_2C_11_2C_5_1C_12_1C_5_2C_12_2C_6_1C_13_1C_6_2C_13_2TraceMulA_1_6A_1_1A_1_2A_1_3A_1_4A_1_5A_2_6A_2_1A_2_2A_2_3A_2_4A_2_5B_6_7B_7_7B_6_8B_7_8B_6_9B_7_9B_6_10B_7_10B_6_11B_7_11B_6_12B_7_12B_6_13B_7_13B_1_7B_8_7B_1_8B_8_8B_1_9B_8_9B_1_10B_8_10B_1_11B_8_11B_1_12B_8_12B_1_13B_8_13B_2_7B_9_7B_2_8B_9_8B_2_9B_9_9B_2_10B_9_10B_2_11B_9_11B_2_12B_9_12B_2_13B_9_13B_3_7B_10_7B_3_8B_10_8B_3_9B_10_9B_3_10B_10_10B_3_11B_10_11B_3_12B_10_12B_3_13B_10_13B_4_7B_11_7B_4_8B_11_8B_4_9B_11_9B_4_10B_11_10B_4_11B_11_11B_4_12B_11_12B_4_13B_11_13B_5_7B_12_7B_5_8B_12_8B_5_9B_12_9B_5_10B_12_10B_5_11B_12_11B_5_12B_12_12B_5_13B_12_13C_7_1C_7_4C_7_2C_7_5C_8_1C_8_4C_8_2C_8_5C_9_1C_9_4C_9_2C_9_5C_10_1C_10_4C_10_2C_10_5C_11_1C_11_4C_11_2C_11_5C_12_1C_12_4C_12_2C_12_5C_13_1C_13_4C_13_2C_13_5TraceMulA_3_7A_3_8A_3_9A_3_10A_3_11A_3_12A_4_7A_4_8A_4_9A_4_10A_4_11A_4_12A_5_7A_5_8A_5_9A_5_10A_5_11A_5_12B_6_1B_7_1B_6_2B_7_2B_6_3B_7_3B_6_4B_7_4B_6_5B_7_5B_6_6B_7_6B_1_1B_8_1B_1_2B_8_2B_1_3B_8_3B_1_4B_8_4B_1_5B_8_5B_1_6B_8_6B_2_1B_9_1B_2_2B_9_2B_2_3B_9_3B_2_4B_9_4B_2_5B_9_5B_2_6B_9_6B_3_1B_10_1B_3_2B_10_2B_3_3B_10_3B_3_4B_10_4B_3_5B_10_5B_3_6B_10_6B_4_1B_11_1B_4_2B_11_2B_4_3B_11_3B_4_4B_11_4B_4_5B_11_5B_4_6B_11_6B_5_1B_12_1B_5_2B_12_2B_5_3B_12_3B_5_4B_12_4B_5_5B_12_5B_5_6B_12_6C_1_3C_1_1C_1_4C_1_2C_1_5C_2_3C_2_1C_2_4C_2_2C_2_5C_3_3C_3_1C_3_4C_3_2C_3_5C_4_3C_4_1C_4_4C_4_2C_4_5C_5_3C_5_1C_5_4C_5_2C_5_5C_6_3C_6_1C_6_4C_6_2C_6_5TraceMulA_3_6A_3_7A_3_1A_3_8A_3_2A_3_9A_3_3A_3_10A_3_4A_3_11A_3_5A_3_12A_4_6A_4_7A_4_1A_4_8A_4_2A_4_9A_4_3A_4_10A_4_4A_4_11A_4_5A_4_12A_5_6A_5_7A_5_1A_5_8A_5_2A_5_9A_5_3A_5_10A_5_4A_5_11A_5_5A_5_12B_6_1B_6_2B_6_3B_6_4B_6_5B_6_6B_1_1B_1_2B_1_3B_1_4B_1_5B_1_6B_2_1B_2_2B_2_3B_2_4B_2_5B_2_6B_3_1B_3_2B_3_3B_3_4B_3_5B_3_6B_4_1B_4_2B_4_3B_4_4B_4_5B_4_6B_5_1B_5_2B_5_3B_5_4B_5_5B_5_6C_1_3C_8_3C_1_4C_8_4C_1_5C_8_5C_2_3C_9_3C_2_4C_9_4C_2_5C_9_5C_3_3C_10_3C_3_4C_10_4C_3_5C_10_5C_4_3C_11_3C_4_4C_11_4C_4_5C_11_5C_5_3C_12_3C_5_4C_12_4C_5_5C_12_5C_13_3C_6_3C_6_4C_13_4C_6_5C_13_5Trace(Mul(Matrix(5, 12, [[A_1_1,A_1_2,A_1_3,A_1_4,A_1_5,A_1_6,A_1_7,A_1_8,A_1_9,A_1_10,A_1_11,A_1_12],[A_2_1,A_2_2,A_2_3,A_2_4,A_2_5,A_2_6,A_2_7,A_2_8,A_2_9,A_2_10,A_2_11,A_2_12],[A_3_1,A_3_2,A_3_3,A_3_4,A_3_5,A_3_6,A_3_7,A_3_8,A_3_9,A_3_10,A_3_11,A_3_12],[A_4_1,A_4_2,A_4_3,A_4_4,A_4_5,A_4_6,A_4_7,A_4_8,A_4_9,A_4_10,A_4_11,A_4_12],[A_5_1,A_5_2,A_5_3,A_5_4,A_5_5,A_5_6,A_5_7,A_5_8,A_5_9,A_5_10,A_5_11,A_5_12]]),Matrix(12, 13, [[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_1_10,B_1_11,B_1_12,B_1_13],[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_2_10,B_2_11,B_2_12,B_2_13],[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_3_10,B_3_11,B_3_12,B_3_13],[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_4_10,B_4_11,B_4_12,B_4_13],[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,B_5_10,B_5_11,B_5_12,B_5_13],[B_6_1,B_6_2,B_6_3,B_6_4,B_6_5,B_6_6,B_6_7,B_6_8,B_6_9,B_6_10,B_6_11,B_6_12,B_6_13],[B_7_1,B_7_2,B_7_3,B_7_4,B_7_5,B_7_6,B_7_7,B_7_8,B_7_9,B_7_10,B_7_11,B_7_12,B_7_13],[B_8_1,B_8_2,B_8_3,B_8_4,B_8_5,B_8_6,B_8_7,B_8_8,B_8_9,B_8_10,B_8_11,B_8_12,B_8_13],[B_9_1,B_9_2,B_9_3,B_9_4,B_9_5,B_9_6,B_9_7,B_9_8,B_9_9,B_9_10,B_9_11,B_9_12,B_9_13],[B_10_1,B_10_2,B_10_3,B_10_4,B_10_5,B_10_6,B_10_7,B_10_8,B_10_9,B_10_10,B_10_11,B_10_12,B_10_13],[B_11_1,B_11_2,B_11_3,B_11_4,B_11_5,B_11_6,B_11_7,B_11_8,B_11_9,B_11_10,B_11_11,B_11_12,B_11_13],[B_12_1,B_12_2,B_12_3,B_12_4,B_12_5,B_12_6,B_12_7,B_12_8,B_12_9,B_12_10,B_12_11,B_12_12,B_12_13]]),Matrix(13, 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],[C_10_1,C_10_2,C_10_3,C_10_4,C_10_5],[C_11_1,C_11_2,C_11_3,C_11_4,C_11_5],[C_12_1,C_12_2,C_12_3,C_12_4,C_12_5],[C_13_1,C_13_2,C_13_3,C_13_4,C_13_5]]))) = Trace(Mul(Matrix(3, 6, [[A_3_7,A_3_8,A_3_9,A_3_10,A_3_11,A_3_12],[A_1_6+A_4_7,A_1_1+A_4_8,A_1_2+A_4_9,A_1_3+A_4_10,A_1_4+A_4_11,A_1_5+A_4_12],[A_2_6+A_5_7,A_2_1+A_5_8,A_2_2+A_5_9,A_2_3+A_5_10,A_2_4+A_5_11,A_2_5+A_5_12]]),Matrix(6, 7, [[B_7_7,B_6_1+B_7_8,B_6_2+B_7_9,B_6_3+B_7_10,B_6_4+B_7_11,B_6_5+B_7_12,B_6_6+B_7_13],[B_8_7,B_1_1+B_8_8,B_1_2+B_8_9,B_1_3+B_8_10,B_1_4+B_8_11,B_1_5+B_8_12,B_1_6+B_8_13],[B_9_7,B_2_1+B_9_8,B_2_2+B_9_9,B_2_3+B_9_10,B_2_4+B_9_11,B_2_5+B_9_12,B_2_6+B_9_13],[B_10_7,B_3_1+B_10_8,B_3_2+B_10_9,B_3_3+B_10_10,B_3_4+B_10_11,B_3_5+B_10_12,B_3_6+B_10_13],[B_11_7,B_4_1+B_11_8,B_4_2+B_11_9,B_4_3+B_11_10,B_4_4+B_11_11,B_4_5+B_11_12,B_4_6+B_11_13],[B_12_7,B_5_1+B_12_8,B_5_2+B_12_9,B_5_3+B_12_10,B_5_4+B_12_11,B_5_5+B_12_12,B_5_6+B_12_13]]),Matrix(7, 3, [[C_7_3,C_7_4,C_7_5],[C_8_3,C_1_1+C_8_4,C_1_2+C_8_5],[C_9_3,C_2_1+C_9_4,C_2_2+C_9_5],[C_10_3,C_3_1+C_10_4,C_3_2+C_10_5],[C_11_3,C_4_1+C_11_4,C_4_2+C_11_5],[C_12_3,C_5_1+C_12_4,C_5_2+C_12_5],[C_13_3,C_6_1+C_13_4,C_6_2+C_13_5]])))+Trace(Mul(Matrix(2, 6, [[A_1_7-A_4_7,A_1_8-A_4_8,A_1_9-A_4_9,A_1_10-A_4_10,A_1_11-A_4_11,A_1_12-A_4_12],[A_2_7-A_5_7,A_2_8-A_5_8,A_2_9-A_5_9,A_2_10-A_5_10,A_2_11-A_5_11,A_2_12-A_5_12]]),Matrix(6, 6, [[B_7_1+B_7_8,B_7_2+B_7_9,B_7_3+B_7_10,B_7_4+B_7_11,B_7_5+B_7_12,B_7_6+B_7_13],[B_8_1+B_8_8,B_8_2+B_8_9,B_8_3+B_8_10,B_8_4+B_8_11,B_8_5+B_8_12,B_8_6+B_8_13],[B_9_1+B_9_8,B_9_2+B_9_9,B_9_3+B_9_10,B_9_4+B_9_11,B_9_5+B_9_12,B_9_6+B_9_13],[B_10_1+B_10_8,B_10_2+B_10_9,B_10_3+B_10_10,B_10_4+B_10_11,B_10_5+B_10_12,B_10_6+B_10_13],[B_11_1+B_11_8,B_11_2+B_11_9,B_11_3+B_11_10,B_11_4+B_11_11,B_11_5+B_11_12,B_11_6+B_11_13],[B_12_1+B_12_8,B_12_2+B_12_9,B_12_3+B_12_10,B_12_4+B_12_11,B_12_5+B_12_12,B_12_6+B_12_13]]),Matrix(6, 2, [[C_1_1,C_1_2],[C_2_1,C_2_2],[C_3_1,C_3_2],[C_4_1,C_4_2],[C_5_1,C_5_2],[C_6_1,C_6_2]])))+Trace(Mul(Matrix(3, 6, [[A_3_6,A_3_1,A_3_2,A_3_3,A_3_4,A_3_5],[-A_1_6+A_4_6,-A_1_1+A_4_1,-A_1_2+A_4_2,-A_1_3+A_4_3,A_4_4-A_1_4,-A_1_5+A_4_5],[-A_2_6+A_5_6,-A_2_1+A_5_1,-A_2_2+A_5_2,-A_2_3+A_5_3,-A_2_4+A_5_4,-A_2_5+A_5_5]]),Matrix(6, 7, [[B_6_7,B_6_1+B_6_8,B_6_2+B_6_9,B_6_3+B_6_10,B_6_4+B_6_11,B_6_5+B_6_12,B_6_6+B_6_13],[B_1_7,B_1_1+B_1_8,B_1_2+B_1_9,B_1_3+B_1_10,B_1_4+B_1_11,B_1_5+B_1_12,B_1_6+B_1_13],[B_2_7,B_2_1+B_2_8,B_2_2+B_2_9,B_2_3+B_2_10,B_2_4+B_2_11,B_2_5+B_2_12,B_2_6+B_2_13],[B_3_7,B_3_1+B_3_8,B_3_2+B_3_9,B_3_3+B_3_10,B_3_4+B_3_11,B_3_5+B_3_12,B_3_6+B_3_13],[B_4_7,B_4_1+B_4_8,B_4_2+B_4_9,B_4_3+B_4_10,B_4_4+B_4_11,B_4_5+B_4_12,B_4_6+B_4_13],[B_5_7,B_5_1+B_5_8,B_5_2+B_5_9,B_5_3+B_5_10,B_5_4+B_5_11,B_5_5+B_5_12,B_5_6+B_5_13]]),Matrix(7, 3, [[C_7_3,C_7_4,C_7_5],[C_8_3,C_8_4,C_8_5],[C_9_3,C_9_4,C_9_5],[C_10_3,C_10_4,C_10_5],[C_11_3,C_11_4,C_11_5],[C_12_3,C_12_4,C_12_5],[C_13_3,C_13_4,C_13_5]])))+Trace(Mul(Matrix(2, 6, [[A_1_6+A_1_7,A_1_1+A_1_8,A_1_2+A_1_9,A_1_3+A_1_10,A_1_4+A_1_11,A_1_5+A_1_12],[A_2_6+A_2_7,A_2_1+A_2_8,A_2_2+A_2_9,A_2_3+A_2_10,A_2_4+A_2_11,A_2_5+A_2_12]]),Matrix(6, 7, [[B_7_7,B_7_8,B_7_9,B_7_10,B_7_11,B_7_12,B_7_13],[B_8_7,B_8_8,B_8_9,B_8_10,B_8_11,B_8_12,B_8_13],[B_9_7,B_9_8,B_9_9,B_9_10,B_9_11,B_9_12,B_9_13],[B_10_7,B_10_8,B_10_9,B_10_10,B_10_11,B_10_12,B_10_13],[B_11_7,B_11_8,B_11_9,B_11_10,B_11_11,B_11_12,B_11_13],[B_12_7,B_12_8,B_12_9,B_12_10,B_12_11,B_12_12,B_12_13]]),Matrix(7, 2, [[C_7_1,C_7_2],[-C_1_1+C_8_1,-C_1_2+C_8_2],[-C_2_1+C_9_1,-C_2_2+C_9_2],[-C_3_1+C_10_1,-C_3_2+C_10_2],[-C_4_1+C_11_1,-C_4_2+C_11_2],[-C_5_1+C_12_1,-C_5_2+C_12_2],[-C_6_1+C_13_1,-C_6_2+C_13_2]])))+Trace(Mul(Matrix(2, 6, [[A_1_6,A_1_1,A_1_2,A_1_3,A_1_4,A_1_5],[A_2_6,A_2_1,A_2_2,A_2_3,A_2_4,A_2_5]]),Matrix(6, 7, [[B_6_7-B_7_7,B_6_8-B_7_8,B_6_9-B_7_9,B_6_10-B_7_10,B_6_11-B_7_11,B_6_12-B_7_12,B_6_13-B_7_13],[B_1_7-B_8_7,B_1_8-B_8_8,B_1_9-B_8_9,B_1_10-B_8_10,B_1_11-B_8_11,B_1_12-B_8_12,B_1_13-B_8_13],[B_2_7-B_9_7,B_2_8-B_9_8,B_2_9-B_9_9,B_2_10-B_9_10,B_2_11-B_9_11,B_2_12-B_9_12,B_2_13-B_9_13],[B_3_7-B_10_7,B_3_8-B_10_8,B_3_9-B_10_9,B_3_10-B_10_10,B_3_11-B_10_11,B_3_12-B_10_12,B_3_13-B_10_13],[B_4_7-B_11_7,B_4_8-B_11_8,B_4_9-B_11_9,B_4_10-B_11_10,B_4_11-B_11_11,B_4_12-B_11_12,B_4_13-B_11_13],[B_5_7-B_12_7,B_5_8-B_12_8,B_5_9-B_12_9,B_5_10-B_12_10,B_5_11-B_12_11,B_5_12-B_12_12,B_5_13-B_12_13]]),Matrix(7, 2, [[C_7_1+C_7_4,C_7_2+C_7_5],[C_8_1+C_8_4,C_8_2+C_8_5],[C_9_1+C_9_4,C_9_2+C_9_5],[C_10_1+C_10_4,C_10_2+C_10_5],[C_11_1+C_11_4,C_11_2+C_11_5],[C_12_1+C_12_4,C_12_2+C_12_5],[C_13_1+C_13_4,C_13_2+C_13_5]])))+Trace(Mul(Matrix(3, 6, [[A_3_7,A_3_8,A_3_9,A_3_10,A_3_11,A_3_12],[A_4_7,A_4_8,A_4_9,A_4_10,A_4_11,A_4_12],[A_5_7,A_5_8,A_5_9,A_5_10,A_5_11,A_5_12]]),Matrix(6, 6, [[-B_6_1+B_7_1,-B_6_2+B_7_2,-B_6_3+B_7_3,-B_6_4+B_7_4,-B_6_5+B_7_5,-B_6_6+B_7_6],[-B_1_1+B_8_1,-B_1_2+B_8_2,-B_1_3+B_8_3,-B_1_4+B_8_4,-B_1_5+B_8_5,-B_1_6+B_8_6],[-B_2_1+B_9_1,-B_2_2+B_9_2,-B_2_3+B_9_3,-B_2_4+B_9_4,-B_2_5+B_9_5,-B_2_6+B_9_6],[-B_3_1+B_10_1,-B_3_2+B_10_2,-B_3_3+B_10_3,-B_3_4+B_10_4,-B_3_5+B_10_5,-B_3_6+B_10_6],[-B_4_1+B_11_1,-B_4_2+B_11_2,-B_4_3+B_11_3,-B_4_4+B_11_4,-B_4_5+B_11_5,-B_4_6+B_11_6],[-B_5_1+B_12_1,-B_5_2+B_12_2,-B_5_3+B_12_3,-B_5_4+B_12_4,-B_5_5+B_12_5,-B_5_6+B_12_6]]),Matrix(6, 3, [[C_1_3,C_1_1+C_1_4,C_1_2+C_1_5],[C_2_3,C_2_1+C_2_4,C_2_2+C_2_5],[C_3_3,C_3_1+C_3_4,C_3_2+C_3_5],[C_4_3,C_4_1+C_4_4,C_4_2+C_4_5],[C_5_3,C_5_1+C_5_4,C_5_2+C_5_5],[C_6_3,C_6_1+C_6_4,C_6_2+C_6_5]])))+Trace(Mul(Matrix(3, 6, [[A_3_6+A_3_7,A_3_1+A_3_8,A_3_2+A_3_9,A_3_3+A_3_10,A_3_4+A_3_11,A_3_5+A_3_12],[A_4_6+A_4_7,A_4_1+A_4_8,A_4_2+A_4_9,A_4_3+A_4_10,A_4_4+A_4_11,A_4_5+A_4_12],[A_5_6+A_5_7,A_5_1+A_5_8,A_5_2+A_5_9,A_5_3+A_5_10,A_5_4+A_5_11,A_5_5+A_5_12]]),Matrix(6, 6, [[B_6_1,B_6_2,B_6_3,B_6_4,B_6_5,B_6_6],[B_1_1,B_1_2,B_1_3,B_1_4,B_1_5,B_1_6],[B_2_1,B_2_2,B_2_3,B_2_4,B_2_5,B_2_6],[B_3_1,B_3_2,B_3_3,B_3_4,B_3_5,B_3_6],[B_4_1,B_4_2,B_4_3,B_4_4,B_4_5,B_4_6],[B_5_1,B_5_2,B_5_3,B_5_4,B_5_5,B_5_6]]),Matrix(6, 3, [[C_1_3-C_8_3,C_1_4-C_8_4,C_1_5-C_8_5],[C_2_3-C_9_3,C_2_4-C_9_4,C_2_5-C_9_5],[C_3_3-C_10_3,C_3_4-C_10_4,C_3_5-C_10_5],[C_4_3-C_11_3,C_4_4-C_11_4,C_4_5-C_11_5],[C_5_3-C_12_3,C_5_4-C_12_4,C_5_5-C_12_5],[-C_13_3+C_6_3,C_6_4-C_13_4,C_6_5-C_13_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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