Description of fast matrix multiplication algorithm: ⟨4×7×13:270⟩

Algorithm type

3X4Y4Z4+X4Y4Z3+2X4Y3Z4+3X3Y3Z4+X3Y3Z3+4X2Y5Z2+X2Y5Z+6X2Y4Z2+XY5Z2+10X2Y3Z2+53X2Y2Z2+8XY4Z+2XY3Z2+X2Y2Z+5X2YZ2+30XY3Z+44XY2Z+XYZ2+94XYZ3X4Y4Z4X4Y4Z32X4Y3Z43X3Y3Z4X3Y3Z34X2Y5Z2X2Y5Z6X2Y4Z2XY5Z210X2Y3Z253X2Y2Z28XY4Z2XY3Z2X2Y2Z5X2YZ230XY3Z44XY2ZXYZ294XYZ3*X^4*Y^4*Z^4+X^4*Y^4*Z^3+2*X^4*Y^3*Z^4+3*X^3*Y^3*Z^4+X^3*Y^3*Z^3+4*X^2*Y^5*Z^2+X^2*Y^5*Z+6*X^2*Y^4*Z^2+X*Y^5*Z^2+10*X^2*Y^3*Z^2+53*X^2*Y^2*Z^2+8*X*Y^4*Z+2*X*Y^3*Z^2+X^2*Y^2*Z+5*X^2*Y*Z^2+30*X*Y^3*Z+44*X*Y^2*Z+X*Y*Z^2+94*X*Y*Z

Algorithm definition

The algorithm ⟨4×7×13:270⟩ could be constructed using the following decomposition:

⟨4×7×13:270⟩ = ⟨2×4×7:46⟩ + ⟨2×4×6:39⟩ + ⟨2×3×7:35⟩ + ⟨2×4×7:46⟩ + ⟨2×3×7:35⟩ + ⟨2×4×6:39⟩ + ⟨2×3×6:30⟩.

This decomposition is defined by the following equality:

TraceMulA_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_1_7A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_2_7A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_3_7A_4_1A_4_2A_4_3A_4_4A_4_5A_4_6A_4_7B_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_13C_1_1C_1_2C_1_3C_1_4C_2_1C_2_2C_2_3C_2_4C_3_1C_3_2C_3_3C_3_4C_4_1C_4_2C_4_3C_4_4C_5_1C_5_2C_5_3C_5_4C_6_1C_6_2C_6_3C_6_4C_7_1C_7_2C_7_3C_7_4C_8_1C_8_2C_8_3C_8_4C_9_1C_9_2C_9_3C_9_4C_10_1C_10_2C_10_3C_10_4C_11_1C_11_2C_11_3C_11_4C_12_1C_12_2C_12_3C_12_4C_13_1C_13_2C_13_3C_13_4=TraceMulA_3_4A_1_1+A_3_5A_1_2+A_3_6A_1_3+A_3_7A_4_4A_2_1+A_4_5A_2_2+A_4_6A_2_3+A_4_7B_4_7B_4_8B_4_9B_4_10B_4_11B_4_12B_4_13B_5_7B_1_1+B_5_8B_1_2+B_5_9B_1_3+B_5_10B_1_4+B_5_11B_1_5+B_5_12B_1_6+B_5_13B_6_7B_2_1+B_6_8B_2_2+B_6_9B_2_3+B_6_10B_2_4+B_6_11B_2_5+B_6_12B_2_6+B_6_13B_7_7B_3_1+B_7_8B_3_2+B_7_9B_3_3+B_7_10B_3_4+B_7_11B_3_5+B_7_12B_3_6+B_7_13C_7_3C_7_4C_1_1+C_8_3C_1_2+C_8_4C_2_1+C_9_3C_2_2+C_9_4C_3_1+C_10_3C_3_2+C_10_4C_4_1+C_11_3C_4_2+C_11_4C_5_1+C_12_3C_5_2+C_12_4C_6_1+C_13_3C_6_2+C_13_4+TraceMulA_1_4-A_3_4A_1_5-A_3_5A_1_6-A_3_6A_1_7-A_3_7A_2_4-A_4_4-A_4_5+A_2_5-A_4_6+A_2_6A_2_7-A_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_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_13B_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_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_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+TraceMul-A_1_1+A_3_1-A_1_2+A_3_2-A_1_3+A_3_3-A_2_1+A_4_1-A_2_2+A_4_2A_4_3-A_2_3B_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_13C_7_3C_7_4C_8_3C_8_4C_9_3C_9_4C_10_3C_10_4C_11_3C_11_4C_12_3C_12_4C_13_3C_13_4+TraceMulA_1_4A_1_1+A_1_5A_1_2+A_1_6A_1_3+A_1_7A_2_4A_2_1+A_2_5A_2_2+A_2_6A_2_3+A_2_7B_4_7B_4_8B_4_9B_4_10B_4_11B_4_12B_4_13B_5_7B_5_8B_5_9B_5_10B_5_11B_5_12B_5_13B_6_7B_6_8B_6_9B_6_10B_6_11B_6_12B_6_13B_7_7B_7_8B_7_9B_7_10B_7_11B_7_12B_7_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_1A_1_2A_1_3A_2_1A_2_2A_2_3B_1_7-B_5_7B_1_8-B_5_8B_1_9-B_5_9B_1_10-B_5_10B_1_11-B_5_11B_1_12-B_5_12B_1_13-B_5_13B_2_7-B_6_7B_2_8-B_6_8B_2_9-B_6_9B_2_10-B_6_10B_2_11-B_6_11B_2_12-B_6_12B_2_13-B_6_13B_3_7-B_7_7B_3_8-B_7_8B_3_9-B_7_9B_3_10-B_7_10B_3_11-B_7_11B_3_12-B_7_12B_3_13-B_7_13C_7_1+C_7_3C_7_2+C_7_4C_8_1+C_8_3C_8_2+C_8_4C_9_1+C_9_3C_9_2+C_9_4C_10_1+C_10_3C_10_2+C_10_4C_11_1+C_11_3C_11_2+C_11_4C_12_1+C_12_3C_12_2+C_12_4C_13_1+C_13_3C_13_2+C_13_4+TraceMulA_3_4A_3_5A_3_6A_3_7A_4_4A_4_5A_4_6A_4_7B_4_1B_4_2B_4_3B_4_4B_4_5B_4_6-B_1_1+B_5_1-B_1_2+B_5_2B_5_3-B_1_3-B_1_4+B_5_4-B_1_5+B_5_5-B_1_6+B_5_6-B_2_1+B_6_1-B_2_2+B_6_2-B_2_3+B_6_3B_6_4-B_2_4-B_2_5+B_6_5-B_2_6+B_6_6-B_3_1+B_7_1-B_3_2+B_7_2-B_3_3+B_7_3B_7_4-B_3_4-B_3_5+B_7_5-B_3_6+B_7_6C_1_1+C_1_3C_1_2+C_1_4C_2_1+C_2_3C_2_2+C_2_4C_3_1+C_3_3C_3_2+C_3_4C_4_1+C_4_3C_4_2+C_4_4C_5_1+C_5_3C_5_4+C_5_2C_6_1+C_6_3C_6_2+C_6_4+TraceMulA_3_1+A_3_5A_3_2+A_3_6A_3_3+A_3_7A_4_1+A_4_5A_4_2+A_4_6A_4_3+A_4_7B_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_6C_1_3-C_8_3C_1_4-C_8_4C_2_3-C_9_3C_2_4-C_9_4C_3_3-C_10_3C_3_4-C_10_4C_4_3-C_11_3C_4_4-C_11_4C_5_3-C_12_3C_5_4-C_12_4-C_13_3+C_6_3C_6_4-C_13_4TraceMulA_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_1_7A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_2_7A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_3_7A_4_1A_4_2A_4_3A_4_4A_4_5A_4_6A_4_7B_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_13C_1_1C_1_2C_1_3C_1_4C_2_1C_2_2C_2_3C_2_4C_3_1C_3_2C_3_3C_3_4C_4_1C_4_2C_4_3C_4_4C_5_1C_5_2C_5_3C_5_4C_6_1C_6_2C_6_3C_6_4C_7_1C_7_2C_7_3C_7_4C_8_1C_8_2C_8_3C_8_4C_9_1C_9_2C_9_3C_9_4C_10_1C_10_2C_10_3C_10_4C_11_1C_11_2C_11_3C_11_4C_12_1C_12_2C_12_3C_12_4C_13_1C_13_2C_13_3C_13_4TraceMulA_3_4A_1_1A_3_5A_1_2A_3_6A_1_3A_3_7A_4_4A_2_1A_4_5A_2_2A_4_6A_2_3A_4_7B_4_7B_4_8B_4_9B_4_10B_4_11B_4_12B_4_13B_5_7B_1_1B_5_8B_1_2B_5_9B_1_3B_5_10B_1_4B_5_11B_1_5B_5_12B_1_6B_5_13B_6_7B_2_1B_6_8B_2_2B_6_9B_2_3B_6_10B_2_4B_6_11B_2_5B_6_12B_2_6B_6_13B_7_7B_3_1B_7_8B_3_2B_7_9B_3_3B_7_10B_3_4B_7_11B_3_5B_7_12B_3_6B_7_13C_7_3C_7_4C_1_1C_8_3C_1_2C_8_4C_2_1C_9_3C_2_2C_9_4C_3_1C_10_3C_3_2C_10_4C_4_1C_11_3C_4_2C_11_4C_5_1C_12_3C_5_2C_12_4C_6_1C_13_3C_6_2C_13_4TraceMulA_1_4A_3_4A_1_5A_3_5A_1_6A_3_6A_1_7A_3_7A_2_4A_4_4A_4_5A_2_5A_4_6A_2_6A_2_7A_4_7B_4_1B_4_8B_4_2B_4_9B_4_3B_4_10B_4_4B_4_11B_4_5B_4_12B_4_6B_4_13B_5_1B_5_8B_5_2B_5_9B_5_3B_5_10B_5_4B_5_11B_5_5B_5_12B_5_6B_5_13B_6_1B_6_8B_6_2B_6_9B_6_3B_6_10B_6_4B_6_11B_6_5B_6_12B_6_6B_6_13B_7_1B_7_8B_7_2B_7_9B_7_3B_7_10B_7_4B_7_11B_7_5B_7_12B_7_6B_7_13C_1_1C_1_2C_2_1C_2_2C_3_1C_3_2C_4_1C_4_2C_5_1C_5_2C_6_1C_6_2TraceMulA_1_1A_3_1A_1_2A_3_2A_1_3A_3_3A_2_1A_4_1A_2_2A_4_2A_4_3A_2_3B_1_7B_1_1B_1_8B_1_2B_1_9B_1_3B_1_10B_1_4B_1_11B_1_5B_1_12B_1_6B_1_13B_2_7B_2_1B_2_8B_2_2B_2_9B_2_3B_2_10B_2_4B_2_11B_2_5B_2_12B_2_6B_2_13B_3_7B_3_1B_3_8B_3_2B_3_9B_3_3B_3_10B_3_4B_3_11B_3_5B_3_12B_3_6B_3_13C_7_3C_7_4C_8_3C_8_4C_9_3C_9_4C_10_3C_10_4C_11_3C_11_4C_12_3C_12_4C_13_3C_13_4TraceMulA_1_4A_1_1A_1_5A_1_2A_1_6A_1_3A_1_7A_2_4A_2_1A_2_5A_2_2A_2_6A_2_3A_2_7B_4_7B_4_8B_4_9B_4_10B_4_11B_4_12B_4_13B_5_7B_5_8B_5_9B_5_10B_5_11B_5_12B_5_13B_6_7B_6_8B_6_9B_6_10B_6_11B_6_12B_6_13B_7_7B_7_8B_7_9B_7_10B_7_11B_7_12B_7_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_1A_1_2A_1_3A_2_1A_2_2A_2_3B_1_7B_5_7B_1_8B_5_8B_1_9B_5_9B_1_10B_5_10B_1_11B_5_11B_1_12B_5_12B_1_13B_5_13B_2_7B_6_7B_2_8B_6_8B_2_9B_6_9B_2_10B_6_10B_2_11B_6_11B_2_12B_6_12B_2_13B_6_13B_3_7B_7_7B_3_8B_7_8B_3_9B_7_9B_3_10B_7_10B_3_11B_7_11B_3_12B_7_12B_3_13B_7_13C_7_1C_7_3C_7_2C_7_4C_8_1C_8_3C_8_2C_8_4C_9_1C_9_3C_9_2C_9_4C_10_1C_10_3C_10_2C_10_4C_11_1C_11_3C_11_2C_11_4C_12_1C_12_3C_12_2C_12_4C_13_1C_13_3C_13_2C_13_4TraceMulA_3_4A_3_5A_3_6A_3_7A_4_4A_4_5A_4_6A_4_7B_4_1B_4_2B_4_3B_4_4B_4_5B_4_6B_1_1B_5_1B_1_2B_5_2B_5_3B_1_3B_1_4B_5_4B_1_5B_5_5B_1_6B_5_6B_2_1B_6_1B_2_2B_6_2B_2_3B_6_3B_6_4B_2_4B_2_5B_6_5B_2_6B_6_6B_3_1B_7_1B_3_2B_7_2B_3_3B_7_3B_7_4B_3_4B_3_5B_7_5B_3_6B_7_6C_1_1C_1_3C_1_2C_1_4C_2_1C_2_3C_2_2C_2_4C_3_1C_3_3C_3_2C_3_4C_4_1C_4_3C_4_2C_4_4C_5_1C_5_3C_5_4C_5_2C_6_1C_6_3C_6_2C_6_4TraceMulA_3_1A_3_5A_3_2A_3_6A_3_3A_3_7A_4_1A_4_5A_4_2A_4_6A_4_3A_4_7B_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_6C_1_3C_8_3C_1_4C_8_4C_2_3C_9_3C_2_4C_9_4C_3_3C_10_3C_3_4C_10_4C_4_3C_11_3C_4_4C_11_4C_5_3C_12_3C_5_4C_12_4C_13_3C_6_3C_6_4C_13_4Trace(Mul(Matrix(4, 7, [[A_1_1,A_1_2,A_1_3,A_1_4,A_1_5,A_1_6,A_1_7],[A_2_1,A_2_2,A_2_3,A_2_4,A_2_5,A_2_6,A_2_7],[A_3_1,A_3_2,A_3_3,A_3_4,A_3_5,A_3_6,A_3_7],[A_4_1,A_4_2,A_4_3,A_4_4,A_4_5,A_4_6,A_4_7]]),Matrix(7, 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]]),Matrix(13, 4, [[C_1_1,C_1_2,C_1_3,C_1_4],[C_2_1,C_2_2,C_2_3,C_2_4],[C_3_1,C_3_2,C_3_3,C_3_4],[C_4_1,C_4_2,C_4_3,C_4_4],[C_5_1,C_5_2,C_5_3,C_5_4],[C_6_1,C_6_2,C_6_3,C_6_4],[C_7_1,C_7_2,C_7_3,C_7_4],[C_8_1,C_8_2,C_8_3,C_8_4],[C_9_1,C_9_2,C_9_3,C_9_4],[C_10_1,C_10_2,C_10_3,C_10_4],[C_11_1,C_11_2,C_11_3,C_11_4],[C_12_1,C_12_2,C_12_3,C_12_4],[C_13_1,C_13_2,C_13_3,C_13_4]]))) = Trace(Mul(Matrix(2, 4, [[A_3_4,A_1_1+A_3_5,A_1_2+A_3_6,A_1_3+A_3_7],[A_4_4,A_2_1+A_4_5,A_2_2+A_4_6,A_2_3+A_4_7]]),Matrix(4, 7, [[B_4_7,B_4_8,B_4_9,B_4_10,B_4_11,B_4_12,B_4_13],[B_5_7,B_1_1+B_5_8,B_1_2+B_5_9,B_1_3+B_5_10,B_1_4+B_5_11,B_1_5+B_5_12,B_1_6+B_5_13],[B_6_7,B_2_1+B_6_8,B_2_2+B_6_9,B_2_3+B_6_10,B_2_4+B_6_11,B_2_5+B_6_12,B_2_6+B_6_13],[B_7_7,B_3_1+B_7_8,B_3_2+B_7_9,B_3_3+B_7_10,B_3_4+B_7_11,B_3_5+B_7_12,B_3_6+B_7_13]]),Matrix(7, 2, [[C_7_3,C_7_4],[C_1_1+C_8_3,C_1_2+C_8_4],[C_2_1+C_9_3,C_2_2+C_9_4],[C_3_1+C_10_3,C_3_2+C_10_4],[C_4_1+C_11_3,C_4_2+C_11_4],[C_5_1+C_12_3,C_5_2+C_12_4],[C_6_1+C_13_3,C_6_2+C_13_4]])))+Trace(Mul(Matrix(2, 4, [[A_1_4-A_3_4,A_1_5-A_3_5,A_1_6-A_3_6,A_1_7-A_3_7],[A_2_4-A_4_4,-A_4_5+A_2_5,-A_4_6+A_2_6,A_2_7-A_4_7]]),Matrix(4, 6, [[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_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],[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_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]]),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(2, 3, [[-A_1_1+A_3_1,-A_1_2+A_3_2,-A_1_3+A_3_3],[-A_2_1+A_4_1,-A_2_2+A_4_2,A_4_3-A_2_3]]),Matrix(3, 7, [[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]]),Matrix(7, 2, [[C_7_3,C_7_4],[C_8_3,C_8_4],[C_9_3,C_9_4],[C_10_3,C_10_4],[C_11_3,C_11_4],[C_12_3,C_12_4],[C_13_3,C_13_4]])))+Trace(Mul(Matrix(2, 4, [[A_1_4,A_1_1+A_1_5,A_1_2+A_1_6,A_1_3+A_1_7],[A_2_4,A_2_1+A_2_5,A_2_2+A_2_6,A_2_3+A_2_7]]),Matrix(4, 7, [[B_4_7,B_4_8,B_4_9,B_4_10,B_4_11,B_4_12,B_4_13],[B_5_7,B_5_8,B_5_9,B_5_10,B_5_11,B_5_12,B_5_13],[B_6_7,B_6_8,B_6_9,B_6_10,B_6_11,B_6_12,B_6_13],[B_7_7,B_7_8,B_7_9,B_7_10,B_7_11,B_7_12,B_7_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, 3, [[A_1_1,A_1_2,A_1_3],[A_2_1,A_2_2,A_2_3]]),Matrix(3, 7, [[B_1_7-B_5_7,B_1_8-B_5_8,B_1_9-B_5_9,B_1_10-B_5_10,B_1_11-B_5_11,B_1_12-B_5_12,B_1_13-B_5_13],[B_2_7-B_6_7,B_2_8-B_6_8,B_2_9-B_6_9,B_2_10-B_6_10,B_2_11-B_6_11,B_2_12-B_6_12,B_2_13-B_6_13],[B_3_7-B_7_7,B_3_8-B_7_8,B_3_9-B_7_9,B_3_10-B_7_10,B_3_11-B_7_11,B_3_12-B_7_12,B_3_13-B_7_13]]),Matrix(7, 2, [[C_7_1+C_7_3,C_7_2+C_7_4],[C_8_1+C_8_3,C_8_2+C_8_4],[C_9_1+C_9_3,C_9_2+C_9_4],[C_10_1+C_10_3,C_10_2+C_10_4],[C_11_1+C_11_3,C_11_2+C_11_4],[C_12_1+C_12_3,C_12_2+C_12_4],[C_13_1+C_13_3,C_13_2+C_13_4]])))+Trace(Mul(Matrix(2, 4, [[A_3_4,A_3_5,A_3_6,A_3_7],[A_4_4,A_4_5,A_4_6,A_4_7]]),Matrix(4, 6, [[B_4_1,B_4_2,B_4_3,B_4_4,B_4_5,B_4_6],[-B_1_1+B_5_1,-B_1_2+B_5_2,B_5_3-B_1_3,-B_1_4+B_5_4,-B_1_5+B_5_5,-B_1_6+B_5_6],[-B_2_1+B_6_1,-B_2_2+B_6_2,-B_2_3+B_6_3,B_6_4-B_2_4,-B_2_5+B_6_5,-B_2_6+B_6_6],[-B_3_1+B_7_1,-B_3_2+B_7_2,-B_3_3+B_7_3,B_7_4-B_3_4,-B_3_5+B_7_5,-B_3_6+B_7_6]]),Matrix(6, 2, [[C_1_1+C_1_3,C_1_2+C_1_4],[C_2_1+C_2_3,C_2_2+C_2_4],[C_3_1+C_3_3,C_3_2+C_3_4],[C_4_1+C_4_3,C_4_2+C_4_4],[C_5_1+C_5_3,C_5_4+C_5_2],[C_6_1+C_6_3,C_6_2+C_6_4]])))+Trace(Mul(Matrix(2, 3, [[A_3_1+A_3_5,A_3_2+A_3_6,A_3_3+A_3_7],[A_4_1+A_4_5,A_4_2+A_4_6,A_4_3+A_4_7]]),Matrix(3, 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]]),Matrix(6, 2, [[C_1_3-C_8_3,C_1_4-C_8_4],[C_2_3-C_9_3,C_2_4-C_9_4],[C_3_3-C_10_3,C_3_4-C_10_4],[C_4_3-C_11_3,C_4_4-C_11_4],[C_5_3-C_12_3,C_5_4-C_12_4],[-C_13_3+C_6_3,C_6_4-C_13_4]])))

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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