Description of fast matrix multiplication algorithm: ⟨3×14×31:985⟩

Algorithm type

176⁢X3⁢Y3⁢Z2+64⁢X2⁢Y3⁢Z3+264⁢X3⁢Y3⁢Z+2⁢X2⁢Y3⁢Z2+96⁢X⁢Y3⁢Z3+5⁢X3⁢Y2⁢Z+105⁢X2⁢Y2⁢Z2+10⁢X⁢Y4⁢Z+5⁢X⁢Y2⁢Z3+25⁢X3⁢Y⁢Z+10⁢X2⁢Y2⁢Z+5⁢X⁢Y3⁢Z+10⁢X⁢Y2⁢Z2+25⁢X⁢Y⁢Z3+19⁢X2⁢Y⁢Z+71⁢X⁢Y2⁢Z+17⁢X⁢Y⁢Z2+76⁢X⁢Y⁢Z176X3Y3Z264X2Y3Z3264X3Y3Z2X2Y3Z296XY3Z35X3Y2Z105X2Y2Z210XY4Z5XY2Z325X3YZ10X2Y2Z5XY3Z10XY2Z225XYZ319X2YZ71XY2Z17XYZ276XYZ176*X^3*Y^3*Z^2+64*X^2*Y^3*Z^3+264*X^3*Y^3*Z+2*X^2*Y^3*Z^2+96*X*Y^3*Z^3+5*X^3*Y^2*Z+105*X^2*Y^2*Z^2+10*X*Y^4*Z+5*X*Y^2*Z^3+25*X^3*Y*Z+10*X^2*Y^2*Z+5*X*Y^3*Z+10*X*Y^2*Z^2+25*X*Y*Z^3+19*X^2*Y*Z+71*X*Y^2*Z+17*X*Y*Z^2+76*X*Y*Z

Algorithm definition

The algorithm ⟨3×14×31:985⟩ could be constructed using the following decomposition:

⟨3×14×31:985⟩ = ⟨3×14×3:95⟩ + ⟨3×14×28:890⟩.

This decomposition is defined by the following equality:

Trace⁡Mul⁡A_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_1_7A_1_8A_1_9A_1_10A_1_11A_1_12A_1_13A_1_14A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_2_7A_2_8A_2_9A_2_10A_2_11A_2_12A_2_13A_2_14A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_3_7A_3_8A_3_9A_3_10A_3_11A_3_12A_3_13A_3_14B_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_1_14B_1_15B_1_16B_1_17B_1_18B_1_19B_1_20B_1_21B_1_22B_1_23B_1_24B_1_25B_1_26B_1_27B_1_28B_1_29B_1_30B_1_31B_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_2_14B_2_15B_2_16B_2_17B_2_18B_2_19B_2_20B_2_21B_2_22B_2_23B_2_24B_2_25B_2_26B_2_27B_2_28B_2_29B_2_30B_2_31B_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_3_14B_3_15B_3_16B_3_17B_3_18B_3_19B_3_20B_3_21B_3_22B_3_23B_3_24B_3_25B_3_26B_3_27B_3_28B_3_29B_3_30B_3_31B_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_4_14B_4_15B_4_16B_4_17B_4_18B_4_19B_4_20B_4_21B_4_22B_4_23B_4_24B_4_25B_4_26B_4_27B_4_28B_4_29B_4_30B_4_31B_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_5_14B_5_15B_5_16B_5_17B_5_18B_5_19B_5_20B_5_21B_5_22B_5_23B_5_24B_5_25B_5_26B_5_27B_5_28B_5_29B_5_30B_5_31B_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_6_14B_6_15B_6_16B_6_17B_6_18B_6_19B_6_20B_6_21B_6_22B_6_23B_6_24B_6_25B_6_26B_6_27B_6_28B_6_29B_6_30B_6_31B_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_7_14B_7_15B_7_16B_7_17B_7_18B_7_19B_7_20B_7_21B_7_22B_7_23B_7_24B_7_25B_7_26B_7_27B_7_28B_7_29B_7_30B_7_31B_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_8_14B_8_15B_8_16B_8_17B_8_18B_8_19B_8_20B_8_21B_8_22B_8_23B_8_24B_8_25B_8_26B_8_27B_8_28B_8_29B_8_30B_8_31B_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_9_14B_9_15B_9_16B_9_17B_9_18B_9_19B_9_20B_9_21B_9_22B_9_23B_9_24B_9_25B_9_26B_9_27B_9_28B_9_29B_9_30B_9_31B_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_10_14B_10_15B_10_16B_10_17B_10_18B_10_19B_10_20B_10_21B_10_22B_10_23B_10_24B_10_25B_10_26B_10_27B_10_28B_10_29B_10_30B_10_31B_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_11_14B_11_15B_11_16B_11_17B_11_18B_11_19B_11_20B_11_21B_11_22B_11_23B_11_24B_11_25B_11_26B_11_27B_11_28B_11_29B_11_30B_11_31B_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_13B_12_14B_12_15B_12_16B_12_17B_12_18B_12_19B_12_20B_12_21B_12_22B_12_23B_12_24B_12_25B_12_26B_12_27B_12_28B_12_29B_12_30B_12_31B_13_1B_13_2B_13_3B_13_4B_13_5B_13_6B_13_7B_13_8B_13_9B_13_10B_13_11B_13_12B_13_13B_13_14B_13_15B_13_16B_13_17B_13_18B_13_19B_13_20B_13_21B_13_22B_13_23B_13_24B_13_25B_13_26B_13_27B_13_28B_13_29B_13_30B_13_31B_14_1B_14_2B_14_3B_14_4B_14_5B_14_6B_14_7B_14_8B_14_9B_14_10B_14_11B_14_12B_14_13B_14_14B_14_15B_14_16B_14_17B_14_18B_14_19B_14_20B_14_21B_14_22B_14_23B_14_24B_14_25B_14_26B_14_27B_14_28B_14_29B_14_30B_14_31C_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_3C_10_1C_10_2C_10_3C_11_1C_11_2C_11_3C_12_1C_12_2C_12_3C_13_1C_13_2C_13_3C_14_1C_14_2C_14_3C_15_1C_15_2C_15_3C_16_1C_16_2C_16_3C_17_1C_17_2C_17_3C_18_1C_18_2C_18_3C_19_1C_19_2C_19_3C_20_1C_20_2C_20_3C_21_1C_21_2C_21_3C_22_1C_22_2C_22_3C_23_1C_23_2C_23_3C_24_1C_24_2C_24_3C_25_1C_25_2C_25_3C_26_1C_26_2C_26_3C_27_1C_27_2C_27_3C_28_1C_28_2C_28_3C_29_1C_29_2C_29_3C_30_1C_30_2C_30_3C_31_1C_31_2C_31_3=Trace⁡Mul⁡A_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_1_7A_1_8A_1_9A_1_10A_1_11A_1_12A_1_13A_1_14A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_2_7A_2_8A_2_9A_2_10A_2_11A_2_12A_2_13A_2_14A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_3_7A_3_8A_3_9A_3_10A_3_11A_3_12A_3_13A_3_14B_1_1B_1_2B_1_3B_2_1B_2_2B_2_3B_3_1B_3_2B_3_3B_4_1B_4_2B_4_3B_5_1B_5_2B_5_3B_6_1B_6_2B_6_3B_7_1B_7_2B_7_3B_8_1B_8_2B_8_3B_9_1B_9_2B_9_3B_10_1B_10_2B_10_3B_11_1B_11_2B_11_3B_12_1B_12_2B_12_3B_13_1B_13_2B_13_3B_14_1B_14_2B_14_3C_1_1C_1_2C_1_3C_2_1C_2_2C_2_3C_3_1C_3_2C_3_3+Trace⁡Mul⁡A_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_1_7A_1_8A_1_9A_1_10A_1_11A_1_12A_1_13A_1_14A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_2_7A_2_8A_2_9A_2_10A_2_11A_2_12A_2_13A_2_14A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_3_7A_3_8A_3_9A_3_10A_3_11A_3_12A_3_13A_3_14B_1_4B_1_5B_1_6B_1_7B_1_8B_1_9B_1_10B_1_11B_1_12B_1_13B_1_14B_1_15B_1_16B_1_17B_1_18B_1_19B_1_20B_1_21B_1_22B_1_23B_1_24B_1_25B_1_26B_1_27B_1_28B_1_29B_1_30B_1_31B_2_4B_2_5B_2_6B_2_7B_2_8B_2_9B_2_10B_2_11B_2_12B_2_13B_2_14B_2_15B_2_16B_2_17B_2_18B_2_19B_2_20B_2_21B_2_22B_2_23B_2_24B_2_25B_2_26B_2_27B_2_28B_2_29B_2_30B_2_31B_3_4B_3_5B_3_6B_3_7B_3_8B_3_9B_3_10B_3_11B_3_12B_3_13B_3_14B_3_15B_3_16B_3_17B_3_18B_3_19B_3_20B_3_21B_3_22B_3_23B_3_24B_3_25B_3_26B_3_27B_3_28B_3_29B_3_30B_3_31B_4_4B_4_5B_4_6B_4_7B_4_8B_4_9B_4_10B_4_11B_4_12B_4_13B_4_14B_4_15B_4_16B_4_17B_4_18B_4_19B_4_20B_4_21B_4_22B_4_23B_4_24B_4_25B_4_26B_4_27B_4_28B_4_29B_4_30B_4_31B_5_4B_5_5B_5_6B_5_7B_5_8B_5_9B_5_10B_5_11B_5_12B_5_13B_5_14B_5_15B_5_16B_5_17B_5_18B_5_19B_5_20B_5_21B_5_22B_5_23B_5_24B_5_25B_5_26B_5_27B_5_28B_5_29B_5_30B_5_31B_6_4B_6_5B_6_6B_6_7B_6_8B_6_9B_6_10B_6_11B_6_12B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2C_7_3C_8_1C_8_2C_8_3C_9_1C_9_2C_9_3C_10_1C_10_2C_10_3C_11_1C_11_2C_11_3C_12_1C_12_2C_12_3C_13_1C_13_2C_13_3C_14_1C_14_2C_14_3C_15_1C_15_2C_15_3C_16_1C_16_2C_16_3C_17_1C_17_2C_17_3C_18_1C_18_2C_18_3C_19_1C_19_2C_19_3C_20_1C_20_2C_20_3C_21_1C_21_2C_21_3C_22_1C_22_2C_22_3C_23_1C_23_2C_23_3C_24_1C_24_2C_24_3C_25_1C_25_2C_25_3C_26_1C_26_2C_26_3C_27_1C_27_2C_27_3C_28_1C_28_2C_28_3C_29_1C_29_2C_29_3C_30_1C_30_2C_30_3C_31_1C_31_2C_31_3TraceMulA_1_1A_1_2A_1_3A_1_4A_1_5A_1_6A_1_7A_1_8A_1_9A_1_10A_1_11A_1_12A_1_13A_1_14A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_2_7A_2_8A_2_9A_2_10A_2_11A_2_12A_2_13A_2_14A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_3_7A_3_8A_3_9A_3_10A_3_11A_3_12A_3_13A_3_14B_1_1B_1_2B_1_3B_2_1B_2_2B_2_3B_3_1B_3_2B_3_3B_4_1B_4_2B_4_3B_5_1B_5_2B_5_3B_6_1B_6_2B_6_3B_7_1B_7_2B_7_3B_8_1B_8_2B_8_3B_9_1B_9_2B_9_3B_10_1B_10_2B_10_3B_11_1B_11_2B_11_3B_12_1B_12_2B_12_3B_13_1B_13_2B_13_3B_14_1B_14_2B_14_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_1_4A_1_5A_1_6A_1_7A_1_8A_1_9A_1_10A_1_11A_1_12A_1_13A_1_14A_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_2_7A_2_8A_2_9A_2_10A_2_11A_2_12A_2_13A_2_14A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_3_7A_3_8A_3_9A_3_10A_3_11A_3_12A_3_13A_3_14B_1_4B_1_5B_1_6B_1_7B_1_8B_1_9B_1_10B_1_11B_1_12B_1_13B_1_14B_1_15B_1_16B_1_17B_1_18B_1_19B_1_20B_1_21B_1_22B_1_23B_1_24B_1_25B_1_26B_1_27B_1_28B_1_29B_1_30B_1_31B_2_4B_2_5B_2_6B_2_7B_2_8B_2_9B_2_10B_2_11B_2_12B_2_13B_2_14B_2_15B_2_16B_2_17B_2_18B_2_19B_2_20B_2_21B_2_22B_2_23B_2_24B_2_25B_2_26B_2_27B_2_28B_2_29B_2_30B_2_31B_3_4B_3_5B_3_6B_3_7B_3_8B_3_9B_3_10B_3_11B_3_12B_3_13B_3_14B_3_15B_3_16B_3_17B_3_18B_3_19B_3_20B_3_21B_3_22B_3_23B_3_24B_3_25B_3_26B_3_27B_3_28B_3_29B_3_30B_3_31B_4_4B_4_5B_4_6B_4_7B_4_8B_4_9B_4_10B_4_11B_4_12B_4_13B_4_14B_4_15B_4_16B_4_17B_4_18B_4_19B_4_20B_4_21B_4_22B_4_23B_4_24B_4_25B_4_26B_4_27B_4_28B_4_29B_4_30B_4_31B_5_4B_5_5B_5_6B_5_7B_5_8B_5_9B_5_10B_5_11B_5_12B_5_13B_5_14B_5_15B_5_16B_5_17B_5_18B_5_19B_5_20B_5_21B_5_22B_5_23B_5_24B_5_25B_5_26B_5_27B_5_28B_5_29B_5_30B_5_31B_6_4B_6_5B_6_6B_6_7B_6_8B_6_9B_6_10B_6_11B_6_12B_6_13B_6_14B_6_15B_6_16B_6_17B_6_18B_6_19B_6_20B_6_21B_6_22B_6_23B_6_24B_6_25B_6_26B_6_27B_6_28B_6_29B_6_30B_6_31B_7_4B_7_5B_7_6B_7_7B_7_8B_7_9B_7_10B_7_11B_7_12B_7_13B_7_14B_7_15B_7_16B_7_17B_7_18B_7_19B_7_20B_7_21B_7_22B_7_23B_7_24B_7_25B_7_26B_7_27B_7_28B_7_29B_7_30B_7_31B_8_4B_8_5B_8_6B_8_7B_8_8B_8_9B_8_10B_8_11B_8_12B_8_13B_8_14B_8_15B_8_16B_8_17B_8_18B_8_19B_8_20B_8_21B_8_22B_8_23B_8_24B_8_25B_8_26B_8_27B_8_28B_8_29B_8_30B_8_31B_9_4B_9_5B_9_6B_9_7B_9_8B_9_9B_9_10B_9_11B_9_12B_9_13B_9_14B_9_15B_9_16B_9_17B_9_18B_9_19B_9_20B_9_21B_9_22B_9_23B_9_24B_9_25B_9_26B_9_27B_9_28B_9_29B_9_30B_9_31B_10_4B_10_5B_10_6B_10_7B_10_8B_10_9B_10_10B_10_11B_10_12B_10_13B_10_14B_10_15B_10_16B_10_17B_10_18B_10_19B_10_20B_10_21B_10_22B_10_23B_10_24B_10_25B_10_26B_10_27B_10_28B_10_29B_10_30B_10_31B_11_4B_11_5B_11_6B_11_7B_11_8B_11_9B_11_10B_11_11B_11_12B_11_13B_11_14B_11_15B_11_16B_11_17B_11_18B_11_19B_11_20B_11_21B_11_22B_11_23B_11_24B_11_25B_11_26B_11_27B_11_28B_11_29B_11_30B_11_31B_12_4B_12_5B_12_6B_12_7B_12_8B_12_9B_12_10B_12_11B_12_12B_12_13B_12_14B_12_15B_12_16B_12_17B_12_18B_12_19B_12_20B_12_21B_12_22B_12_23B_12_24B_12_25B_12_26B_12_27B_12_28B_12_29B_12_30B_12_31B_13_4B_13_5B_13_6B_13_7B_13_8B_13_9B_13_10B_13_11B_13_12B_13_13B_13_14B_13_15B_13_16B_13_17B_13_18B_13_19B_13_20B_13_21B_13_22B_13_23B_13_24B_13_25B_13_26B_13_27B_13_28B_13_29B_13_30B_13_31B_14_4B_14_5B_14_6B_14_7B_14_8B_14_9B_14_10B_14_11B_14_12B_14_13B_14_14B_14_15B_14_16B_14_17B_14_18B_14_19B_14_20B_14_21B_14_22B_14_23B_14_24B_14_25B_14_26B_14_27B_14_28B_14_29B_14_30B_14_31C_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_3C_10_1C_10_2C_10_3C_11_1C_11_2C_11_3C_12_1C_12_2C_12_3C_13_1C_13_2C_13_3C_14_1C_14_2C_14_3C_15_1C_15_2C_15_3C_16_1C_16_2C_16_3C_17_1C_17_2C_17_3C_18_1C_18_2C_18_3C_19_1C_19_2C_19_3C_20_1C_20_2C_20_3C_21_1C_21_2C_21_3C_22_1C_22_2C_22_3C_23_1C_23_2C_23_3C_24_1C_24_2C_24_3C_25_1C_25_2C_25_3C_26_1C_26_2C_26_3C_27_1C_27_2C_27_3C_28_1C_28_2C_28_3C_29_1C_29_2C_29_3C_30_1C_30_2C_30_3C_31_1C_31_2C_31_3Trace(Mul(Matrix(3, 14, [[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_1_13,A_1_14],[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_2_13,A_2_14],[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_3_13,A_3_14]]),Matrix(14, 31, [[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_1_14,B_1_15,B_1_16,B_1_17,B_1_18,B_1_19,B_1_20,B_1_21,B_1_22,B_1_23,B_1_24,B_1_25,B_1_26,B_1_27,B_1_28,B_1_29,B_1_30,B_1_31],[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_2_14,B_2_15,B_2_16,B_2_17,B_2_18,B_2_19,B_2_20,B_2_21,B_2_22,B_2_23,B_2_24,B_2_25,B_2_26,B_2_27,B_2_28,B_2_29,B_2_30,B_2_31],[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_3_14,B_3_15,B_3_16,B_3_17,B_3_18,B_3_19,B_3_20,B_3_21,B_3_22,B_3_23,B_3_24,B_3_25,B_3_26,B_3_27,B_3_28,B_3_29,B_3_30,B_3_31],[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_4_14,B_4_15,B_4_16,B_4_17,B_4_18,B_4_19,B_4_20,B_4_21,B_4_22,B_4_23,B_4_24,B_4_25,B_4_26,B_4_27,B_4_28,B_4_29,B_4_30,B_4_31],[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_5_14,B_5_15,B_5_16,B_5_17,B_5_18,B_5_19,B_5_20,B_5_21,B_5_22,B_5_23,B_5_24,B_5_25,B_5_26,B_5_27,B_5_28,B_5_29,B_5_30,B_5_31],[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_6_14,B_6_15,B_6_16,B_6_17,B_6_18,B_6_19,B_6_20,B_6_21,B_6_22,B_6_23,B_6_24,B_6_25,B_6_26,B_6_27,B_6_28,B_6_29,B_6_30,B_6_31],[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_7_14,B_7_15,B_7_16,B_7_17,B_7_18,B_7_19,B_7_20,B_7_21,B_7_22,B_7_23,B_7_24,B_7_25,B_7_26,B_7_27,B_7_28,B_7_29,B_7_30,B_7_31],[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_8_14,B_8_15,B_8_16,B_8_17,B_8_18,B_8_19,B_8_20,B_8_21,B_8_22,B_8_23,B_8_24,B_8_25,B_8_26,B_8_27,B_8_28,B_8_29,B_8_30,B_8_31],[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_9_14,B_9_15,B_9_16,B_9_17,B_9_18,B_9_19,B_9_20,B_9_21,B_9_22,B_9_23,B_9_24,B_9_25,B_9_26,B_9_27,B_9_28,B_9_29,B_9_30,B_9_31],[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_10_14,B_10_15,B_10_16,B_10_17,B_10_18,B_10_19,B_10_20,B_10_21,B_10_22,B_10_23,B_10_24,B_10_25,B_10_26,B_10_27,B_10_28,B_10_29,B_10_30,B_10_31],[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_11_14,B_11_15,B_11_16,B_11_17,B_11_18,B_11_19,B_11_20,B_11_21,B_11_22,B_11_23,B_11_24,B_11_25,B_11_26,B_11_27,B_11_28,B_11_29,B_11_30,B_11_31],[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,B_12_14,B_12_15,B_12_16,B_12_17,B_12_18,B_12_19,B_12_20,B_12_21,B_12_22,B_12_23,B_12_24,B_12_25,B_12_26,B_12_27,B_12_28,B_12_29,B_12_30,B_12_31],[B_13_1,B_13_2,B_13_3,B_13_4,B_13_5,B_13_6,B_13_7,B_13_8,B_13_9,B_13_10,B_13_11,B_13_12,B_13_13,B_13_14,B_13_15,B_13_16,B_13_17,B_13_18,B_13_19,B_13_20,B_13_21,B_13_22,B_13_23,B_13_24,B_13_25,B_13_26,B_13_27,B_13_28,B_13_29,B_13_30,B_13_31],[B_14_1,B_14_2,B_14_3,B_14_4,B_14_5,B_14_6,B_14_7,B_14_8,B_14_9,B_14_10,B_14_11,B_14_12,B_14_13,B_14_14,B_14_15,B_14_16,B_14_17,B_14_18,B_14_19,B_14_20,B_14_21,B_14_22,B_14_23,B_14_24,B_14_25,B_14_26,B_14_27,B_14_28,B_14_29,B_14_30,B_14_31]]),Matrix(31, 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],[C_10_1,C_10_2,C_10_3],[C_11_1,C_11_2,C_11_3],[C_12_1,C_12_2,C_12_3],[C_13_1,C_13_2,C_13_3],[C_14_1,C_14_2,C_14_3],[C_15_1,C_15_2,C_15_3],[C_16_1,C_16_2,C_16_3],[C_17_1,C_17_2,C_17_3],[C_18_1,C_18_2,C_18_3],[C_19_1,C_19_2,C_19_3],[C_20_1,C_20_2,C_20_3],[C_21_1,C_21_2,C_21_3],[C_22_1,C_22_2,C_22_3],[C_23_1,C_23_2,C_23_3],[C_24_1,C_24_2,C_24_3],[C_25_1,C_25_2,C_25_3],[C_26_1,C_26_2,C_26_3],[C_27_1,C_27_2,C_27_3],[C_28_1,C_28_2,C_28_3],[C_29_1,C_29_2,C_29_3],[C_30_1,C_30_2,C_30_3],[C_31_1,C_31_2,C_31_3]]))) = Trace(Mul(Matrix(3, 14, [[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_1_13,A_1_14],[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_2_13,A_2_14],[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_3_13,A_3_14]]),Matrix(14, 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],[B_4_1,B_4_2,B_4_3],[B_5_1,B_5_2,B_5_3],[B_6_1,B_6_2,B_6_3],[B_7_1,B_7_2,B_7_3],[B_8_1,B_8_2,B_8_3],[B_9_1,B_9_2,B_9_3],[B_10_1,B_10_2,B_10_3],[B_11_1,B_11_2,B_11_3],[B_12_1,B_12_2,B_12_3],[B_13_1,B_13_2,B_13_3],[B_14_1,B_14_2,B_14_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, 14, [[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_1_13,A_1_14],[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_2_13,A_2_14],[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_3_13,A_3_14]]),Matrix(14, 28, [[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_1_14,B_1_15,B_1_16,B_1_17,B_1_18,B_1_19,B_1_20,B_1_21,B_1_22,B_1_23,B_1_24,B_1_25,B_1_26,B_1_27,B_1_28,B_1_29,B_1_30,B_1_31],[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_2_14,B_2_15,B_2_16,B_2_17,B_2_18,B_2_19,B_2_20,B_2_21,B_2_22,B_2_23,B_2_24,B_2_25,B_2_26,B_2_27,B_2_28,B_2_29,B_2_30,B_2_31],[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_3_14,B_3_15,B_3_16,B_3_17,B_3_18,B_3_19,B_3_20,B_3_21,B_3_22,B_3_23,B_3_24,B_3_25,B_3_26,B_3_27,B_3_28,B_3_29,B_3_30,B_3_31],[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_4_14,B_4_15,B_4_16,B_4_17,B_4_18,B_4_19,B_4_20,B_4_21,B_4_22,B_4_23,B_4_24,B_4_25,B_4_26,B_4_27,B_4_28,B_4_29,B_4_30,B_4_31],[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_5_14,B_5_15,B_5_16,B_5_17,B_5_18,B_5_19,B_5_20,B_5_21,B_5_22,B_5_23,B_5_24,B_5_25,B_5_26,B_5_27,B_5_28,B_5_29,B_5_30,B_5_31],[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_6_14,B_6_15,B_6_16,B_6_17,B_6_18,B_6_19,B_6_20,B_6_21,B_6_22,B_6_23,B_6_24,B_6_25,B_6_26,B_6_27,B_6_28,B_6_29,B_6_30,B_6_31],[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_7_14,B_7_15,B_7_16,B_7_17,B_7_18,B_7_19,B_7_20,B_7_21,B_7_22,B_7_23,B_7_24,B_7_25,B_7_26,B_7_27,B_7_28,B_7_29,B_7_30,B_7_31],[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_8_14,B_8_15,B_8_16,B_8_17,B_8_18,B_8_19,B_8_20,B_8_21,B_8_22,B_8_23,B_8_24,B_8_25,B_8_26,B_8_27,B_8_28,B_8_29,B_8_30,B_8_31],[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_9_14,B_9_15,B_9_16,B_9_17,B_9_18,B_9_19,B_9_20,B_9_21,B_9_22,B_9_23,B_9_24,B_9_25,B_9_26,B_9_27,B_9_28,B_9_29,B_9_30,B_9_31],[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_10_14,B_10_15,B_10_16,B_10_17,B_10_18,B_10_19,B_10_20,B_10_21,B_10_22,B_10_23,B_10_24,B_10_25,B_10_26,B_10_27,B_10_28,B_10_29,B_10_30,B_10_31],[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_11_14,B_11_15,B_11_16,B_11_17,B_11_18,B_11_19,B_11_20,B_11_21,B_11_22,B_11_23,B_11_24,B_11_25,B_11_26,B_11_27,B_11_28,B_11_29,B_11_30,B_11_31],[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,B_12_14,B_12_15,B_12_16,B_12_17,B_12_18,B_12_19,B_12_20,B_12_21,B_12_22,B_12_23,B_12_24,B_12_25,B_12_26,B_12_27,B_12_28,B_12_29,B_12_30,B_12_31],[B_13_4,B_13_5,B_13_6,B_13_7,B_13_8,B_13_9,B_13_10,B_13_11,B_13_12,B_13_13,B_13_14,B_13_15,B_13_16,B_13_17,B_13_18,B_13_19,B_13_20,B_13_21,B_13_22,B_13_23,B_13_24,B_13_25,B_13_26,B_13_27,B_13_28,B_13_29,B_13_30,B_13_31],[B_14_4,B_14_5,B_14_6,B_14_7,B_14_8,B_14_9,B_14_10,B_14_11,B_14_12,B_14_13,B_14_14,B_14_15,B_14_16,B_14_17,B_14_18,B_14_19,B_14_20,B_14_21,B_14_22,B_14_23,B_14_24,B_14_25,B_14_26,B_14_27,B_14_28,B_14_29,B_14_30,B_14_31]]),Matrix(28, 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],[C_10_1,C_10_2,C_10_3],[C_11_1,C_11_2,C_11_3],[C_12_1,C_12_2,C_12_3],[C_13_1,C_13_2,C_13_3],[C_14_1,C_14_2,C_14_3],[C_15_1,C_15_2,C_15_3],[C_16_1,C_16_2,C_16_3],[C_17_1,C_17_2,C_17_3],[C_18_1,C_18_2,C_18_3],[C_19_1,C_19_2,C_19_3],[C_20_1,C_20_2,C_20_3],[C_21_1,C_21_2,C_21_3],[C_22_1,C_22_2,C_22_3],[C_23_1,C_23_2,C_23_3],[C_24_1,C_24_2,C_24_3],[C_25_1,C_25_2,C_25_3],[C_26_1,C_26_2,C_26_3],[C_27_1,C_27_2,C_27_3],[C_28_1,C_28_2,C_28_3],[C_29_1,C_29_2,C_29_3],[C_30_1,C_30_2,C_30_3],[C_31_1,C_31_2,C_31_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.

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