Description of fast matrix multiplication algorithm: ⟨3×16×16:576⟩

Algorithm type

2XY9Z+36X3Y4Z3+12X2Y6Z2+6XY6Z2+48X2Y4Z2+16XY6Z+12X2Y4Z+60X2Y2Z2+44XY4Z+6X2Y2Z+36XY3Z+18XY2Z2+150XY2Z+130XYZ2XY9Z36X3Y4Z312X2Y6Z26XY6Z248X2Y4Z216XY6Z12X2Y4Z60X2Y2Z244XY4Z6X2Y2Z36XY3Z18XY2Z2150XY2Z130XYZ2*X*Y^9*Z+36*X^3*Y^4*Z^3+12*X^2*Y^6*Z^2+6*X*Y^6*Z^2+48*X^2*Y^4*Z^2+16*X*Y^6*Z+12*X^2*Y^4*Z+60*X^2*Y^2*Z^2+44*X*Y^4*Z+6*X^2*Y^2*Z+36*X*Y^3*Z+18*X*Y^2*Z^2+150*X*Y^2*Z+130*X*Y*Z

Algorithm definition

The algorithm ⟨3×16×16:576⟩ could be constructed using the following decomposition:

⟨3×16×16:576⟩ = ⟨2×4×4:26⟩ + ⟨1×4×4:16⟩ + ⟨1×4×4:16⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨1×4×4:16⟩ + ⟨1×4×4:16⟩ + ⟨1×4×4:16⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨1×4×4:16⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨1×4×4:16⟩ + ⟨1×4×4:16⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨1×4×4:16⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨1×4×4:16⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩.

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_1_13A_1_14A_1_15A_1_16A_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_2_15A_2_16A_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_14A_3_15A_3_16B_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_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_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_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_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_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_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_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_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_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_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_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_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_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_15_1B_15_2B_15_3B_15_4B_15_5B_15_6B_15_7B_15_8B_15_9B_15_10B_15_11B_15_12B_15_13B_15_14B_15_15B_15_16B_16_1B_16_2B_16_3B_16_4B_16_5B_16_6B_16_7B_16_8B_16_9B_16_10B_16_11B_16_12B_16_13B_16_14B_16_15B_16_16C_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_3=TraceMulA_2_1A_2_2A_2_3A_2_4A_3_1A_3_2A_3_3A_3_4-B_5_1+B_1_1-B_1_9B_1_2-B_5_2-B_1_10B_1_3-B_5_3-B_1_11B_1_4-B_5_4-B_1_12B_2_1-B_6_1-B_2_9B_2_2-B_6_2-B_2_10B_2_3-B_6_3-B_2_11B_2_4-B_6_4-B_2_12B_3_1-B_7_1-B_3_9B_3_2-B_7_2-B_3_10B_3_3-B_7_3-B_3_11B_3_4-B_7_4-B_3_12B_4_1-B_8_1-B_4_9B_4_2-B_8_2-B_4_10B_4_3-B_8_3-B_4_11B_4_4-B_8_4-B_4_12C_1_2C_1_1+C_1_3C_2_2C_2_1+C_2_3C_3_2C_3_1+C_3_3C_4_2C_4_1+C_4_3+TraceMulA_1_5A_1_6A_1_7A_1_8B_13_1-B_1_5+B_5_5-B_13_5-B_5_9B_13_2-B_1_6+B_5_6-B_13_6-B_5_10B_13_3-B_1_7+B_5_7-B_13_7-B_5_11B_13_4-B_1_8+B_5_8-B_13_8-B_5_12B_14_1-B_2_5+B_6_5-B_14_5-B_6_9B_14_2-B_2_6+B_6_6-B_14_6-B_6_10B_14_3-B_2_7+B_6_7-B_14_7-B_6_11B_14_4-B_2_8+B_6_8-B_14_8-B_6_12B_15_1-B_3_5+B_7_5-B_15_5-B_7_9B_15_2-B_3_6+B_7_6-B_15_6-B_7_10B_15_3-B_3_7+B_7_7-B_15_7-B_7_11B_15_4-B_3_8+B_7_8-B_15_8-B_7_12B_16_1-B_4_5+B_8_5-B_16_5-B_8_9B_16_2-B_4_6+B_8_6-B_16_6-B_8_10B_16_3-B_4_7+B_8_7-B_16_7-B_8_11B_16_4-B_4_8+B_8_8-B_16_8-B_8_12C_5_1+C_5_3C_6_1+C_6_3C_7_1+C_7_3C_8_1+C_8_3+TraceMulA_1_9A_1_10A_1_11A_1_12-B_9_5-B_1_9+B_9_9-B_13_9-B_9_6-B_1_10+B_9_10-B_13_10-B_9_7-B_1_11+B_9_11-B_13_11-B_9_8-B_1_12+B_9_12-B_13_12-B_10_5-B_2_9+B_10_9-B_14_9-B_10_6-B_2_10+B_10_10-B_14_10-B_10_7-B_2_11+B_10_11-B_14_11-B_10_8-B_2_12+B_10_12-B_14_12-B_11_5-B_3_9+B_11_9-B_15_9-B_11_6-B_3_10+B_11_10-B_15_10-B_11_7-B_3_11+B_11_11-B_15_11-B_11_8-B_3_12+B_11_12-B_15_12-B_12_5-B_4_9+B_12_9-B_16_9-B_12_6-B_4_10+B_12_10-B_16_10-B_12_7-B_4_11+B_12_11-B_16_11-B_12_8-B_4_12+B_12_12-B_16_12C_9_1C_10_1C_11_1C_12_1+TraceMulA_2_9A_2_10A_2_11A_2_12A_3_9A_3_10A_3_11A_3_12-B_9_1-B_5_9+B_9_9-B_9_13-B_9_2-B_5_10+B_9_10-B_9_14-B_9_3-B_5_11+B_9_11-B_9_15-B_9_4-B_5_12+B_9_12-B_9_16-B_10_1-B_6_9+B_10_9-B_10_13-B_10_2-B_6_10+B_10_10-B_10_14-B_10_3-B_6_11+B_10_11-B_10_15-B_10_4-B_6_12+B_10_12-B_10_16-B_11_1-B_7_9+B_11_9-B_11_13-B_11_2-B_7_10+B_11_10-B_11_14-B_11_3-B_7_11+B_11_11-B_11_15-B_11_4-B_7_12+B_11_12-B_11_16-B_12_1-B_8_9+B_12_9-B_12_13-B_12_2-B_8_10+B_12_10-B_12_14-B_12_3-B_8_11+B_12_11-B_12_15-B_12_4-B_8_12+B_12_12-B_12_16C_9_2C_9_3C_10_2C_10_3C_11_2C_11_3C_12_2C_12_3+TraceMulA_2_13A_2_14A_2_15A_2_16A_3_13A_3_14A_3_15A_3_16B_13_1-B_13_9+B_1_13-B_5_13+B_13_13B_13_2-B_13_10+B_1_14-B_5_14+B_13_14B_13_3-B_13_11+B_1_15-B_5_15+B_13_15B_13_4-B_13_12+B_1_16-B_5_16+B_13_16B_14_1-B_14_9+B_2_13-B_6_13+B_14_13B_14_2-B_14_10+B_2_14-B_6_14+B_14_14B_14_3-B_14_11+B_2_15-B_6_15+B_14_15B_14_4-B_14_12+B_2_16-B_6_16+B_14_16B_15_1-B_15_9+B_3_13-B_7_13+B_15_13B_15_2-B_15_10+B_3_14-B_7_14+B_15_14B_15_3-B_15_11+B_3_15-B_7_15+B_15_15B_15_4-B_15_12+B_3_16-B_7_16+B_15_16B_16_1-B_16_9+B_4_13-B_8_13+B_16_13B_16_2-B_16_10+B_4_14-B_8_14+B_16_14B_16_3-B_16_11+B_4_15-B_8_15+B_16_15B_16_4-B_16_12+B_4_16-B_8_16+B_16_16C_13_2C_13_1+C_13_3C_14_2C_14_1+C_14_3C_15_2C_15_1+C_15_3C_16_2C_16_1+C_16_3+TraceMulA_1_1+A_1_5A_1_2+A_1_6A_1_3+A_1_7A_1_4+A_1_8B_13_1+B_1_5B_13_2+B_1_6B_13_3+B_1_7B_13_4+B_1_8B_14_1+B_2_5B_14_2+B_2_6B_14_3+B_2_7B_14_4+B_2_8B_15_1+B_3_5B_15_2+B_3_6B_15_3+B_3_7B_15_4+B_3_8B_16_1+B_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B_8_16C_5_2C_5_3C_6_2C_6_3C_7_2C_7_3C_8_2C_8_3TraceMulA_2_5A_2_9A_2_6A_2_10A_2_7A_2_11A_2_8A_2_12A_3_5A_3_9A_3_6A_3_10A_3_7A_3_11A_3_8A_3_12B_5_9B_5_10B_5_11B_5_12B_6_9B_6_10B_6_11B_6_12B_7_9B_7_10B_7_11B_7_12B_8_9B_8_10B_8_11B_8_12C_5_2C_9_2C_5_3C_9_3C_5_1C_9_1C_6_2C_10_2C_6_3C_10_3C_6_1C_10_1C_7_2C_11_2C_7_3C_11_3C_7_1C_11_1C_8_2C_12_2C_8_1C_12_1C_8_3C_12_3TraceMulA_2_5A_2_13A_2_6A_2_14A_2_7A_2_15A_2_8A_2_16A_3_5A_3_13A_3_6A_3_14A_3_7A_3_15A_3_8A_3_16B_1_13B_5_13B_1_14B_5_14B_1_15B_5_15B_1_16B_5_16B_2_13B_6_13B_2_14B_6_14B_2_15B_6_15B_2_16B_6_16B_3_13B_7_13B_3_14B_7_14B_3_15B_7_15B_3_16B_7_16B_4_13B_8_13B_4_14B_8_14B_4_15B_8_15B_4_16B_8_16C_5_2C_13_2C_5_3C_13_3C_6_2C_14_2C_6_3C_14_3C_15_2C_7_2C_15_3C_7_3C_8_2C_16_2C_8_3C_16_3TraceMulA_1_9A_1_13A_1_10A_1_14A_1_11A_1_15A_1_12A_1_16B_13_9B_13_10B_13_11B_13_12B_14_9B_14_10B_14_11B_14_12B_15_9B_15_10B_15_11B_15_12B_16_9B_16_10B_16_11B_16_12C_9_1C_13_1C_9_3C_13_3C_10_1C_14_1C_10_3C_14_3C_11_1C_15_1C_11_3C_15_3C_12_1C_16_1C_12_3C_16_3TraceMulA_1_13A_3_13A_1_14A_3_14A_1_15A_3_15A_1_16A_3_16B_13_1B_13_5B_1_13B_9_13B_13_13B_13_2B_13_6B_1_14B_9_14B_13_14B_13_3B_13_7B_1_15B_9_15B_13_15B_13_4B_13_8B_1_16B_9_16B_13_16B_14_1B_14_5B_2_13B_10_13B_14_13B_14_2B_14_6B_2_14B_10_14B_14_14B_14_3B_14_7B_2_15B_10_15B_14_15B_14_4B_14_8B_2_16B_10_16B_14_16B_15_1B_15_5B_3_13B_11_13B_15_13B_15_2B_15_6B_3_14B_11_14B_15_14B_15_3B_15_7B_3_15B_11_15B_15_15B_15_4B_15_8B_3_16B_11_16B_15_16B_16_1B_16_5B_4_13B_12_13B_16_13B_16_2B_16_6B_4_14B_12_14B_16_14B_16_3B_16_7B_4_15B_12_15B_16_15B_16_4B_16_8B_4_16B_12_16B_16_16C_13_1C_14_1C_15_1C_16_1TraceMulA_2_13A_2_14A_2_15A_2_16A_1_5A_3_13A_1_6A_3_14A_1_7A_3_15A_1_8A_3_16B_13_1B_13_5B_1_13B_5_13B_13_2B_13_6B_1_14B_5_14B_13_3B_13_7B_1_15B_5_15B_13_4B_13_8B_1_16B_5_16B_14_1B_14_5B_2_13B_6_13B_14_2B_14_6B_2_14B_6_14B_14_3B_14_7B_2_15B_6_15B_14_4B_14_8B_2_16B_6_16B_15_1B_15_5B_3_13B_7_13B_15_2B_15_6B_3_14B_7_14B_15_3B_15_7B_3_15B_7_15B_15_4B_15_8B_3_16B_7_16B_16_1B_16_5B_4_13B_8_13B_16_2B_16_6B_4_14B_8_14B_16_3B_16_7B_4_15B_8_15B_16_4B_16_8B_4_16B_8_16C_5_2C_13_1C_5_3C_6_2C_14_1C_6_3C_7_2C_15_1C_7_3C_8_2C_16_1C_8_3TraceMulA_2_1A_2_2A_2_3A_2_4A_1_1A_3_1A_1_9A_1_2A_3_2A_1_10A_1_3A_3_3A_1_11A_1_4A_3_4A_1_12B_9_1B_1_9B_9_2B_1_10B_9_3B_1_11B_9_4B_1_12B_10_1B_2_9B_10_2B_2_10B_10_3B_2_11B_10_4B_2_12B_11_1B_3_9B_11_2B_3_10B_11_3B_3_11B_11_4B_3_12B_12_1B_4_9B_12_2B_4_10B_12_3B_4_11B_12_4B_4_12C_1_2C_9_2C_1_1C_1_3C_9_3C_2_2C_10_2C_2_1C_2_3C_10_3C_3_2C_11_2C_3_1C_3_3C_11_3C_4_2C_12_2C_4_1C_4_3C_12_3TraceMulA_2_5A_2_9A_2_6A_2_10A_2_7A_2_11A_2_8A_2_12A_1_5A_3_5A_3_9A_1_6A_3_6A_3_10A_1_7A_3_7A_3_11A_1_8A_3_8A_3_12B_9_5B_5_9B_9_6B_5_10B_9_7B_5_11B_9_8B_5_12B_10_5B_6_9B_10_6B_6_10B_10_7B_6_11B_10_8B_6_12B_11_5B_7_9B_11_6B_7_10B_11_7B_7_11B_11_8B_7_12B_12_5B_8_9B_12_6B_8_10B_12_7B_8_11B_12_8B_8_12C_5_2C_5_1C_9_1C_5_3C_6_2C_6_1C_10_1C_6_3C_7_2C_11_1C_7_1C_7_3C_8_2C_8_1C_12_1C_8_3TraceMulA_2_13A_2_14A_2_15A_2_16A_1_9A_1_13A_3_13A_1_10A_1_14A_3_14A_1_11A_1_15A_3_15A_1_12A_1_16A_3_16B_13_9B_9_13B_13_10B_9_14B_13_11B_9_15B_13_12B_9_16B_14_9B_10_13B_14_10B_10_14B_14_11B_10_15B_14_12B_10_16B_15_9B_11_13B_15_10B_11_14B_15_11B_11_15B_15_12B_11_16B_16_9B_12_13B_16_10B_12_14B_16_11B_12_15B_16_12B_12_16C_9_2C_13_2C_13_1C_9_3C_13_3C_10_2C_14_2C_14_1C_10_3C_14_3C_11_2C_15_2C_15_1C_11_3C_15_3C_12_2C_16_2C_16_1C_12_3C_16_3TraceMulA_1_5A_3_5A_1_9A_3_9A_1_6A_3_6A_1_10A_3_10A_1_7A_3_7A_1_11A_3_11A_1_8A_3_8A_1_12A_3_12B_9_5B_9_6B_9_7B_9_8B_10_5B_10_6B_10_7B_10_8B_11_5B_11_6B_11_7B_11_8B_12_5B_12_6B_12_7B_12_8C_5_1C_9_1C_6_1C_10_1C_11_1C_7_1C_8_1C_12_1TraceMulA_2_13A_2_14A_2_15A_2_16A_1_1A_1_5A_1_13A_3_13A_1_2A_1_6A_1_14A_3_14A_1_3A_1_7A_1_15A_3_15A_1_4A_1_8A_1_16A_3_16B_13_1B_1_13B_13_2B_1_14B_13_3B_1_15B_13_4B_1_16B_14_1B_2_13B_14_2B_2_14B_14_3B_2_15B_14_4B_2_16B_15_1B_3_13B_15_2B_3_14B_15_3B_3_15B_15_4B_3_16B_16_1B_4_13B_16_2B_4_14B_16_3B_4_15B_16_4B_4_16C_1_2C_5_2C_13_2C_13_1C_1_3C_5_3C_13_3C_2_2C_6_2C_14_2C_14_1C_2_3C_6_3C_14_3C_3_2C_7_2C_15_2C_15_1C_3_3C_7_3C_15_3C_4_2C_8_2C_16_2C_16_1C_4_3C_8_3C_16_3TraceMulA_2_1A_2_5A_2_13A_2_2A_2_6A_2_14A_2_3A_2_7A_2_15A_2_4A_2_8A_2_16A_1_1A_3_1A_1_5A_3_5A_1_13A_3_13A_1_2A_3_2A_1_6A_3_6A_1_14A_3_14A_1_3A_3_3A_1_7A_3_7A_1_15A_3_15A_1_4A_3_4A_1_8A_3_8A_1_16A_3_16B_1_13B_1_14B_1_15B_1_16B_2_13B_2_14B_2_15B_2_16B_3_13B_3_14B_3_15B_3_16B_4_13B_4_14B_4_15B_4_16C_1_2C_5_2C_13_2C_1_3C_5_3C_13_3C_2_2C_6_2C_14_2C_2_3C_6_3C_14_3C_3_2C_7_2C_15_2C_3_3C_7_3C_15_3C_4_2C_8_2C_16_2C_4_3C_8_3C_16_3TraceMulA_1_1A_1_5A_1_13A_1_2A_1_6A_1_14A_1_3A_1_7A_1_15A_1_4A_1_8A_1_16B_13_1B_13_2B_13_3B_13_4B_14_1B_14_2B_14_3B_14_4B_15_1B_15_2B_15_3B_15_4B_16_1B_16_2B_16_3B_16_4C_1_1C_5_1C_13_1C_1_3C_5_3C_13_3C_2_1C_6_1C_14_1C_2_3C_6_3C_14_3C_3_1C_7_1C_15_1C_3_3C_7_3C_15_3C_4_1C_8_1C_16_1C_4_3C_8_3C_16_3TraceMulA_2_1A_2_9A_2_2A_2_10A_2_3A_2_11A_2_4A_2_12A_1_1A_3_1A_1_9A_3_9A_1_2A_3_2A_1_10A_3_10A_1_3A_3_3A_1_11A_3_11A_1_4A_3_4A_1_12A_3_12B_9_1B_9_2B_9_3B_9_4B_10_1B_10_2B_10_3B_10_4B_11_1B_11_2B_11_3B_11_4B_12_1B_12_2B_12_3B_12_4C_1_2C_9_2C_1_3C_9_3C_2_2C_10_2C_2_3C_10_3C_3_2C_11_2C_3_3C_11_3C_4_2C_12_2C_4_3C_12_3TraceMulA_2_9A_2_13A_2_10A_2_14A_2_11A_2_15A_2_12A_2_16A_1_9A_3_9A_1_13A_3_13A_1_10A_3_10A_1_14A_3_14A_1_11A_3_11A_1_15A_3_15A_1_12A_3_12A_1_16A_3_16B_9_13B_9_14B_9_15B_9_16B_10_13B_10_14B_10_15B_10_16B_11_13B_11_14B_11_15B_11_16B_12_13B_12_14B_12_15B_12_16C_9_2C_13_2C_9_3C_13_3C_10_2C_14_2C_10_3C_14_3C_11_2C_15_2C_11_3C_15_3C_12_2C_16_2C_12_3C_16_3Trace(Mul(Matrix(3, 16, [[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_1_15,A_1_16],[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_2_15,A_2_16],[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,A_3_15,A_3_16]]),Matrix(16, 16, [[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_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_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_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_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_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_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_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_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_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_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_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_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_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_15_1,B_15_2,B_15_3,B_15_4,B_15_5,B_15_6,B_15_7,B_15_8,B_15_9,B_15_10,B_15_11,B_15_12,B_15_13,B_15_14,B_15_15,B_15_16],[B_16_1,B_16_2,B_16_3,B_16_4,B_16_5,B_16_6,B_16_7,B_16_8,B_16_9,B_16_10,B_16_11,B_16_12,B_16_13,B_16_14,B_16_15,B_16_16]]),Matrix(16, 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]]))) = Trace(Mul(Matrix(2, 4, [[A_2_1,A_2_2,A_2_3,A_2_4],[A_3_1,A_3_2,A_3_3,A_3_4]]),Matrix(4, 4, [[-B_5_1+B_1_1-B_1_9,B_1_2-B_5_2-B_1_10,B_1_3-B_5_3-B_1_11,B_1_4-B_5_4-B_1_12],[B_2_1-B_6_1-B_2_9,B_2_2-B_6_2-B_2_10,B_2_3-B_6_3-B_2_11,B_2_4-B_6_4-B_2_12],[B_3_1-B_7_1-B_3_9,B_3_2-B_7_2-B_3_10,B_3_3-B_7_3-B_3_11,B_3_4-B_7_4-B_3_12],[B_4_1-B_8_1-B_4_9,B_4_2-B_8_2-B_4_10,B_4_3-B_8_3-B_4_11,B_4_4-B_8_4-B_4_12]]),Matrix(4, 2, [[C_1_2,C_1_1+C_1_3],[C_2_2,C_2_1+C_2_3],[C_3_2,C_3_1+C_3_3],[C_4_2,C_4_1+C_4_3]])))+Trace(Mul(Matrix(1, 4, [[A_1_5,A_1_6,A_1_7,A_1_8]]),Matrix(4, 4, [[B_13_1-B_1_5+B_5_5-B_13_5-B_5_9,B_13_2-B_1_6+B_5_6-B_13_6-B_5_10,B_13_3-B_1_7+B_5_7-B_13_7-B_5_11,B_13_4-B_1_8+B_5_8-B_13_8-B_5_12],[B_14_1-B_2_5+B_6_5-B_14_5-B_6_9,B_14_2-B_2_6+B_6_6-B_14_6-B_6_10,B_14_3-B_2_7+B_6_7-B_14_7-B_6_11,B_14_4-B_2_8+B_6_8-B_14_8-B_6_12],[B_15_1-B_3_5+B_7_5-B_15_5-B_7_9,B_15_2-B_3_6+B_7_6-B_15_6-B_7_10,B_15_3-B_3_7+B_7_7-B_15_7-B_7_11,B_15_4-B_3_8+B_7_8-B_15_8-B_7_12],[B_16_1-B_4_5+B_8_5-B_16_5-B_8_9,B_16_2-B_4_6+B_8_6-B_16_6-B_8_10,B_16_3-B_4_7+B_8_7-B_16_7-B_8_11,B_16_4-B_4_8+B_8_8-B_16_8-B_8_12]]),Matrix(4, 1, [[C_5_1+C_5_3],[C_6_1+C_6_3],[C_7_1+C_7_3],[C_8_1+C_8_3]])))+Trace(Mul(Matrix(1, 4, [[A_1_9,A_1_10,A_1_11,A_1_12]]),Matrix(4, 4, [[-B_9_5-B_1_9+B_9_9-B_13_9,-B_9_6-B_1_10+B_9_10-B_13_10,-B_9_7-B_1_11+B_9_11-B_13_11,-B_9_8-B_1_12+B_9_12-B_13_12],[-B_10_5-B_2_9+B_10_9-B_14_9,-B_10_6-B_2_10+B_10_10-B_14_10,-B_10_7-B_2_11+B_10_11-B_14_11,-B_10_8-B_2_12+B_10_12-B_14_12],[-B_11_5-B_3_9+B_11_9-B_15_9,-B_11_6-B_3_10+B_11_10-B_15_10,-B_11_7-B_3_11+B_11_11-B_15_11,-B_11_8-B_3_12+B_11_12-B_15_12],[-B_12_5-B_4_9+B_12_9-B_16_9,-B_12_6-B_4_10+B_12_10-B_16_10,-B_12_7-B_4_11+B_12_11-B_16_11,-B_12_8-B_4_12+B_12_12-B_16_12]]),Matrix(4, 1, [[C_9_1],[C_10_1],[C_11_1],[C_12_1]])))+Trace(Mul(Matrix(2, 4, [[A_2_9,A_2_10,A_2_11,A_2_12],[A_3_9,A_3_10,A_3_11,A_3_12]]),Matrix(4, 4, [[-B_9_1-B_5_9+B_9_9-B_9_13,-B_9_2-B_5_10+B_9_10-B_9_14,-B_9_3-B_5_11+B_9_11-B_9_15,-B_9_4-B_5_12+B_9_12-B_9_16],[-B_10_1-B_6_9+B_10_9-B_10_13,-B_10_2-B_6_10+B_10_10-B_10_14,-B_10_3-B_6_11+B_10_11-B_10_15,-B_10_4-B_6_12+B_10_12-B_10_16],[-B_11_1-B_7_9+B_11_9-B_11_13,-B_11_2-B_7_10+B_11_10-B_11_14,-B_11_3-B_7_11+B_11_11-B_11_15,-B_11_4-B_7_12+B_11_12-B_11_16],[-B_12_1-B_8_9+B_12_9-B_12_13,-B_12_2-B_8_10+B_12_10-B_12_14,-B_12_3-B_8_11+B_12_11-B_12_15,-B_12_4-B_8_12+B_12_12-B_12_16]]),Matrix(4, 2, [[C_9_2,C_9_3],[C_10_2,C_10_3],[C_11_2,C_11_3],[C_12_2,C_12_3]])))+Trace(Mul(Matrix(2, 4, [[A_2_13,A_2_14,A_2_15,A_2_16],[A_3_13,A_3_14,A_3_15,A_3_16]]),Matrix(4, 4, [[B_13_1-B_13_9+B_1_13-B_5_13+B_13_13,B_13_2-B_13_10+B_1_14-B_5_14+B_13_14,B_13_3-B_13_11+B_1_15-B_5_15+B_13_15,B_13_4-B_13_12+B_1_16-B_5_16+B_13_16],[B_14_1-B_14_9+B_2_13-B_6_13+B_14_13,B_14_2-B_14_10+B_2_14-B_6_14+B_14_14,B_14_3-B_14_11+B_2_15-B_6_15+B_14_15,B_14_4-B_14_12+B_2_16-B_6_16+B_14_16],[B_15_1-B_15_9+B_3_13-B_7_13+B_15_13,B_15_2-B_15_10+B_3_14-B_7_14+B_15_14,B_15_3-B_15_11+B_3_15-B_7_15+B_15_15,B_15_4-B_15_12+B_3_16-B_7_16+B_15_16],[B_16_1-B_16_9+B_4_13-B_8_13+B_16_13,B_16_2-B_16_10+B_4_14-B_8_14+B_16_14,B_16_3-B_16_11+B_4_15-B_8_15+B_16_15,B_16_4-B_16_12+B_4_16-B_8_16+B_16_16]]),Matrix(4, 2, [[C_13_2,C_13_1+C_13_3],[C_14_2,C_14_1+C_14_3],[C_15_2,C_15_1+C_15_3],[C_16_2,C_16_1+C_16_3]])))+Trace(Mul(Matrix(1, 4, [[A_1_1+A_1_5,A_1_2+A_1_6,A_1_3+A_1_7,A_1_4+A_1_8]]),Matrix(4, 4, [[B_13_1+B_1_5,B_13_2+B_1_6,B_13_3+B_1_7,B_13_4+B_1_8],[B_14_1+B_2_5,B_14_2+B_2_6,B_14_3+B_2_7,B_14_4+B_2_8],[B_15_1+B_3_5,B_15_2+B_3_6,B_15_3+B_3_7,B_15_4+B_3_8],[B_16_1+B_4_5,B_16_2+B_4_6,B_16_3+B_4_7,B_16_4+B_4_8]]),Matrix(4, 1, [[C_1_1+C_5_1],[C_2_1+C_6_1],[C_3_1+C_7_1],[C_4_1+C_8_1]])))+Trace(Mul(Matrix(1, 4, [[A_1_1+A_1_9,A_1_2+A_1_10,A_1_3+A_1_11,A_1_4+A_1_12]]),Matrix(4, 4, [[B_1_9,B_1_10,B_1_11,B_1_12],[B_2_9,B_2_10,B_2_11,B_2_12],[B_3_9,B_3_10,B_3_11,B_3_12],[B_4_9,B_4_10,B_4_11,B_4_12]]),Matrix(4, 1, [[C_1_3+C_9_3+C_1_1+C_9_1],[C_2_3+C_10_3+C_2_1+C_10_1],[C_3_3+C_11_3+C_3_1+C_11_1],[C_4_1+C_12_1+C_4_3+C_12_3]])))+Trace(Mul(Matrix(1, 4, [[A_1_1-A_3_1,A_1_2-A_3_2,A_1_3-A_3_3,A_1_4-A_3_4]]),Matrix(4, 4, [[B_1_1-B_9_1-B_1_5,B_1_2-B_9_2-B_1_6,B_1_3-B_9_3-B_1_7,B_1_4-B_9_4-B_1_8],[B_2_1-B_10_1-B_2_5,B_2_2-B_10_2-B_2_6,B_2_3-B_10_3-B_2_7,B_2_4-B_10_4-B_2_8],[B_3_1-B_11_1-B_3_5,B_3_2-B_11_2-B_3_6,B_3_3-B_11_3-B_3_7,B_3_4-B_11_4-B_3_8],[B_4_1-B_12_1-B_4_5,B_4_2-B_12_2-B_4_6,B_4_3-B_12_3-B_4_7,B_4_4-B_12_4-B_4_8]]),Matrix(4, 1, [[C_1_1],[C_2_1],[C_3_1],[C_4_1]])))+Trace(Mul(Matrix(2, 4, [[A_2_1,A_2_2,A_2_3,A_2_4],[A_3_1+A_1_5,A_3_2+A_1_6,A_3_3+A_1_7,A_3_4+A_1_8]]),Matrix(4, 4, [[B_5_1-B_1_5,B_5_2-B_1_6,B_5_3-B_1_7,B_5_4-B_1_8],[B_6_1-B_2_5,B_6_2-B_2_6,B_6_3-B_2_7,B_6_4-B_2_8],[B_7_1-B_3_5,B_7_2-B_3_6,B_7_3-B_3_7,B_7_4-B_3_8],[B_8_1-B_4_5,B_8_2-B_4_6,B_8_3-B_4_7,B_8_4-B_4_8]]),Matrix(4, 2, [[-C_5_2,C_1_1-C_5_3],[-C_6_2,C_2_1-C_6_3],[-C_7_2,C_3_1-C_7_3],[-C_8_2,C_4_1-C_8_3]])))+Trace(Mul(Matrix(2, 4, [[A_2_1+A_2_5,A_2_2+A_2_6,A_2_3+A_2_7,A_2_4+A_2_8],[A_3_1+A_3_5,A_3_2+A_3_6,A_3_3+A_3_7,A_3_4+A_3_8]]),Matrix(4, 4, [[B_5_1+B_1_13,B_5_2+B_1_14,B_5_3+B_1_15,B_5_4+B_1_16],[B_6_1+B_2_13,B_6_2+B_2_14,B_6_3+B_2_15,B_6_4+B_2_16],[B_7_1+B_3_13,B_7_2+B_3_14,B_7_3+B_3_15,B_7_4+B_3_16],[B_8_1+B_4_13,B_8_2+B_4_14,B_8_3+B_4_15,B_8_4+B_4_16]]),Matrix(4, 2, [[C_1_2+C_5_2,C_1_3+C_5_3],[C_2_2+C_6_2,C_2_3+C_6_3],[C_3_2+C_7_2,C_3_3+C_7_3],[C_4_2+C_8_2,C_4_3+C_8_3]])))+Trace(Mul(Matrix(1, 4, [[A_1_5+A_1_13,A_1_6+A_1_14,A_1_7+A_1_15,A_1_8+A_1_16]]),Matrix(4, 4, [[-B_13_1+B_13_5,-B_13_2+B_13_6,-B_13_3+B_13_7,-B_13_4+B_13_8],[-B_14_1+B_14_5,-B_14_2+B_14_6,-B_14_3+B_14_7,-B_14_4+B_14_8],[-B_15_1+B_15_5,-B_15_2+B_15_6,-B_15_3+B_15_7,-B_15_4+B_15_8],[-B_16_1+B_16_5,-B_16_2+B_16_6,B_16_7-B_16_3,-B_16_4+B_16_8]]),Matrix(4, 1, [[C_5_1+C_13_1],[C_6_1+C_14_1],[C_15_1+C_7_1],[C_8_1+C_16_1]])))+Trace(Mul(Matrix(2, 4, [[-A_2_5,-A_2_6,-A_2_7,-A_2_8],[A_1_5-A_3_5,A_1_6-A_3_6,A_1_7-A_3_7,A_1_8-A_3_8]]),Matrix(4, 4, [[B_5_1-B_5_5+B_9_5-B_1_13+B_5_13,B_5_2-B_5_6+B_9_6-B_1_14+B_5_14,B_5_3-B_5_7+B_9_7-B_1_15+B_5_15,B_5_4-B_5_8+B_9_8-B_1_16+B_5_16],[B_6_1-B_6_5+B_10_5-B_2_13+B_6_13,B_6_2-B_6_6+B_10_6-B_2_14+B_6_14,B_6_3-B_6_7+B_10_7-B_2_15+B_6_15,B_6_4-B_6_8+B_10_8-B_2_16+B_6_16],[B_7_1-B_7_5+B_11_5-B_3_13+B_7_13,B_7_2-B_7_6+B_11_6-B_3_14+B_7_14,B_7_3-B_7_7+B_11_7-B_3_15+B_7_15,B_7_4-B_7_8+B_11_8-B_3_16+B_7_16],[B_8_1-B_8_5+B_12_5-B_4_13+B_8_13,B_8_2-B_8_6+B_12_6-B_4_14+B_8_14,B_8_3-B_8_7+B_12_7-B_4_15+B_8_15,B_8_4-B_8_8+B_12_8-B_4_16+B_8_16]]),Matrix(4, 2, [[C_5_2,C_5_3],[C_6_2,C_6_3],[C_7_2,C_7_3],[C_8_2,C_8_3]])))+Trace(Mul(Matrix(2, 4, [[A_2_5+A_2_9,A_2_6+A_2_10,A_2_7+A_2_11,A_2_8+A_2_12],[A_3_5+A_3_9,A_3_6+A_3_10,A_3_7+A_3_11,A_3_8+A_3_12]]),Matrix(4, 4, [[B_5_9,B_5_10,B_5_11,B_5_12],[B_6_9,B_6_10,B_6_11,B_6_12],[B_7_9,B_7_10,B_7_11,B_7_12],[B_8_9,B_8_10,B_8_11,B_8_12]]),Matrix(4, 2, [[C_5_2+C_9_2,C_5_3+C_9_3+C_5_1+C_9_1],[C_6_2+C_10_2,C_6_3+C_10_3+C_6_1+C_10_1],[C_7_2+C_11_2,C_7_3+C_11_3+C_7_1+C_11_1],[C_8_2+C_12_2,C_8_1+C_12_1+C_8_3+C_12_3]])))+Trace(Mul(Matrix(2, 4, [[A_2_5+A_2_13,A_2_6+A_2_14,A_2_7+A_2_15,A_2_8+A_2_16],[A_3_5+A_3_13,A_3_6+A_3_14,A_3_7+A_3_15,A_3_8+A_3_16]]),Matrix(4, 4, [[-B_1_13+B_5_13,-B_1_14+B_5_14,-B_1_15+B_5_15,-B_1_16+B_5_16],[-B_2_13+B_6_13,-B_2_14+B_6_14,-B_2_15+B_6_15,-B_2_16+B_6_16],[-B_3_13+B_7_13,-B_3_14+B_7_14,-B_3_15+B_7_15,-B_3_16+B_7_16],[-B_4_13+B_8_13,-B_4_14+B_8_14,-B_4_15+B_8_15,-B_4_16+B_8_16]]),Matrix(4, 2, [[C_5_2+C_13_2,C_5_3+C_13_3],[C_6_2+C_14_2,C_6_3+C_14_3],[C_15_2+C_7_2,C_15_3+C_7_3],[C_8_2+C_16_2,C_8_3+C_16_3]])))+Trace(Mul(Matrix(1, 4, [[A_1_9+A_1_13,A_1_10+A_1_14,A_1_11+A_1_15,A_1_12+A_1_16]]),Matrix(4, 4, [[B_13_9,B_13_10,B_13_11,B_13_12],[B_14_9,B_14_10,B_14_11,B_14_12],[B_15_9,B_15_10,B_15_11,B_15_12],[B_16_9,B_16_10,B_16_11,B_16_12]]),Matrix(4, 1, [[C_9_1+C_13_1+C_9_3+C_13_3],[C_10_1+C_14_1+C_10_3+C_14_3],[C_11_1+C_15_1+C_11_3+C_15_3],[C_12_1+C_16_1+C_12_3+C_16_3]])))+Trace(Mul(Matrix(1, 4, [[A_1_13-A_3_13,A_1_14-A_3_14,A_1_15-A_3_15,A_1_16-A_3_16]]),Matrix(4, 4, [[B_13_1-B_13_5+B_1_13-B_9_13+B_13_13,B_13_2-B_13_6+B_1_14-B_9_14+B_13_14,B_13_3-B_13_7+B_1_15-B_9_15+B_13_15,B_13_4-B_13_8+B_1_16-B_9_16+B_13_16],[B_14_1-B_14_5+B_2_13-B_10_13+B_14_13,B_14_2-B_14_6+B_2_14-B_10_14+B_14_14,B_14_3-B_14_7+B_2_15-B_10_15+B_14_15,B_14_4-B_14_8+B_2_16-B_10_16+B_14_16],[B_15_1-B_15_5+B_3_13-B_11_13+B_15_13,B_15_2-B_15_6+B_3_14-B_11_14+B_15_14,B_15_3-B_15_7+B_3_15-B_11_15+B_15_15,B_15_4-B_15_8+B_3_16-B_11_16+B_15_16],[B_16_1-B_16_5+B_4_13-B_12_13+B_16_13,B_16_2-B_16_6+B_4_14-B_12_14+B_16_14,B_16_3-B_16_7+B_4_15-B_12_15+B_16_15,B_16_4-B_16_8+B_4_16-B_12_16+B_16_16]]),Matrix(4, 1, [[C_13_1],[C_14_1],[C_15_1],[C_16_1]])))+Trace(Mul(Matrix(2, 4, [[A_2_13,A_2_14,A_2_15,A_2_16],[A_1_5+A_3_13,A_1_6+A_3_14,A_1_7+A_3_15,A_1_8+A_3_16]]),Matrix(4, 4, [[B_13_1-B_13_5-B_1_13+B_5_13,B_13_2-B_13_6-B_1_14+B_5_14,B_13_3-B_13_7-B_1_15+B_5_15,B_13_4-B_13_8-B_1_16+B_5_16],[B_14_1-B_14_5-B_2_13+B_6_13,B_14_2-B_14_6-B_2_14+B_6_14,B_14_3-B_14_7-B_2_15+B_6_15,B_14_4-B_14_8-B_2_16+B_6_16],[B_15_1-B_15_5-B_3_13+B_7_13,B_15_2-B_15_6-B_3_14+B_7_14,B_15_3-B_15_7-B_3_15+B_7_15,B_15_4-B_15_8-B_3_16+B_7_16],[B_16_1-B_16_5-B_4_13+B_8_13,B_16_2-B_16_6-B_4_14+B_8_14,B_16_3-B_16_7-B_4_15+B_8_15,B_16_4-B_16_8-B_4_16+B_8_16]]),Matrix(4, 2, [[-C_5_2,C_13_1-C_5_3],[-C_6_2,C_14_1-C_6_3],[-C_7_2,C_15_1-C_7_3],[-C_8_2,C_16_1-C_8_3]])))+Trace(Mul(Matrix(2, 4, [[-A_2_1,-A_2_2,-A_2_3,-A_2_4],[A_1_1-A_3_1+A_1_9,A_1_2-A_3_2+A_1_10,A_1_3-A_3_3+A_1_11,A_1_4-A_3_4+A_1_12]]),Matrix(4, 4, [[B_9_1-B_1_9,B_9_2-B_1_10,B_9_3-B_1_11,B_9_4-B_1_12],[B_10_1-B_2_9,B_10_2-B_2_10,B_10_3-B_2_11,B_10_4-B_2_12],[B_11_1-B_3_9,B_11_2-B_3_10,B_11_3-B_3_11,B_11_4-B_3_12],[B_12_1-B_4_9,B_12_2-B_4_10,B_12_3-B_4_11,B_12_4-B_4_12]]),Matrix(4, 2, [[C_1_2+C_9_2,C_1_1+C_1_3+C_9_3],[C_2_2+C_10_2,C_2_1+C_2_3+C_10_3],[C_3_2+C_11_2,C_3_1+C_3_3+C_11_3],[C_4_2+C_12_2,C_4_1+C_4_3+C_12_3]])))+Trace(Mul(Matrix(2, 4, [[-A_2_5-A_2_9,-A_2_6-A_2_10,-A_2_7-A_2_11,-A_2_8-A_2_12],[A_1_5-A_3_5-A_3_9,A_1_6-A_3_6-A_3_10,A_1_7-A_3_7-A_3_11,A_1_8-A_3_8-A_3_12]]),Matrix(4, 4, [[-B_9_5+B_5_9,-B_9_6+B_5_10,-B_9_7+B_5_11,-B_9_8+B_5_12],[-B_10_5+B_6_9,-B_10_6+B_6_10,-B_10_7+B_6_11,-B_10_8+B_6_12],[-B_11_5+B_7_9,-B_11_6+B_7_10,-B_11_7+B_7_11,-B_11_8+B_7_12],[-B_12_5+B_8_9,-B_12_6+B_8_10,-B_12_7+B_8_11,-B_12_8+B_8_12]]),Matrix(4, 2, [[C_5_2,C_5_1+C_9_1+C_5_3],[C_6_2,C_6_1+C_10_1+C_6_3],[C_7_2,C_11_1+C_7_1+C_7_3],[C_8_2,C_8_1+C_12_1+C_8_3]])))+Trace(Mul(Matrix(2, 4, [[-A_2_13,-A_2_14,-A_2_15,-A_2_16],[A_1_9+A_1_13-A_3_13,A_1_10+A_1_14-A_3_14,A_1_11+A_1_15-A_3_15,A_1_12+A_1_16-A_3_16]]),Matrix(4, 4, [[-B_13_9+B_9_13,-B_13_10+B_9_14,-B_13_11+B_9_15,-B_13_12+B_9_16],[-B_14_9+B_10_13,-B_14_10+B_10_14,-B_14_11+B_10_15,-B_14_12+B_10_16],[-B_15_9+B_11_13,-B_15_10+B_11_14,-B_15_11+B_11_15,-B_15_12+B_11_16],[-B_16_9+B_12_13,-B_16_10+B_12_14,-B_16_11+B_12_15,-B_16_12+B_12_16]]),Matrix(4, 2, [[C_9_2+C_13_2,C_13_1+C_9_3+C_13_3],[C_10_2+C_14_2,C_14_1+C_10_3+C_14_3],[C_11_2+C_15_2,C_15_1+C_11_3+C_15_3],[C_12_2+C_16_2,C_16_1+C_12_3+C_16_3]])))+Trace(Mul(Matrix(1, 4, [[A_1_5-A_3_5+A_1_9-A_3_9,A_1_6-A_3_6+A_1_10-A_3_10,A_1_7-A_3_7+A_1_11-A_3_11,A_1_8-A_3_8+A_1_12-A_3_12]]),Matrix(4, 4, [[B_9_5,B_9_6,B_9_7,B_9_8],[B_10_5,B_10_6,B_10_7,B_10_8],[B_11_5,B_11_6,B_11_7,B_11_8],[B_12_5,B_12_6,B_12_7,B_12_8]]),Matrix(4, 1, [[C_5_1+C_9_1],[C_6_1+C_10_1],[C_11_1+C_7_1],[C_8_1+C_12_1]])))+Trace(Mul(Matrix(2, 4, [[A_2_13,A_2_14,A_2_15,A_2_16],[A_1_1+A_1_5-A_1_13+A_3_13,A_1_2+A_1_6-A_1_14+A_3_14,A_1_3+A_1_7-A_1_15+A_3_15,A_1_4+A_1_8-A_1_16+A_3_16]]),Matrix(4, 4, [[-B_13_1+B_1_13,-B_13_2+B_1_14,-B_13_3+B_1_15,-B_13_4+B_1_16],[-B_14_1+B_2_13,-B_14_2+B_2_14,-B_14_3+B_2_15,-B_14_4+B_2_16],[-B_15_1+B_3_13,-B_15_2+B_3_14,-B_15_3+B_3_15,-B_15_4+B_3_16],[-B_16_1+B_4_13,-B_16_2+B_4_14,-B_16_3+B_4_15,-B_16_4+B_4_16]]),Matrix(4, 2, [[-C_1_2-C_5_2+C_13_2,C_13_1-C_1_3-C_5_3+C_13_3],[-C_2_2-C_6_2+C_14_2,C_14_1-C_2_3-C_6_3+C_14_3],[-C_3_2-C_7_2+C_15_2,C_15_1-C_3_3-C_7_3+C_15_3],[-C_4_2-C_8_2+C_16_2,C_16_1-C_4_3-C_8_3+C_16_3]])))+Trace(Mul(Matrix(2, 4, [[-A_2_1-A_2_5+A_2_13,-A_2_2-A_2_6+A_2_14,-A_2_3-A_2_7+A_2_15,-A_2_4-A_2_8+A_2_16],[A_1_1-A_3_1+A_1_5-A_3_5-A_1_13+A_3_13,A_1_2-A_3_2+A_1_6-A_3_6-A_1_14+A_3_14,A_1_3-A_3_3+A_1_7-A_3_7-A_1_15+A_3_15,A_1_4-A_3_4+A_1_8-A_3_8-A_1_16+A_3_16]]),Matrix(4, 4, [[B_1_13,B_1_14,B_1_15,B_1_16],[B_2_13,B_2_14,B_2_15,B_2_16],[B_3_13,B_3_14,B_3_15,B_3_16],[B_4_13,B_4_14,B_4_15,B_4_16]]),Matrix(4, 2, [[C_1_2+C_5_2-C_13_2,C_1_3+C_5_3-C_13_3],[C_2_2+C_6_2-C_14_2,C_2_3+C_6_3-C_14_3],[C_3_2+C_7_2-C_15_2,C_3_3+C_7_3-C_15_3],[C_4_2+C_8_2-C_16_2,C_4_3+C_8_3-C_16_3]])))+Trace(Mul(Matrix(1, 4, [[-A_1_1-A_1_5+A_1_13,-A_1_2-A_1_6+A_1_14,-A_1_3-A_1_7+A_1_15,-A_1_4-A_1_8+A_1_16]]),Matrix(4, 4, [[B_13_1,B_13_2,B_13_3,B_13_4],[B_14_1,B_14_2,B_14_3,B_14_4],[B_15_1,B_15_2,B_15_3,B_15_4],[B_16_1,B_16_2,B_16_3,B_16_4]]),Matrix(4, 1, [[C_1_1+C_5_1-C_13_1+C_1_3+C_5_3-C_13_3],[C_2_1+C_6_1-C_14_1+C_2_3+C_6_3-C_14_3],[C_3_1+C_7_1-C_15_1+C_3_3+C_7_3-C_15_3],[C_4_1+C_8_1-C_16_1+C_4_3+C_8_3-C_16_3]])))+Trace(Mul(Matrix(2, 4, [[A_2_1+A_2_9,A_2_2+A_2_10,A_2_3+A_2_11,A_2_4+A_2_12],[-A_1_1+A_3_1-A_1_9+A_3_9,-A_1_2+A_3_2-A_1_10+A_3_10,-A_1_3+A_3_3-A_1_11+A_3_11,-A_1_4+A_3_4-A_1_12+A_3_12]]),Matrix(4, 4, [[B_9_1,B_9_2,B_9_3,B_9_4],[B_10_1,B_10_2,B_10_3,B_10_4],[B_11_1,B_11_2,B_11_3,B_11_4],[B_12_1,B_12_2,B_12_3,B_12_4]]),Matrix(4, 2, [[C_1_2+C_9_2,C_1_3+C_9_3],[C_2_2+C_10_2,C_2_3+C_10_3],[C_3_2+C_11_2,C_3_3+C_11_3],[C_4_2+C_12_2,C_4_3+C_12_3]])))+Trace(Mul(Matrix(2, 4, [[A_2_9+A_2_13,A_2_10+A_2_14,A_2_11+A_2_15,A_2_12+A_2_16],[-A_1_9+A_3_9-A_1_13+A_3_13,-A_1_10+A_3_10-A_1_14+A_3_14,-A_1_11+A_3_11-A_1_15+A_3_15,-A_1_12+A_3_12-A_1_16+A_3_16]]),Matrix(4, 4, [[B_9_13,B_9_14,B_9_15,B_9_16],[B_10_13,B_10_14,B_10_15,B_10_16],[B_11_13,B_11_14,B_11_15,B_11_16],[B_12_13,B_12_14,B_12_15,B_12_16]]),Matrix(4, 2, [[C_9_2+C_13_2,C_9_3+C_13_3],[C_10_2+C_14_2,C_10_3+C_14_3],[C_11_2+C_15_2,C_11_3+C_15_3],[C_12_2+C_16_2,C_12_3+C_16_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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