Description of fast matrix multiplication algorithm: ⟨5×9×15:474⟩

Algorithm type

9X4Y4Z4+12X4Y4Z3+6X3Y4Z3+6X2Y2Z5+60X2Y4Z2+6XY6Z+15X2Y2Z3+63X2Y2Z2+36XY4Z+9XY2Z3+12X2Y2Z+6XY3Z+21XY2Z2+6XYZ3+114XY2Z+18XYZ2+75XYZ9X4Y4Z412X4Y4Z36X3Y4Z36X2Y2Z560X2Y4Z26XY6Z15X2Y2Z363X2Y2Z236XY4Z9XY2Z312X2Y2Z6XY3Z21XY2Z26XYZ3114XY2Z18XYZ275XYZ9*X^4*Y^4*Z^4+12*X^4*Y^4*Z^3+6*X^3*Y^4*Z^3+6*X^2*Y^2*Z^5+60*X^2*Y^4*Z^2+6*X*Y^6*Z+15*X^2*Y^2*Z^3+63*X^2*Y^2*Z^2+36*X*Y^4*Z+9*X*Y^2*Z^3+12*X^2*Y^2*Z+6*X*Y^3*Z+21*X*Y^2*Z^2+6*X*Y*Z^3+114*X*Y^2*Z+18*X*Y*Z^2+75*X*Y*Z

Algorithm definition

The algorithm ⟨5×9×15:474⟩ could be constructed using the following decomposition:

⟨5×9×15:474⟩ = ⟨3×3×5:36⟩ + ⟨2×3×5:25⟩ + ⟨2×3×5:25⟩ + ⟨3×3×5:36⟩ + ⟨2×3×5:25⟩ + ⟨2×3×5:25⟩ + ⟨2×3×5:25⟩ + ⟨3×3×5:36⟩ + ⟨3×3×5:36⟩ + ⟨3×3×5:36⟩ + ⟨3×3×5:36⟩ + ⟨3×3×5:36⟩ + ⟨3×3×5:36⟩ + ⟨2×3×5:25⟩ + ⟨3×3×5:36⟩.

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_2_1A_2_2A_2_3A_2_4A_2_5A_2_6A_2_7A_2_8A_2_9A_3_1A_3_2A_3_3A_3_4A_3_5A_3_6A_3_7A_3_8A_3_9A_4_1A_4_2A_4_3A_4_4A_4_5A_4_6A_4_7A_4_8A_4_9A_5_1A_5_2A_5_3A_5_4A_5_5A_5_6A_5_7A_5_8A_5_9B_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_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_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_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_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_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_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_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_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_15C_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_5C_14_1C_14_2C_14_3C_14_4C_14_5C_15_1C_15_2C_15_3C_15_4C_15_5=TraceMulA_3_1A_3_2A_3_3A_4_1A_4_2A_4_3A_5_1A_5_2A_5_3B_1_1-B_4_1-B_1_11B_1_2-B_4_2-B_1_12B_1_3-B_4_3-B_1_13B_1_4-B_4_4-B_1_14B_1_5-B_4_5-B_1_15B_2_1-B_5_1-B_2_11B_2_2-B_5_2-B_2_12B_2_3-B_5_3-B_2_13B_2_4-B_5_4-B_2_14B_2_5-B_5_5-B_2_15-B_6_1+B_3_1-B_3_11B_3_2-B_6_2-B_3_12B_3_3-B_6_3-B_3_13B_3_4-B_6_4-B_3_14B_3_5-B_6_5-B_3_15C_1_3C_1_1+C_1_4C_1_2+C_1_5C_2_3C_2_1+C_2_4C_2_2+C_2_5C_3_3C_3_1+C_3_4C_3_2+C_3_5C_4_3C_4_1+C_4_4C_4_2+C_4_5C_5_3C_5_1+C_5_4C_5_2+C_5_5+TraceMulA_1_4A_1_5A_1_6A_2_4A_2_5A_2_6-B_1_6+B_4_6-B_4_11-B_1_7+B_4_7-B_4_12-B_1_8+B_4_8-B_4_13-B_1_9+B_4_9-B_4_14-B_1_10+B_4_10-B_4_15-B_2_6+B_5_6-B_5_11-B_2_7+B_5_7-B_5_12-B_2_8+B_5_8-B_5_13-B_2_9+B_5_9-B_5_14-B_2_10+B_5_10-B_5_15-B_3_6+B_6_6-B_6_11-B_3_7+B_6_7-B_6_12-B_3_8+B_6_8-B_6_13-B_3_9+B_6_9-B_6_14-B_3_10+B_6_10-B_6_15C_6_1+C_6_4C_6_2+C_6_5C_7_1+C_7_4C_7_2+C_7_5C_8_1+C_8_4C_8_2+C_8_5C_9_1+C_9_4C_9_2+C_9_5C_10_1+C_10_4C_10_2+C_10_5+TraceMulA_1_7A_1_8A_1_9A_2_7A_2_8A_2_9-B_7_6-B_1_11+B_7_11-B_7_7-B_1_12+B_7_12-B_7_8-B_1_13+B_7_13-B_7_9-B_1_14+B_7_14-B_7_10-B_1_15+B_7_15-B_8_6-B_2_11+B_8_11-B_8_7-B_2_12+B_8_12-B_8_8-B_2_13+B_8_13-B_8_9-B_2_14+B_8_14-B_8_10-B_2_15+B_8_15-B_9_6-B_3_11+B_9_11-B_9_7-B_3_12+B_9_12-B_9_8-B_3_13+B_9_13-B_9_9-B_3_14+B_9_14-B_9_10-B_3_15+B_9_15C_11_1C_11_2C_12_1C_12_2C_13_1C_13_2C_14_1C_14_2C_15_1C_15_2+TraceMulA_3_7A_3_8A_3_9A_4_7A_4_8A_4_9A_5_7A_5_8A_5_9-B_7_1-B_4_11+B_7_11-B_7_2-B_4_12+B_7_12-B_7_3-B_4_13+B_7_13-B_7_4-B_4_14+B_7_14-B_7_5-B_4_15+B_7_15-B_8_1-B_5_11+B_8_11-B_8_2-B_5_12+B_8_12-B_8_3-B_5_13+B_8_13-B_8_4-B_5_14+B_8_14-B_8_5-B_5_15+B_8_15-B_9_1-B_6_11+B_9_11-B_9_2-B_6_12+B_9_12-B_9_3-B_6_13+B_9_13-B_9_4-B_6_14+B_9_14-B_9_5-B_6_15+B_9_15C_11_3C_11_4C_11_5C_12_3C_12_4C_12_5C_13_3C_13_4C_13_5C_14_3C_14_4C_14_5C_15_3C_15_4C_15_5+TraceMulA_1_1+A_1_4A_1_2+A_1_5A_1_3+A_1_6A_2_1+A_2_4A_2_2+A_2_5A_2_3+A_2_6B_1_6B_1_7B_1_8B_1_9B_1_10B_2_6B_2_7B_2_8B_2_9B_2_10B_3_6B_3_7B_3_8B_3_9B_3_10C_1_1+C_6_1C_1_2+C_6_2C_2_1+C_7_1C_2_2+C_7_2C_3_1+C_8_1C_3_2+C_8_2C_4_1+C_9_1C_4_2+C_9_2C_5_1+C_10_1C_5_2+C_10_2+TraceMulA_1_1+A_1_7A_1_2+A_1_8A_1_3+A_1_9A_2_1+A_2_7A_2_2+A_2_8A_2_3+A_2_9B_1_11B_1_12B_1_13B_1_14B_1_15B_2_11B_2_12B_2_13B_2_14B_2_15B_3_11B_3_12B_3_13B_3_14B_3_15C_1_1+C_11_1+C_1_4+C_11_4C_1_2+C_11_2+C_1_5+C_11_5C_2_1+C_12_1+C_2_4+C_12_4C_2_2+C_12_2+C_2_5+C_12_5C_3_1+C_13_1+C_3_4+C_13_4C_3_2+C_13_2+C_3_5+C_13_5C_4_1+C_14_1+C_4_4+C_14_4C_4_2+C_14_2+C_4_5+C_14_5C_5_1+C_15_1+C_5_4+C_15_4C_5_2+C_15_2+C_5_5+C_15_5+TraceMulA_1_1-A_4_1A_1_2-A_4_2A_1_3-A_4_3A_2_1-A_5_1A_2_2-A_5_2A_2_3-A_5_3B_1_1-B_7_1-B_1_6B_1_2-B_7_2-B_1_7B_1_3-B_7_3-B_1_8B_1_4-B_7_4-B_1_9B_1_5-B_7_5-B_1_10B_2_1-B_8_1-B_2_6B_2_2-B_8_2-B_2_7B_2_3-B_8_3-B_2_8B_2_4-B_8_4-B_2_9B_2_5-B_8_5-B_2_10B_3_1-B_9_1-B_3_6B_3_2-B_9_2-B_3_7B_3_3-B_9_3-B_3_8B_3_4-B_9_4-B_3_9B_3_5-B_9_5-B_3_10C_1_1C_1_2C_2_1C_2_2C_3_1C_3_2C_4_1C_4_2C_5_1C_5_2+TraceMulA_3_1A_3_2A_3_3A_4_1+A_1_4A_4_2+A_1_5A_4_3+A_1_6A_5_1+A_2_4A_5_2+A_2_5A_5_3+A_2_6B_4_1-B_1_6B_4_2-B_1_7B_4_3-B_1_8B_4_4-B_1_9B_4_5-B_1_10B_5_1-B_2_6B_5_2-B_2_7B_5_3-B_2_8B_5_4-B_2_9B_5_5-B_2_10B_6_1-B_3_6B_6_2-B_3_7B_6_3-B_3_8B_6_4-B_3_9B_6_5-B_3_10-C_6_3C_1_1-C_6_4C_1_2-C_6_5-C_7_3C_2_1-C_7_4C_2_2-C_7_5-C_8_3C_3_1-C_8_4C_3_2-C_8_5-C_9_3C_4_1-C_9_4C_4_2-C_9_5-C_10_3C_5_1-C_10_4C_5_2-C_10_5+TraceMulA_3_1+A_3_4A_3_2+A_3_5A_3_3+A_3_6A_4_1+A_4_4A_4_2+A_4_5A_4_3+A_4_6A_5_1+A_5_4A_5_2+A_5_5A_5_3+A_5_6B_4_1B_4_2B_4_3B_4_4B_4_5B_5_1B_5_2B_5_3B_5_4B_5_5B_6_1B_6_2B_6_3B_6_4B_6_5C_1_3+C_6_3C_1_4+C_6_4C_1_5+C_6_5C_2_3+C_7_3C_2_4+C_7_4C_2_5+C_7_5C_3_3+C_8_3C_3_4+C_8_4C_3_5+C_8_5C_4_3+C_9_3C_4_4+C_9_4C_4_5+C_9_5C_5_3+C_10_3C_5_4+C_10_4C_5_5+C_10_5+TraceMul-A_3_4-A_3_5-A_3_6A_1_4-A_4_4A_1_5-A_4_5A_1_6-A_4_6A_2_4-A_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_7B_1_12B_7_12B_7_8B_1_13B_7_13B_7_9B_1_14B_7_14B_7_10B_1_15B_7_15B_8_6B_2_11B_8_11B_8_7B_2_12B_8_12B_8_8B_2_13B_8_13B_8_9B_2_14B_8_14B_8_10B_2_15B_8_15B_9_6B_3_11B_9_11B_9_7B_3_12B_9_12B_9_8B_3_13B_9_13B_9_9B_3_14B_9_14B_9_10B_3_15B_9_15C_11_1C_11_2C_12_1C_12_2C_13_1C_13_2C_14_1C_14_2C_15_1C_15_2TraceMulA_3_7A_3_8A_3_9A_4_7A_4_8A_4_9A_5_7A_5_8A_5_9B_7_1B_4_11B_7_11B_7_2B_4_12B_7_12B_7_3B_4_13B_7_13B_7_4B_4_14B_7_14B_7_5B_4_15B_7_15B_8_1B_5_11B_8_11B_8_2B_5_12B_8_12B_8_3B_5_13B_8_13B_8_4B_5_14B_8_14B_8_5B_5_15B_8_15B_9_1B_6_11B_9_11B_9_2B_6_12B_9_12B_9_3B_6_13B_9_13B_9_4B_6_14B_9_14B_9_5B_6_15B_9_15C_11_3C_11_4C_11_5C_12_3C_12_4C_12_5C_13_3C_13_4C_13_5C_14_3C_14_4C_14_5C_15_3C_15_4C_15_5TraceMulA_1_1A_1_4A_1_2A_1_5A_1_3A_1_6A_2_1A_2_4A_2_2A_2_5A_2_3A_2_6B_1_6B_1_7B_1_8B_1_9B_1_10B_2_6B_2_7B_2_8B_2_9B_2_10B_3_6B_3_7B_3_8B_3_9B_3_10C_1_1C_6_1C_1_2C_6_2C_2_1C_7_1C_2_2C_7_2C_3_1C_8_1C_3_2C_8_2C_4_1C_9_1C_4_2C_9_2C_5_1C_10_1C_5_2C_10_2TraceMulA_1_1A_1_7A_1_2A_1_8A_1_3A_1_9A_2_1A_2_7A_2_2A_2_8A_2_3A_2_9B_1_11B_1_12B_1_13B_1_14B_1_15B_2_11B_2_12B_2_13B_2_14B_2_15B_3_11B_3_12B_3_13B_3_14B_3_15C_1_1C_11_1C_1_4C_11_4C_1_2C_11_2C_1_5C_11_5C_2_1C_12_1C_2_4C_12_4C_2_2C_12_2C_2_5C_12_5C_3_1C_13_1C_3_4C_13_4C_3_2C_13_2C_3_5C_13_5C_4_1C_14_1C_4_4C_14_4C_4_2C_14_2C_4_5C_14_5C_5_1C_15_1C_5_4C_15_4C_5_2C_15_2C_5_5C_15_5TraceMulA_1_1A_4_1A_1_2A_4_2A_1_3A_4_3A_2_1A_5_1A_2_2A_5_2A_2_3A_5_3B_1_1B_7_1B_1_6B_1_2B_7_2B_1_7B_1_3B_7_3B_1_8B_1_4B_7_4B_1_9B_1_5B_7_5B_1_10B_2_1B_8_1B_2_6B_2_2B_8_2B_2_7B_2_3B_8_3B_2_8B_2_4B_8_4B_2_9B_2_5B_8_5B_2_10B_3_1B_9_1B_3_6B_3_2B_9_2B_3_7B_3_3B_9_3B_3_8B_3_4B_9_4B_3_9B_3_5B_9_5B_3_10C_1_1C_1_2C_2_1C_2_2C_3_1C_3_2C_4_1C_4_2C_5_1C_5_2TraceMulA_3_1A_3_2A_3_3A_4_1A_1_4A_4_2A_1_5A_4_3A_1_6A_5_1A_2_4A_5_2A_2_5A_5_3A_2_6B_4_1B_1_6B_4_2B_1_7B_4_3B_1_8B_4_4B_1_9B_4_5B_1_10B_5_1B_2_6B_5_2B_2_7B_5_3B_2_8B_5_4B_2_9B_5_5B_2_10B_6_1B_3_6B_6_2B_3_7B_6_3B_3_8B_6_4B_3_9B_6_5B_3_10C_6_3C_1_1C_6_4C_1_2C_6_5C_7_3C_2_1C_7_4C_2_2C_7_5C_8_3C_3_1C_8_4C_3_2C_8_5C_9_3C_4_1C_9_4C_4_2C_9_5C_10_3C_5_1C_10_4C_5_2C_10_5TraceMulA_3_1A_3_4A_3_2A_3_5A_3_3A_3_6A_4_1A_4_4A_4_2A_4_5A_4_3A_4_6A_5_1A_5_4A_5_2A_5_5A_5_3A_5_6B_4_1B_4_2B_4_3B_4_4B_4_5B_5_1B_5_2B_5_3B_5_4B_5_5B_6_1B_6_2B_6_3B_6_4B_6_5C_1_3C_6_3C_1_4C_6_4C_1_5C_6_5C_2_3C_7_3C_2_4C_7_4C_2_5C_7_5C_3_3C_8_3C_3_4C_8_4C_3_5C_8_5C_4_3C_9_3C_4_4C_9_4C_4_5C_9_5C_5_3C_10_3C_5_4C_10_4C_5_5C_10_5TraceMulA_3_4A_3_5A_3_6A_1_4A_4_4A_1_5A_4_5A_1_6A_4_6A_2_4A_5_4A_2_5A_5_5A_2_6A_5_6B_4_1B_4_6B_7_6B_4_2B_4_7B_7_7B_4_3B_4_8B_7_8B_4_4B_4_9B_7_9B_4_5B_4_10B_7_10B_5_1B_5_6B_8_6B_5_2B_5_7B_8_7B_5_3B_5_8B_8_8B_5_4B_5_9B_8_9B_5_5B_5_10B_8_10B_6_1B_6_6B_9_6B_6_2B_6_7B_9_7B_6_3B_6_8B_9_8B_6_4B_6_9B_9_9B_6_5B_6_10B_9_10C_6_3C_6_4C_6_5C_7_3C_7_4C_7_5C_8_3C_8_4C_8_5C_9_3C_9_4C_9_5C_10_3C_10_4C_10_5TraceMulA_3_4A_3_7A_3_5A_3_8A_3_6A_3_9A_4_4A_4_7A_4_5A_4_8A_4_6A_4_9A_5_4A_5_7A_5_5A_5_8A_5_6A_5_9B_4_11B_4_12B_4_13B_4_14B_4_15B_5_11B_5_12B_5_13B_5_14B_5_15B_6_11B_6_12B_6_13B_6_14B_6_15C_6_3C_11_3C_6_1C_11_1C_6_4C_11_4C_6_2C_11_2C_6_5C_11_5C_7_3C_12_3C_7_1C_12_1C_7_4C_12_4C_7_2C_12_2C_7_5C_12_5C_8_3C_13_3C_8_1C_13_1C_8_4C_13_4C_8_2C_13_2C_8_5C_13_5C_9_3C_14_3C_9_1C_14_1C_9_4C_14_4C_9_2C_14_2C_9_5C_14_5C_10_3C_15_3C_10_1C_15_1C_10_4C_15_4C_10_2C_15_2C_10_5C_15_5TraceMulA_3_1A_3_2A_3_3A_1_1A_4_1A_1_7A_1_2A_4_2A_1_8A_1_3A_4_3A_1_9A_2_1A_5_1A_2_7A_2_2A_5_2A_2_8A_2_3A_5_3A_2_9B_7_1B_1_11B_7_2B_1_12B_7_3B_1_13B_7_4B_1_14B_7_5B_1_15B_8_1B_2_11B_8_2B_2_12B_8_3B_2_13B_8_4B_2_14B_8_5B_2_15B_9_1B_3_11B_9_2B_3_12B_9_3B_3_13B_9_4B_3_14B_9_5B_3_15C_1_3C_11_3C_1_1C_1_4C_11_4C_1_2C_1_5C_11_5C_2_3C_12_3C_2_1C_2_4C_12_4C_2_2C_2_5C_12_5C_3_3C_13_3C_3_1C_3_4C_13_4C_3_2C_3_5C_13_5C_4_3C_14_3C_4_1C_4_4C_14_4C_4_2C_4_5C_14_5C_5_3C_15_3C_5_1C_5_4C_15_4C_5_2C_5_5C_15_5TraceMulA_3_4A_3_7A_3_5A_3_8A_3_6A_3_9A_1_4A_4_4A_4_7A_1_5A_4_5A_4_8A_1_6A_4_6A_4_9A_2_4A_5_4A_5_7A_2_5A_5_5A_5_8A_2_6A_5_6A_5_9B_7_6B_4_11B_7_7B_4_12B_7_8B_4_13B_7_9B_4_14B_7_10B_4_15B_8_6B_5_11B_8_7B_5_12B_8_8B_5_13B_8_9B_5_14B_8_10B_5_15B_9_6B_6_11B_9_7B_6_12B_9_8B_6_13B_9_9B_6_14B_9_10B_6_15C_6_3C_6_1C_11_1C_6_4C_6_2C_11_2C_6_5C_7_3C_7_1C_12_1C_7_4C_7_2C_12_2C_7_5C_8_3C_8_1C_13_1C_8_4C_8_2C_13_2C_8_5C_9_3C_9_1C_14_1C_9_4C_9_2C_14_2C_9_5C_10_3C_10_1C_15_1C_10_4C_10_2C_15_2C_10_5TraceMulA_1_4A_4_4A_1_7A_4_7A_1_5A_4_5A_1_8A_4_8A_1_6A_4_6A_1_9A_4_9A_2_4A_5_4A_2_7A_5_7A_2_5A_5_5A_2_8A_5_8A_2_6A_5_6A_2_9A_5_9B_7_6B_7_7B_7_8B_7_9B_7_10B_8_6B_8_7B_8_8B_8_9B_8_10B_9_6B_9_7B_9_8B_9_9B_9_10C_6_1C_11_1C_6_2C_11_2C_7_1C_12_1C_7_2C_12_2C_8_1C_13_1C_8_2C_13_2C_9_1C_14_1C_9_2C_14_2C_10_1C_15_1C_10_2C_15_2TraceMulA_3_1A_3_7A_3_2A_3_8A_3_3A_3_9A_1_1A_4_1A_1_7A_4_7A_1_2A_4_2A_1_8A_4_8A_1_3A_4_3A_1_9A_4_9A_2_1A_5_1A_2_7A_5_7A_2_2A_5_2A_2_8A_5_8A_2_3A_5_3A_2_9A_5_9B_7_1B_7_2B_7_3B_7_4B_7_5B_8_1B_8_2B_8_3B_8_4B_8_5B_9_1B_9_2B_9_3B_9_4B_9_5C_1_3C_11_3C_1_4C_11_4C_1_5C_11_5C_2_3C_12_3C_2_4C_12_4C_2_5C_12_5C_3_3C_13_3C_3_4C_13_4C_3_5C_13_5C_4_3C_14_3C_4_4C_14_4C_4_5C_14_5C_5_3C_15_3C_5_4C_15_4C_5_5C_15_5Trace(Mul(Matrix(5, 9, [[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_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_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_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_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]]),Matrix(9, 15, [[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_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_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_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_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_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_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_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_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]]),Matrix(15, 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],[C_14_1,C_14_2,C_14_3,C_14_4,C_14_5],[C_15_1,C_15_2,C_15_3,C_15_4,C_15_5]]))) = Trace(Mul(Matrix(3, 3, [[A_3_1,A_3_2,A_3_3],[A_4_1,A_4_2,A_4_3],[A_5_1,A_5_2,A_5_3]]),Matrix(3, 5, [[B_1_1-B_4_1-B_1_11,B_1_2-B_4_2-B_1_12,B_1_3-B_4_3-B_1_13,B_1_4-B_4_4-B_1_14,B_1_5-B_4_5-B_1_15],[B_2_1-B_5_1-B_2_11,B_2_2-B_5_2-B_2_12,B_2_3-B_5_3-B_2_13,B_2_4-B_5_4-B_2_14,B_2_5-B_5_5-B_2_15],[-B_6_1+B_3_1-B_3_11,B_3_2-B_6_2-B_3_12,B_3_3-B_6_3-B_3_13,B_3_4-B_6_4-B_3_14,B_3_5-B_6_5-B_3_15]]),Matrix(5, 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]])))+Trace(Mul(Matrix(2, 3, [[A_1_4,A_1_5,A_1_6],[A_2_4,A_2_5,A_2_6]]),Matrix(3, 5, [[-B_1_6+B_4_6-B_4_11,-B_1_7+B_4_7-B_4_12,-B_1_8+B_4_8-B_4_13,-B_1_9+B_4_9-B_4_14,-B_1_10+B_4_10-B_4_15],[-B_2_6+B_5_6-B_5_11,-B_2_7+B_5_7-B_5_12,-B_2_8+B_5_8-B_5_13,-B_2_9+B_5_9-B_5_14,-B_2_10+B_5_10-B_5_15],[-B_3_6+B_6_6-B_6_11,-B_3_7+B_6_7-B_6_12,-B_3_8+B_6_8-B_6_13,-B_3_9+B_6_9-B_6_14,-B_3_10+B_6_10-B_6_15]]),Matrix(5, 2, [[C_6_1+C_6_4,C_6_2+C_6_5],[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]])))+Trace(Mul(Matrix(2, 3, [[A_1_7,A_1_8,A_1_9],[A_2_7,A_2_8,A_2_9]]),Matrix(3, 5, [[-B_7_6-B_1_11+B_7_11,-B_7_7-B_1_12+B_7_12,-B_7_8-B_1_13+B_7_13,-B_7_9-B_1_14+B_7_14,-B_7_10-B_1_15+B_7_15],[-B_8_6-B_2_11+B_8_11,-B_8_7-B_2_12+B_8_12,-B_8_8-B_2_13+B_8_13,-B_8_9-B_2_14+B_8_14,-B_8_10-B_2_15+B_8_15],[-B_9_6-B_3_11+B_9_11,-B_9_7-B_3_12+B_9_12,-B_9_8-B_3_13+B_9_13,-B_9_9-B_3_14+B_9_14,-B_9_10-B_3_15+B_9_15]]),Matrix(5, 2, [[C_11_1,C_11_2],[C_12_1,C_12_2],[C_13_1,C_13_2],[C_14_1,C_14_2],[C_15_1,C_15_2]])))+Trace(Mul(Matrix(3, 3, [[A_3_7,A_3_8,A_3_9],[A_4_7,A_4_8,A_4_9],[A_5_7,A_5_8,A_5_9]]),Matrix(3, 5, [[-B_7_1-B_4_11+B_7_11,-B_7_2-B_4_12+B_7_12,-B_7_3-B_4_13+B_7_13,-B_7_4-B_4_14+B_7_14,-B_7_5-B_4_15+B_7_15],[-B_8_1-B_5_11+B_8_11,-B_8_2-B_5_12+B_8_12,-B_8_3-B_5_13+B_8_13,-B_8_4-B_5_14+B_8_14,-B_8_5-B_5_15+B_8_15],[-B_9_1-B_6_11+B_9_11,-B_9_2-B_6_12+B_9_12,-B_9_3-B_6_13+B_9_13,-B_9_4-B_6_14+B_9_14,-B_9_5-B_6_15+B_9_15]]),Matrix(5, 3, [[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],[C_14_3,C_14_4,C_14_5],[C_15_3,C_15_4,C_15_5]])))+Trace(Mul(Matrix(2, 3, [[A_1_1+A_1_4,A_1_2+A_1_5,A_1_3+A_1_6],[A_2_1+A_2_4,A_2_2+A_2_5,A_2_3+A_2_6]]),Matrix(3, 5, [[B_1_6,B_1_7,B_1_8,B_1_9,B_1_10],[B_2_6,B_2_7,B_2_8,B_2_9,B_2_10],[B_3_6,B_3_7,B_3_8,B_3_9,B_3_10]]),Matrix(5, 2, [[C_1_1+C_6_1,C_1_2+C_6_2],[C_2_1+C_7_1,C_2_2+C_7_2],[C_3_1+C_8_1,C_3_2+C_8_2],[C_4_1+C_9_1,C_4_2+C_9_2],[C_5_1+C_10_1,C_5_2+C_10_2]])))+Trace(Mul(Matrix(2, 3, [[A_1_1+A_1_7,A_1_2+A_1_8,A_1_3+A_1_9],[A_2_1+A_2_7,A_2_2+A_2_8,A_2_3+A_2_9]]),Matrix(3, 5, [[B_1_11,B_1_12,B_1_13,B_1_14,B_1_15],[B_2_11,B_2_12,B_2_13,B_2_14,B_2_15],[B_3_11,B_3_12,B_3_13,B_3_14,B_3_15]]),Matrix(5, 2, [[C_1_1+C_11_1+C_1_4+C_11_4,C_1_2+C_11_2+C_1_5+C_11_5],[C_2_1+C_12_1+C_2_4+C_12_4,C_2_2+C_12_2+C_2_5+C_12_5],[C_3_1+C_13_1+C_3_4+C_13_4,C_3_2+C_13_2+C_3_5+C_13_5],[C_4_1+C_14_1+C_4_4+C_14_4,C_4_2+C_14_2+C_4_5+C_14_5],[C_5_1+C_15_1+C_5_4+C_15_4,C_5_2+C_15_2+C_5_5+C_15_5]])))+Trace(Mul(Matrix(2, 3, [[A_1_1-A_4_1,A_1_2-A_4_2,A_1_3-A_4_3],[A_2_1-A_5_1,A_2_2-A_5_2,A_2_3-A_5_3]]),Matrix(3, 5, [[B_1_1-B_7_1-B_1_6,B_1_2-B_7_2-B_1_7,B_1_3-B_7_3-B_1_8,B_1_4-B_7_4-B_1_9,B_1_5-B_7_5-B_1_10],[B_2_1-B_8_1-B_2_6,B_2_2-B_8_2-B_2_7,B_2_3-B_8_3-B_2_8,B_2_4-B_8_4-B_2_9,B_2_5-B_8_5-B_2_10],[B_3_1-B_9_1-B_3_6,B_3_2-B_9_2-B_3_7,B_3_3-B_9_3-B_3_8,B_3_4-B_9_4-B_3_9,B_3_5-B_9_5-B_3_10]]),Matrix(5, 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]])))+Trace(Mul(Matrix(3, 3, [[A_3_1,A_3_2,A_3_3],[A_4_1+A_1_4,A_4_2+A_1_5,A_4_3+A_1_6],[A_5_1+A_2_4,A_5_2+A_2_5,A_5_3+A_2_6]]),Matrix(3, 5, [[B_4_1-B_1_6,B_4_2-B_1_7,B_4_3-B_1_8,B_4_4-B_1_9,B_4_5-B_1_10],[B_5_1-B_2_6,B_5_2-B_2_7,B_5_3-B_2_8,B_5_4-B_2_9,B_5_5-B_2_10],[B_6_1-B_3_6,B_6_2-B_3_7,B_6_3-B_3_8,B_6_4-B_3_9,B_6_5-B_3_10]]),Matrix(5, 3, [[-C_6_3,C_1_1-C_6_4,C_1_2-C_6_5],[-C_7_3,C_2_1-C_7_4,C_2_2-C_7_5],[-C_8_3,C_3_1-C_8_4,C_3_2-C_8_5],[-C_9_3,C_4_1-C_9_4,C_4_2-C_9_5],[-C_10_3,C_5_1-C_10_4,C_5_2-C_10_5]])))+Trace(Mul(Matrix(3, 3, [[A_3_1+A_3_4,A_3_2+A_3_5,A_3_3+A_3_6],[A_4_1+A_4_4,A_4_2+A_4_5,A_4_3+A_4_6],[A_5_1+A_5_4,A_5_2+A_5_5,A_5_3+A_5_6]]),Matrix(3, 5, [[B_4_1,B_4_2,B_4_3,B_4_4,B_4_5],[B_5_1,B_5_2,B_5_3,B_5_4,B_5_5],[B_6_1,B_6_2,B_6_3,B_6_4,B_6_5]]),Matrix(5, 3, [[C_1_3+C_6_3,C_1_4+C_6_4,C_1_5+C_6_5],[C_2_3+C_7_3,C_2_4+C_7_4,C_2_5+C_7_5],[C_3_3+C_8_3,C_3_4+C_8_4,C_3_5+C_8_5],[C_4_3+C_9_3,C_4_4+C_9_4,C_4_5+C_9_5],[C_5_3+C_10_3,C_5_4+C_10_4,C_5_5+C_10_5]])))+Trace(Mul(Matrix(3, 3, [[-A_3_4,-A_3_5,-A_3_6],[A_1_4-A_4_4,A_1_5-A_4_5,A_1_6-A_4_6],[A_2_4-A_5_4,A_2_5-A_5_5,A_2_6-A_5_6]]),Matrix(3, 5, [[B_4_1-B_4_6+B_7_6,B_4_2-B_4_7+B_7_7,B_4_3-B_4_8+B_7_8,B_4_4-B_4_9+B_7_9,B_4_5-B_4_10+B_7_10],[B_5_1-B_5_6+B_8_6,B_5_2-B_5_7+B_8_7,B_5_3-B_5_8+B_8_8,B_5_4-B_5_9+B_8_9,B_5_5-B_5_10+B_8_10],[B_6_1-B_6_6+B_9_6,B_6_2-B_6_7+B_9_7,B_6_3-B_6_8+B_9_8,B_6_4-B_6_9+B_9_9,B_6_5-B_6_10+B_9_10]]),Matrix(5, 3, [[C_6_3,C_6_4,C_6_5],[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]])))+Trace(Mul(Matrix(3, 3, [[A_3_4+A_3_7,A_3_5+A_3_8,A_3_6+A_3_9],[A_4_4+A_4_7,A_4_5+A_4_8,A_4_6+A_4_9],[A_5_4+A_5_7,A_5_5+A_5_8,A_5_6+A_5_9]]),Matrix(3, 5, [[B_4_11,B_4_12,B_4_13,B_4_14,B_4_15],[B_5_11,B_5_12,B_5_13,B_5_14,B_5_15],[B_6_11,B_6_12,B_6_13,B_6_14,B_6_15]]),Matrix(5, 3, [[C_6_3+C_11_3,C_6_1+C_11_1+C_6_4+C_11_4,C_6_2+C_11_2+C_6_5+C_11_5],[C_7_3+C_12_3,C_7_1+C_12_1+C_7_4+C_12_4,C_7_2+C_12_2+C_7_5+C_12_5],[C_8_3+C_13_3,C_8_1+C_13_1+C_8_4+C_13_4,C_8_2+C_13_2+C_8_5+C_13_5],[C_9_3+C_14_3,C_9_1+C_14_1+C_9_4+C_14_4,C_9_2+C_14_2+C_9_5+C_14_5],[C_10_3+C_15_3,C_10_1+C_15_1+C_10_4+C_15_4,C_10_2+C_15_2+C_10_5+C_15_5]])))+Trace(Mul(Matrix(3, 3, [[-A_3_1,-A_3_2,-A_3_3],[A_1_1-A_4_1+A_1_7,A_1_2-A_4_2+A_1_8,A_1_3-A_4_3+A_1_9],[A_2_1-A_5_1+A_2_7,A_2_2-A_5_2+A_2_8,A_2_3-A_5_3+A_2_9]]),Matrix(3, 5, [[B_7_1-B_1_11,B_7_2-B_1_12,B_7_3-B_1_13,B_7_4-B_1_14,B_7_5-B_1_15],[B_8_1-B_2_11,B_8_2-B_2_12,B_8_3-B_2_13,B_8_4-B_2_14,B_8_5-B_2_15],[B_9_1-B_3_11,B_9_2-B_3_12,B_9_3-B_3_13,B_9_4-B_3_14,B_9_5-B_3_15]]),Matrix(5, 3, [[C_1_3+C_11_3,C_1_1+C_1_4+C_11_4,C_1_2+C_1_5+C_11_5],[C_2_3+C_12_3,C_2_1+C_2_4+C_12_4,C_2_2+C_2_5+C_12_5],[C_3_3+C_13_3,C_3_1+C_3_4+C_13_4,C_3_2+C_3_5+C_13_5],[C_4_3+C_14_3,C_4_1+C_4_4+C_14_4,C_4_2+C_4_5+C_14_5],[C_5_3+C_15_3,C_5_1+C_5_4+C_15_4,C_5_2+C_5_5+C_15_5]])))+Trace(Mul(Matrix(3, 3, [[-A_3_4-A_3_7,-A_3_5-A_3_8,-A_3_6-A_3_9],[A_1_4-A_4_4-A_4_7,A_1_5-A_4_5-A_4_8,A_1_6-A_4_6-A_4_9],[A_2_4-A_5_4-A_5_7,A_2_5-A_5_5-A_5_8,A_2_6-A_5_6-A_5_9]]),Matrix(3, 5, [[-B_7_6+B_4_11,-B_7_7+B_4_12,-B_7_8+B_4_13,-B_7_9+B_4_14,-B_7_10+B_4_15],[-B_8_6+B_5_11,-B_8_7+B_5_12,-B_8_8+B_5_13,-B_8_9+B_5_14,-B_8_10+B_5_15],[-B_9_6+B_6_11,-B_9_7+B_6_12,-B_9_8+B_6_13,-B_9_9+B_6_14,-B_9_10+B_6_15]]),Matrix(5, 3, [[C_6_3,C_6_1+C_11_1+C_6_4,C_6_2+C_11_2+C_6_5],[C_7_3,C_7_1+C_12_1+C_7_4,C_7_2+C_12_2+C_7_5],[C_8_3,C_8_1+C_13_1+C_8_4,C_8_2+C_13_2+C_8_5],[C_9_3,C_9_1+C_14_1+C_9_4,C_9_2+C_14_2+C_9_5],[C_10_3,C_10_1+C_15_1+C_10_4,C_10_2+C_15_2+C_10_5]])))+Trace(Mul(Matrix(2, 3, [[A_1_4-A_4_4+A_1_7-A_4_7,A_1_5-A_4_5+A_1_8-A_4_8,A_1_6-A_4_6+A_1_9-A_4_9],[A_2_4-A_5_4+A_2_7-A_5_7,A_2_5-A_5_5+A_2_8-A_5_8,A_2_6-A_5_6+A_2_9-A_5_9]]),Matrix(3, 5, [[B_7_6,B_7_7,B_7_8,B_7_9,B_7_10],[B_8_6,B_8_7,B_8_8,B_8_9,B_8_10],[B_9_6,B_9_7,B_9_8,B_9_9,B_9_10]]),Matrix(5, 2, [[C_6_1+C_11_1,C_6_2+C_11_2],[C_7_1+C_12_1,C_7_2+C_12_2],[C_8_1+C_13_1,C_8_2+C_13_2],[C_9_1+C_14_1,C_9_2+C_14_2],[C_10_1+C_15_1,C_10_2+C_15_2]])))+Trace(Mul(Matrix(3, 3, [[A_3_1+A_3_7,A_3_2+A_3_8,A_3_3+A_3_9],[-A_1_1+A_4_1-A_1_7+A_4_7,-A_1_2+A_4_2-A_1_8+A_4_8,-A_1_3+A_4_3-A_1_9+A_4_9],[-A_2_1+A_5_1-A_2_7+A_5_7,-A_2_2+A_5_2-A_2_8+A_5_8,-A_2_3+A_5_3-A_2_9+A_5_9]]),Matrix(3, 5, [[B_7_1,B_7_2,B_7_3,B_7_4,B_7_5],[B_8_1,B_8_2,B_8_3,B_8_4,B_8_5],[B_9_1,B_9_2,B_9_3,B_9_4,B_9_5]]),Matrix(5, 3, [[C_1_3+C_11_3,C_1_4+C_11_4,C_1_5+C_11_5],[C_2_3+C_12_3,C_2_4+C_12_4,C_2_5+C_12_5],[C_3_3+C_13_3,C_3_4+C_13_4,C_3_5+C_13_5],[C_4_3+C_14_3,C_4_4+C_14_4,C_4_5+C_14_5],[C_5_3+C_15_3,C_5_4+C_15_4,C_5_5+C_15_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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