Description of fast matrix multiplication algorithm: ⟨5×9×13:417⟩

Algorithm type

4X4Y4Z4+6X4Y4Z3+6X4Y3Z3+7X3Y4Z3+X2Y6Z2+X5Y2Z2+2X3Y3Z3+X2Y6Z+X2Y4Z3+X2Y2Z5+36X2Y4Z2+2X2Y2Z4+4X2YZ5+13XY6Z+XY4Z3+XY2Z5+X2Y4Z+10X2Y3Z2+11X2Y2Z3+XY5Z+4XY4Z2+2X3Y2Z+4X2Y3Z+55X2Y2Z2+14XY4Z+7XY2Z3+3X3YZ+9X2Y2Z+21XY3Z+22XY2Z2+8XYZ3+87XY2Z+12XYZ2+59XYZ4X4Y4Z46X4Y4Z36X4Y3Z37X3Y4Z3X2Y6Z2X5Y2Z22X3Y3Z3X2Y6ZX2Y4Z3X2Y2Z536X2Y4Z22X2Y2Z44X2YZ513XY6ZXY4Z3XY2Z5X2Y4Z10X2Y3Z211X2Y2Z3XY5Z4XY4Z22X3Y2Z4X2Y3Z55X2Y2Z214XY4Z7XY2Z33X3YZ9X2Y2Z21XY3Z22XY2Z28XYZ387XY2Z12XYZ259XYZ4*X^4*Y^4*Z^4+6*X^4*Y^4*Z^3+6*X^4*Y^3*Z^3+7*X^3*Y^4*Z^3+X^2*Y^6*Z^2+X^5*Y^2*Z^2+2*X^3*Y^3*Z^3+X^2*Y^6*Z+X^2*Y^4*Z^3+X^2*Y^2*Z^5+36*X^2*Y^4*Z^2+2*X^2*Y^2*Z^4+4*X^2*Y*Z^5+13*X*Y^6*Z+X*Y^4*Z^3+X*Y^2*Z^5+X^2*Y^4*Z+10*X^2*Y^3*Z^2+11*X^2*Y^2*Z^3+X*Y^5*Z+4*X*Y^4*Z^2+2*X^3*Y^2*Z+4*X^2*Y^3*Z+55*X^2*Y^2*Z^2+14*X*Y^4*Z+7*X*Y^2*Z^3+3*X^3*Y*Z+9*X^2*Y^2*Z+21*X*Y^3*Z+22*X*Y^2*Z^2+8*X*Y*Z^3+87*X*Y^2*Z+12*X*Y*Z^2+59*X*Y*Z

Algorithm definition

The algorithm ⟨5×9×13:417⟩ could be constructed using the following decomposition:

⟨5×9×13:417⟩ = ⟨3×3×4:29⟩ + ⟨3×3×5:36⟩ + ⟨2×3×5:25⟩ + ⟨2×3×5:25⟩ + ⟨2×3×4:20⟩ + ⟨2×3×4:20⟩ + ⟨3×3×4:29⟩ + ⟨3×3×5:36⟩ + ⟨3×3×4:29⟩ + ⟨3×3×4:29⟩ + ⟨2×3×4:20⟩ + ⟨3×3×4:29⟩ + ⟨2×3×5:25⟩ + ⟨3×3×4:29⟩ + ⟨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_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_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_4_7A_4_8A_4_9A_5_7A_5_8A_5_9B_1_4+B_7_4+B_7_5B_1_1+B_7_1+B_7_6B_1_2+B_7_2+B_7_7B_1_3+B_7_3+B_7_8B_2_4+B_8_4+B_8_5B_2_1+B_8_1+B_8_6B_2_2+B_8_2+B_8_7B_2_3+B_8_3+B_8_8B_3_4+B_9_4+B_9_5B_3_1+B_9_1+B_9_6B_3_2+B_9_2+B_9_7B_3_3+B_9_3+B_9_8C_4_3C_4_1+C_4_4C_4_2+C_4_5C_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_5+TraceMul-A_3_7-A_3_8-A_3_9A_1_1-A_4_7A_1_2-A_4_8A_1_3-A_4_9A_2_1-A_5_7A_2_2-A_5_8A_2_3-A_5_9B_7_9B_1_4+B_7_10B_1_1+B_7_11B_1_2+B_7_12B_1_3+B_7_13B_8_9B_2_4+B_8_10B_2_1+B_8_11B_2_2+B_8_12B_2_3+B_8_13B_9_9B_3_4+B_9_10B_3_1+B_9_11B_3_2+B_9_12B_3_3+B_9_13-C_9_3-C_9_4-C_9_5-C_10_3C_4_1-C_10_4C_4_2-C_10_5-C_11_3C_1_1-C_11_4C_1_2-C_11_5-C_12_3C_2_1-C_12_4C_2_2-C_12_5-C_13_3C_3_1-C_13_4C_3_2-C_13_5+TraceMulA_1_1A_1_2A_1_3A_2_1A_2_2A_2_3-B_1_9-B_7_9B_1_5-B_1_10-B_7_10B_1_6-B_1_11-B_7_11B_1_7-B_1_12-B_7_12B_1_8-B_1_13-B_7_13-B_2_9-B_8_9B_2_5-B_2_10-B_8_10B_2_6-B_2_11-B_8_11B_2_7-B_2_12-B_8_12B_2_8-B_2_13-B_8_13-B_3_9-B_9_9B_3_5-B_3_10-B_9_10B_3_6-B_3_11-B_9_11B_3_7-B_3_12-B_9_12B_3_8-B_3_13-B_9_13-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+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_7_9B_7_10B_7_11B_7_12B_7_13B_8_9B_8_10B_8_11B_8_12B_8_13B_9_9B_9_10B_9_11B_9_12B_9_13-C_9_1-C_9_2-C_4_1-C_10_1-C_4_2-C_10_2-C_1_1-C_11_1-C_1_2-C_11_2-C_2_1-C_12_1-C_2_2-C_12_2-C_3_1-C_13_1-C_3_2-C_13_2+TraceMulA_1_7-A_4_7A_1_8-A_4_8A_1_9-A_4_9A_2_7-A_5_7A_2_8-A_5_8A_2_9-A_5_9B_7_4+B_4_4-B_7_10B_4_1+B_7_1-B_7_11B_4_2+B_7_2-B_7_12B_4_3+B_7_3-B_7_13B_5_4+B_8_4-B_8_10B_5_1+B_8_1-B_8_11B_5_2+B_8_2-B_8_12B_5_3+B_8_3-B_8_13B_6_4+B_9_4-B_9_10B_6_1+B_9_1-B_9_11B_6_2+B_9_2-B_9_12B_6_3+B_9_3-B_9_13C_4_1C_4_2C_1_1C_1_2C_2_1C_2_2C_3_1C_3_2+TraceMulA_1_4A_1_5A_1_6A_2_4A_2_5A_2_6B_7_5+B_4_5-B_4_10B_4_6+B_7_6-B_4_11B_7_7+B_4_7-B_4_12B_4_8+B_7_8-B_4_13B_5_5+B_8_5-B_5_10B_5_6+B_8_6-B_5_11B_5_7+B_8_7-B_5_12B_5_8+B_8_8-B_5_13B_6_5+B_9_5-B_6_10B_6_6+B_9_6-B_6_11B_6_7+B_9_7-B_6_12B_6_8+B_9_8-B_6_13C_5_1C_5_2C_6_1C_6_2C_7_1C_7_2C_8_1C_8_2+TraceMulA_3_7A_3_8A_3_9A_1_4-A_1_7+A_4_7A_1_5-A_1_8+A_4_8A_1_6-A_1_9+A_4_9A_2_4-A_2_7+A_5_7A_2_5-A_2_8+A_5_8A_2_6-A_2_9+A_5_9B_4_4-B_7_5B_4_1-B_7_6B_4_2-B_7_7B_4_3-B_7_8B_5_4-B_8_5B_5_1-B_8_6B_5_2-B_8_7B_5_3-B_8_8B_6_4-B_9_5B_6_1-B_9_6B_6_2-B_9_7B_6_3-B_9_8C_4_3-C_5_3C_4_1+C_4_4-C_5_4C_4_2+C_4_5-C_5_5C_1_3-C_6_3C_1_1+C_1_4-C_6_4C_1_2+C_1_5-C_6_5C_2_3-C_7_3C_2_1+C_2_4-C_7_4C_2_2+C_2_5-C_7_5C_3_3-C_8_3C_3_1+C_3_4-C_8_4C_3_2+C_3_5-C_8_5+TraceMul-A_3_1+A_3_4-A_3_2+A_3_5-A_3_3+A_3_6A_1_1-A_4_1+A_4_4A_1_2-A_4_2+A_4_5A_1_3-A_4_3+A_4_6A_2_1-A_5_1+A_5_4A_2_2-A_5_2+A_5_5A_2_3-A_5_3+A_5_6B_4_9B_1_5+B_4_10B_4_11+B_1_6B_1_7+B_4_12B_1_8+B_4_13B_5_9B_2_5+B_5_10B_5_11+B_2_6B_2_7+B_5_12B_2_8+B_5_13B_6_9B_3_5+B_6_10B_6_11+B_3_6B_3_7+B_6_12B_3_8+B_6_13C_9_3C_9_1+C_9_4C_9_2+C_9_5C_10_3C_5_1+C_10_1+C_10_4C_5_2+C_10_2+C_10_5C_11_3C_6_1+C_11_1+C_11_4C_6_2+C_11_2+C_11_5C_12_3C_7_1+C_12_1+C_12_4C_7_2+C_12_2+C_12_5C_13_3C_8_1+C_13_1+C_13_4C_8_2+C_13_2+C_13_5+TraceMulA_3_1-A_3_4A_3_2-A_3_5-A_3_6+A_3_3A_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_1_5B_1_6B_1_7B_1_8B_2_5B_2_6B_2_7B_2_8B_3_5B_3_6B_3_7B_3_8C_5_3+C_10_3C_5_1+C_10_1+C_5_4+C_10_4C_5_2+C_10_2+C_5_5+C_10_5C_6_3+C_11_3C_6_1+C_11_1+C_6_4+C_11_4C_6_2+C_11_2+C_6_5+C_11_5C_7_3+C_12_3C_7_1+C_12_1+C_7_4+C_12_4C_7_2+C_12_2+C_7_5+C_12_5C_8_3+C_13_3C_8_1+C_13_1+C_8_4+C_13_4C_8_2+C_13_2+C_8_5+C_13_5+TraceMul-A_3_4+A_3_7-A_3_5+A_3_8-A_3_6+A_3_9A_1_4-A_4_4-A_1_7+A_4_7A_1_5-A_4_5-A_1_8+A_4_8A_1_6-A_4_6-A_1_9+A_4_9A_2_4-A_5_4-A_2_7+A_5_7A_2_5-A_5_5-A_2_8+A_5_8A_2_6-A_5_6-A_2_9+A_5_9B_4_4B_4_1B_4_2B_4_3B_5_4B_5_1B_5_2B_5_3B_6_4B_6_1B_6_2B_6_3-C_4_3+C_5_3-C_4_4+C_5_4-C_4_5+C_5_5-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+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_2_9A_5_9B_4_4B_7_5B_4_1B_7_6B_4_2B_7_7B_4_3B_7_8B_5_4B_8_5B_5_1B_8_6B_5_2B_8_7B_5_3B_8_8B_6_4B_9_5B_6_1B_9_6B_6_2B_9_7B_6_3B_9_8C_4_3C_5_3C_4_1C_4_4C_5_4C_4_2C_4_5C_5_5C_1_3C_6_3C_1_1C_1_4C_6_4C_1_2C_1_5C_6_5C_2_3C_7_3C_2_1C_2_4C_7_4C_2_2C_2_5C_7_5C_3_3C_8_3C_3_1C_3_4C_8_4C_3_2C_3_5C_8_5TraceMulA_3_1A_3_4A_3_2A_3_5A_3_3A_3_6A_1_1A_4_1A_4_4A_1_2A_4_2A_4_5A_1_3A_4_3A_4_6A_2_1A_5_1A_5_4A_2_2A_5_2A_5_5A_2_3A_5_3A_5_6B_4_9B_1_5B_4_10B_4_11B_1_6B_1_7B_4_12B_1_8B_4_13B_5_9B_2_5B_5_10B_5_11B_2_6B_2_7B_5_12B_2_8B_5_13B_6_9B_3_5B_6_10B_6_11B_3_6B_3_7B_6_12B_3_8B_6_13C_9_3C_9_1C_9_4C_9_2C_9_5C_10_3C_5_1C_10_1C_10_4C_5_2C_10_2C_10_5C_11_3C_6_1C_11_1C_11_4C_6_2C_11_2C_11_5C_12_3C_7_1C_12_1C_12_4C_7_2C_12_2C_12_5C_13_3C_8_1C_13_1C_13_4C_8_2C_13_2C_13_5TraceMulA_3_1A_3_4A_3_2A_3_5A_3_6A_3_3A_4_1A_4_4A_4_2A_4_5A_4_3A_4_6A_5_1A_5_4A_5_2A_5_5A_5_3A_5_6B_1_5B_1_6B_1_7B_1_8B_2_5B_2_6B_2_7B_2_8B_3_5B_3_6B_3_7B_3_8C_5_3C_10_3C_5_1C_10_1C_5_4C_10_4C_5_2C_10_2C_5_5C_10_5C_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_5TraceMulA_3_4A_3_7A_3_5A_3_8A_3_6A_3_9A_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_4_4B_4_1B_4_2B_4_3B_5_4B_5_1B_5_2B_5_3B_6_4B_6_1B_6_2B_6_3C_4_3C_5_3C_4_4C_5_4C_4_5C_5_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_5TraceMulA_1_4A_1_7A_1_5A_1_8A_1_6A_1_9A_2_4A_2_7A_2_5A_2_8A_2_6A_2_9B_7_5B_7_6B_7_7B_7_8B_8_5B_8_6B_8_7B_8_8B_9_5B_9_6B_9_7B_9_8C_4_1C_5_1C_4_4C_5_4C_4_2C_5_2C_4_5C_5_5C_1_1C_6_1C_1_4C_6_4C_1_2C_6_2C_1_5C_6_5C_2_1C_7_1C_2_4C_7_4C_2_2C_7_2C_2_5C_7_5C_3_1C_8_1C_3_4C_8_4C_3_2C_8_2C_3_5C_8_5TraceMulA_3_4A_3_5A_3_6A_4_4A_4_5A_4_6A_5_4A_5_5A_5_6B_4_4B_1_5B_4_5B_4_1B_1_6B_4_6B_4_2B_1_7B_4_7B_4_3B_1_8B_4_8B_5_4B_2_5B_5_5B_5_1B_2_6B_5_6B_5_2B_2_7B_5_7B_5_3B_2_8B_5_8B_6_4B_3_5B_6_5B_6_1B_3_6B_6_6B_6_2B_3_7B_6_7B_6_3B_3_8B_6_8C_5_3C_5_4C_5_5C_6_3C_6_4C_6_5C_7_3C_7_4C_7_5C_8_3C_8_4C_8_5TraceMulA_1_1A_4_1A_1_4A_4_4A_1_2A_4_2A_1_5A_4_5A_1_3A_4_3A_1_6A_4_6A_2_1A_5_1A_2_4A_5_4A_2_2A_5_2A_2_5A_5_5A_2_3A_5_3A_2_6A_5_6B_4_9B_4_10B_4_11B_4_12B_4_13B_5_9B_5_10B_5_11B_5_12B_5_13B_6_9B_6_10B_6_11B_6_12B_6_13C_9_1C_9_2C_5_1C_10_1C_5_2C_10_2C_6_1C_11_1C_6_2C_11_2C_7_1C_12_1C_7_2C_12_2C_8_1C_13_1C_8_2C_13_2TraceMulA_3_1A_3_7A_3_2A_3_8A_3_3A_3_9A_4_1A_4_7A_4_2A_4_8A_4_3A_4_9A_5_1A_5_7A_5_2A_5_8A_5_3A_5_9B_1_4B_1_1B_1_2B_1_3B_2_4B_2_1B_2_2B_2_3B_3_4B_3_1B_3_2B_3_3C_4_3C_10_3C_4_4C_10_4C_4_5C_10_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_5TraceMulA_3_1A_3_2A_3_3A_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_9B_4_9B_1_4B_1_10B_4_10B_1_1B_1_11B_4_11B_1_2B_1_12B_4_12B_1_3B_1_13B_4_13B_2_9B_5_9B_2_4B_2_10B_5_10B_2_1B_2_11B_5_11B_2_2B_2_12B_5_12B_2_3B_2_13B_5_13B_3_9B_6_9B_3_4B_3_10B_6_10B_3_1B_3_11B_6_11B_3_2B_3_12B_6_12B_3_3B_3_13B_6_13C_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_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, 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]]),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, 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, 4, [[B_1_4+B_7_4+B_7_5,B_1_1+B_7_1+B_7_6,B_1_2+B_7_2+B_7_7,B_1_3+B_7_3+B_7_8],[B_2_4+B_8_4+B_8_5,B_2_1+B_8_1+B_8_6,B_2_2+B_8_2+B_8_7,B_2_3+B_8_3+B_8_8],[B_3_4+B_9_4+B_9_5,B_3_1+B_9_1+B_9_6,B_3_2+B_9_2+B_9_7,B_3_3+B_9_3+B_9_8]]),Matrix(4, 3, [[C_4_3,C_4_1+C_4_4,C_4_2+C_4_5],[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]])))+Trace(Mul(Matrix(3, 3, [[-A_3_7,-A_3_8,-A_3_9],[A_1_1-A_4_7,A_1_2-A_4_8,A_1_3-A_4_9],[A_2_1-A_5_7,A_2_2-A_5_8,A_2_3-A_5_9]]),Matrix(3, 5, [[B_7_9,B_1_4+B_7_10,B_1_1+B_7_11,B_1_2+B_7_12,B_1_3+B_7_13],[B_8_9,B_2_4+B_8_10,B_2_1+B_8_11,B_2_2+B_8_12,B_2_3+B_8_13],[B_9_9,B_3_4+B_9_10,B_3_1+B_9_11,B_3_2+B_9_12,B_3_3+B_9_13]]),Matrix(5, 3, [[-C_9_3,-C_9_4,-C_9_5],[-C_10_3,C_4_1-C_10_4,C_4_2-C_10_5],[-C_11_3,C_1_1-C_11_4,C_1_2-C_11_5],[-C_12_3,C_2_1-C_12_4,C_2_2-C_12_5],[-C_13_3,C_3_1-C_13_4,C_3_2-C_13_5]])))+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, 5, [[-B_1_9-B_7_9,B_1_5-B_1_10-B_7_10,B_1_6-B_1_11-B_7_11,B_1_7-B_1_12-B_7_12,B_1_8-B_1_13-B_7_13],[-B_2_9-B_8_9,B_2_5-B_2_10-B_8_10,B_2_6-B_2_11-B_8_11,B_2_7-B_2_12-B_8_12,B_2_8-B_2_13-B_8_13],[-B_3_9-B_9_9,B_3_5-B_3_10-B_9_10,B_3_6-B_3_11-B_9_11,B_3_7-B_3_12-B_9_12,B_3_8-B_3_13-B_9_13]]),Matrix(5, 2, [[-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(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_7_9,B_7_10,B_7_11,B_7_12,B_7_13],[B_8_9,B_8_10,B_8_11,B_8_12,B_8_13],[B_9_9,B_9_10,B_9_11,B_9_12,B_9_13]]),Matrix(5, 2, [[-C_9_1,-C_9_2],[-C_4_1-C_10_1,-C_4_2-C_10_2],[-C_1_1-C_11_1,-C_1_2-C_11_2],[-C_2_1-C_12_1,-C_2_2-C_12_2],[-C_3_1-C_13_1,-C_3_2-C_13_2]])))+Trace(Mul(Matrix(2, 3, [[A_1_7-A_4_7,A_1_8-A_4_8,A_1_9-A_4_9],[A_2_7-A_5_7,A_2_8-A_5_8,A_2_9-A_5_9]]),Matrix(3, 4, [[B_7_4+B_4_4-B_7_10,B_4_1+B_7_1-B_7_11,B_4_2+B_7_2-B_7_12,B_4_3+B_7_3-B_7_13],[B_5_4+B_8_4-B_8_10,B_5_1+B_8_1-B_8_11,B_5_2+B_8_2-B_8_12,B_5_3+B_8_3-B_8_13],[B_6_4+B_9_4-B_9_10,B_6_1+B_9_1-B_9_11,B_6_2+B_9_2-B_9_12,B_6_3+B_9_3-B_9_13]]),Matrix(4, 2, [[C_4_1,C_4_2],[C_1_1,C_1_2],[C_2_1,C_2_2],[C_3_1,C_3_2]])))+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, 4, [[B_7_5+B_4_5-B_4_10,B_4_6+B_7_6-B_4_11,B_7_7+B_4_7-B_4_12,B_4_8+B_7_8-B_4_13],[B_5_5+B_8_5-B_5_10,B_5_6+B_8_6-B_5_11,B_5_7+B_8_7-B_5_12,B_5_8+B_8_8-B_5_13],[B_6_5+B_9_5-B_6_10,B_6_6+B_9_6-B_6_11,B_6_7+B_9_7-B_6_12,B_6_8+B_9_8-B_6_13]]),Matrix(4, 2, [[C_5_1,C_5_2],[C_6_1,C_6_2],[C_7_1,C_7_2],[C_8_1,C_8_2]])))+Trace(Mul(Matrix(3, 3, [[A_3_7,A_3_8,A_3_9],[A_1_4-A_1_7+A_4_7,A_1_5-A_1_8+A_4_8,A_1_6-A_1_9+A_4_9],[A_2_4-A_2_7+A_5_7,A_2_5-A_2_8+A_5_8,A_2_6-A_2_9+A_5_9]]),Matrix(3, 4, [[B_4_4-B_7_5,B_4_1-B_7_6,B_4_2-B_7_7,B_4_3-B_7_8],[B_5_4-B_8_5,B_5_1-B_8_6,B_5_2-B_8_7,B_5_3-B_8_8],[B_6_4-B_9_5,B_6_1-B_9_6,B_6_2-B_9_7,B_6_3-B_9_8]]),Matrix(4, 3, [[C_4_3-C_5_3,C_4_1+C_4_4-C_5_4,C_4_2+C_4_5-C_5_5],[C_1_3-C_6_3,C_1_1+C_1_4-C_6_4,C_1_2+C_1_5-C_6_5],[C_2_3-C_7_3,C_2_1+C_2_4-C_7_4,C_2_2+C_2_5-C_7_5],[C_3_3-C_8_3,C_3_1+C_3_4-C_8_4,C_3_2+C_3_5-C_8_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_1_1-A_4_1+A_4_4,A_1_2-A_4_2+A_4_5,A_1_3-A_4_3+A_4_6],[A_2_1-A_5_1+A_5_4,A_2_2-A_5_2+A_5_5,A_2_3-A_5_3+A_5_6]]),Matrix(3, 5, [[B_4_9,B_1_5+B_4_10,B_4_11+B_1_6,B_1_7+B_4_12,B_1_8+B_4_13],[B_5_9,B_2_5+B_5_10,B_5_11+B_2_6,B_2_7+B_5_12,B_2_8+B_5_13],[B_6_9,B_3_5+B_6_10,B_6_11+B_3_6,B_3_7+B_6_12,B_3_8+B_6_13]]),Matrix(5, 3, [[C_9_3,C_9_1+C_9_4,C_9_2+C_9_5],[C_10_3,C_5_1+C_10_1+C_10_4,C_5_2+C_10_2+C_10_5],[C_11_3,C_6_1+C_11_1+C_11_4,C_6_2+C_11_2+C_11_5],[C_12_3,C_7_1+C_12_1+C_12_4,C_7_2+C_12_2+C_12_5],[C_13_3,C_8_1+C_13_1+C_13_4,C_8_2+C_13_2+C_13_5]])))+Trace(Mul(Matrix(3, 3, [[A_3_1-A_3_4,A_3_2-A_3_5,-A_3_6+A_3_3],[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, 4, [[B_1_5,B_1_6,B_1_7,B_1_8],[B_2_5,B_2_6,B_2_7,B_2_8],[B_3_5,B_3_6,B_3_7,B_3_8]]),Matrix(4, 3, [[C_5_3+C_10_3,C_5_1+C_10_1+C_5_4+C_10_4,C_5_2+C_10_2+C_5_5+C_10_5],[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]])))+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_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, 4, [[B_4_4,B_4_1,B_4_2,B_4_3],[B_5_4,B_5_1,B_5_2,B_5_3],[B_6_4,B_6_1,B_6_2,B_6_3]]),Matrix(4, 3, [[-C_4_3+C_5_3,-C_4_4+C_5_4,-C_4_5+C_5_5],[-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]])))+Trace(Mul(Matrix(2, 3, [[A_1_4-A_1_7,A_1_5-A_1_8,A_1_6-A_1_9],[A_2_4-A_2_7,A_2_5-A_2_8,A_2_6-A_2_9]]),Matrix(3, 4, [[B_7_5,B_7_6,B_7_7,B_7_8],[B_8_5,B_8_6,B_8_7,B_8_8],[B_9_5,B_9_6,B_9_7,B_9_8]]),Matrix(4, 2, [[C_4_1-C_5_1+C_4_4-C_5_4,C_4_2-C_5_2+C_4_5-C_5_5],[C_1_1-C_6_1+C_1_4-C_6_4,C_1_2-C_6_2+C_1_5-C_6_5],[C_2_1-C_7_1+C_2_4-C_7_4,C_2_2-C_7_2+C_2_5-C_7_5],[C_3_1-C_8_1+C_3_4-C_8_4,C_3_2-C_8_2+C_3_5-C_8_5]])))+Trace(Mul(Matrix(3, 3, [[A_3_4,A_3_5,A_3_6],[A_4_4,A_4_5,A_4_6],[A_5_4,A_5_5,A_5_6]]),Matrix(3, 4, [[B_4_4+B_1_5+B_4_5,B_4_1+B_1_6+B_4_6,B_4_2+B_1_7+B_4_7,B_4_3+B_1_8+B_4_8],[B_5_4+B_2_5+B_5_5,B_5_1+B_2_6+B_5_6,B_5_2+B_2_7+B_5_7,B_5_3+B_2_8+B_5_8],[B_6_4+B_3_5+B_6_5,B_6_1+B_3_6+B_6_6,B_6_2+B_3_7+B_6_7,B_6_3+B_3_8+B_6_8]]),Matrix(4, 3, [[C_5_3,C_5_4,C_5_5],[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]])))+Trace(Mul(Matrix(2, 3, [[A_1_1-A_4_1-A_1_4+A_4_4,A_1_2-A_4_2-A_1_5+A_4_5,A_1_3-A_4_3-A_1_6+A_4_6],[A_2_1-A_5_1-A_2_4+A_5_4,A_2_2-A_5_2-A_2_5+A_5_5,A_2_3-A_5_3-A_2_6+A_5_6]]),Matrix(3, 5, [[B_4_9,B_4_10,B_4_11,B_4_12,B_4_13],[B_5_9,B_5_10,B_5_11,B_5_12,B_5_13],[B_6_9,B_6_10,B_6_11,B_6_12,B_6_13]]),Matrix(5, 2, [[-C_9_1,-C_9_2],[-C_5_1-C_10_1,-C_5_2-C_10_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]])))+Trace(Mul(Matrix(3, 3, [[A_3_1-A_3_7,A_3_2-A_3_8,A_3_3-A_3_9],[A_4_1-A_4_7,A_4_2-A_4_8,A_4_3-A_4_9],[A_5_1-A_5_7,A_5_2-A_5_8,A_5_3-A_5_9]]),Matrix(3, 4, [[B_1_4,B_1_1,B_1_2,B_1_3],[B_2_4,B_2_1,B_2_2,B_2_3],[B_3_4,B_3_1,B_3_2,B_3_3]]),Matrix(4, 3, [[C_4_3+C_10_3,C_4_4+C_10_4,C_4_5+C_10_5],[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]])))+Trace(Mul(Matrix(3, 3, [[-A_3_1,-A_3_2,-A_3_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_9-B_4_9,B_1_4-B_1_10-B_4_10,B_1_1-B_1_11-B_4_11,B_1_2-B_1_12-B_4_12,B_1_3-B_1_13-B_4_13],[-B_2_9-B_5_9,B_2_4-B_2_10-B_5_10,B_2_1-B_2_11-B_5_11,B_2_2-B_2_12-B_5_12,B_2_3-B_2_13-B_5_13],[-B_3_9-B_6_9,B_3_4-B_3_10-B_6_10,B_3_1-B_3_11-B_6_11,B_3_2-B_3_12-B_6_12,B_3_3-B_3_13-B_6_13]]),Matrix(5, 3, [[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]])))

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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