Description of fast matrix multiplication algorithm: ⟨5×7×16:404⟩

Algorithm type

6X4Y4Z4+3X3Y5Z4+2X3Y4Z5+6X3Y4Z4+X4Y4Z2+3X3Y3Z4+X2Y6Z2+6X2Y5Z2+X2Y4Z3+XY6Z2+XY4Z4+10X2Y4Z2+32X2Y3Z3+5X2Y3Z2+3X2Y2Z3+3XY4Z2+48XY3Z3+2XY2Z4+70X2Y2Z2+2XY3Z2+X2Y2Z+2X2YZ2+24XY3Z+6XY2Z2+12X2YZ+49XY2Z+4XYZ2+100XYZ6X4Y4Z43X3Y5Z42X3Y4Z56X3Y4Z4X4Y4Z23X3Y3Z4X2Y6Z26X2Y5Z2X2Y4Z3XY6Z2XY4Z410X2Y4Z232X2Y3Z35X2Y3Z23X2Y2Z33XY4Z248XY3Z32XY2Z470X2Y2Z22XY3Z2X2Y2Z2X2YZ224XY3Z6XY2Z212X2YZ49XY2Z4XYZ2100XYZ6*X^4*Y^4*Z^4+3*X^3*Y^5*Z^4+2*X^3*Y^4*Z^5+6*X^3*Y^4*Z^4+X^4*Y^4*Z^2+3*X^3*Y^3*Z^4+X^2*Y^6*Z^2+6*X^2*Y^5*Z^2+X^2*Y^4*Z^3+X*Y^6*Z^2+X*Y^4*Z^4+10*X^2*Y^4*Z^2+32*X^2*Y^3*Z^3+5*X^2*Y^3*Z^2+3*X^2*Y^2*Z^3+3*X*Y^4*Z^2+48*X*Y^3*Z^3+2*X*Y^2*Z^4+70*X^2*Y^2*Z^2+2*X*Y^3*Z^2+X^2*Y^2*Z+2*X^2*Y*Z^2+24*X*Y^3*Z+6*X*Y^2*Z^2+12*X^2*Y*Z+49*X*Y^2*Z+4*X*Y*Z^2+100*X*Y*Z

Algorithm definition

The algorithm ⟨5×7×16:404⟩ could be constructed using the following decomposition:

⟨5×7×16:404⟩ = ⟨3×3×8:55⟩ + ⟨3×3×8:55⟩ + ⟨2×4×8:52⟩ + ⟨3×4×8:75⟩ + ⟨2×4×8:52⟩ + ⟨2×3×8:40⟩ + ⟨3×4×8:75⟩.

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_7A_5_1A_5_2A_5_3A_5_4A_5_5A_5_6A_5_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_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_16C_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_5C_16_1C_16_2C_16_3C_16_4C_16_5=TraceMulA_1_3-A_1_2A_1_4-A_1_6A_1_5-A_1_7A_2_3-A_2_2A_2_4-A_2_6A_2_5-A_2_7A_3_3-A_3_2A_3_4-A_3_6A_3_5-A_3_7-B_3_1+B_3_9-B_3_2+B_3_10-B_3_12+B_3_11B_3_3-B_3_4-B_3_5+B_3_7B_3_8-B_3_6-B_3_13+B_3_15-B_3_14+B_3_16-B_4_1+B_4_9-B_4_2+B_4_10-B_4_12+B_4_11B_4_3-B_4_4B_4_7-B_4_5B_4_8-B_4_6-B_4_13+B_4_15-B_4_14+B_4_16-B_5_1+B_5_9-B_5_2+B_5_10-B_5_12+B_5_11B_5_3-B_5_4B_5_7-B_5_5B_5_8-B_5_6-B_5_13+B_5_15-B_5_14+B_5_16C_9_1C_9_2-C_9_4C_9_3-C_9_5C_10_1C_10_2-C_10_4C_10_3-C_10_5C_11_1C_11_2-C_11_4C_11_3-C_11_5C_3_1C_3_2-C_3_4C_3_3-C_3_5C_7_1C_7_2-C_7_4C_7_3-C_7_5C_8_1C_8_2-C_8_4C_8_3-C_8_5C_15_1C_15_2-C_15_4C_15_3-C_15_5C_16_1C_16_2-C_16_4C_16_3-C_16_5+TraceMulA_1_3A_1_4A_1_5A_2_3+A_4_3A_2_4+A_4_4A_2_5+A_4_5A_3_3+A_5_3A_3_4+A_5_4A_3_5+A_5_5B_3_9+B_2_9B_3_10+B_2_10B_3_11+B_2_11B_3_3+B_2_3B_3_7+B_2_7B_3_8+B_2_8B_3_15+B_2_15B_3_16+B_2_16B_4_9+B_6_9B_4_10+B_6_10B_4_11+B_6_11B_4_3+B_6_3B_4_7+B_6_7B_4_8+B_6_8B_4_15+B_6_15B_4_16+B_6_16B_5_9+B_7_9B_5_10+B_7_10B_5_11+B_7_11B_5_3+B_7_3B_5_7+B_7_7B_5_8+B_7_8B_5_15+B_7_15B_5_16+B_7_16C_1_1+C_9_1C_1_2+C_9_2C_1_3+C_9_3C_2_1+C_10_1C_2_2+C_10_2C_2_3+C_10_3C_12_1+C_11_1C_12_2+C_11_2C_12_3+C_11_3C_3_1+C_4_1C_3_2+C_4_2C_4_3+C_3_3C_7_1+C_5_1C_7_2+C_5_2C_5_3+C_7_3C_8_1+C_6_1C_8_2+C_6_2C_6_3+C_8_3C_13_1+C_15_1C_13_2+C_15_2C_13_3+C_15_3C_14_1+C_16_1C_14_2+C_16_2C_14_3+C_16_3+TraceMulA_4_1A_4_2A_4_6A_4_7A_5_1A_5_2A_5_6A_5_7B_1_1B_1_2B_1_12B_1_4B_1_5B_1_6B_1_13B_1_14B_2_1B_2_2B_2_12B_2_4B_2_5B_2_6B_2_13B_2_14B_6_1B_6_2B_6_12B_6_4B_6_5B_6_6B_6_13B_6_14B_7_1B_7_2B_7_12B_7_4B_7_5B_7_6B_7_13B_7_14C_1_4C_1_5C_2_4C_2_5C_12_4C_12_5C_4_4C_4_5C_5_4C_5_5C_6_4C_6_5C_13_4C_13_5C_14_4C_14_5+TraceMulA_1_1A_1_2A_1_6A_1_7A_2_1A_2_2A_2_6A_2_7A_3_1A_3_2A_3_6A_3_7B_1_1-B_1_9B_1_2-B_1_10B_1_12-B_1_11B_1_4-B_1_3B_1_5-B_1_7B_1_6-B_1_8B_1_13-B_1_15B_1_14-B_1_16B_3_1+B_2_1-B_3_9-B_2_9B_3_2+B_2_2-B_3_10-B_2_10B_3_12+B_2_12-B_3_11-B_2_11B_3_4+B_2_4-B_3_3-B_2_3B_3_5+B_2_5-B_3_7-B_2_7B_3_6+B_2_6-B_3_8-B_2_8B_3_13+B_2_13-B_3_15-B_2_15B_3_14+B_2_14-B_3_16-B_2_16B_4_1+B_6_1-B_4_9-B_6_9B_4_2+B_6_2-B_4_10-B_6_10B_4_12+B_6_12-B_4_11-B_6_11B_4_4+B_6_4-B_4_3-B_6_3B_4_5+B_6_5-B_4_7-B_6_7B_4_6+B_6_6-B_4_8-B_6_8B_4_13+B_6_13-B_4_15-B_6_15B_4_14+B_6_14-B_4_16-B_6_16B_5_1+B_7_1-B_5_9-B_7_9B_5_2+B_7_2-B_5_10-B_7_10B_5_12+B_7_12-B_5_11-B_7_11B_5_4+B_7_4-B_5_3-B_7_3B_5_5+B_7_5-B_5_7-B_7_7B_5_6+B_7_6-B_5_8-B_7_8B_5_13+B_7_13-B_5_15-B_7_15B_5_14+B_7_14-B_5_16-B_7_16C_1_1C_1_2C_1_3C_2_1C_2_2C_2_3C_12_1C_12_2C_12_3C_4_1C_4_2C_4_3C_5_1C_5_2C_5_3C_6_1C_6_2C_6_3C_13_1C_13_2C_13_3C_14_1C_14_2C_14_3+TraceMulA_2_1+A_4_1-A_2_3-A_4_3+A_2_2+A_4_2-A_2_4-A_4_4+A_2_6+A_4_6-A_2_5-A_4_5+A_2_7+A_4_7A_3_1+A_5_1-A_3_3-A_5_3+A_3_2+A_5_2-A_3_4-A_5_4+A_3_6+A_5_6-A_3_5-A_5_5+A_3_7+A_5_7B_1_9B_1_10B_1_11B_1_3B_1_7B_1_8B_1_15B_1_16B_2_9B_2_10B_2_11B_2_3B_2_7B_2_8B_2_15B_2_16B_6_9B_6_10B_6_11B_6_3B_6_7B_6_8B_6_15B_6_16B_7_9B_7_10B_7_11B_7_3B_7_7B_7_8B_7_15B_7_16C_9_4C_9_5C_10_4C_10_5C_11_4C_11_5C_3_4C_3_5C_7_4C_7_5C_8_4C_8_5C_15_4C_15_5C_16_4C_16_5+TraceMulA_4_3A_4_4A_4_5A_5_3A_5_4A_5_5B_3_1B_3_2B_3_12B_3_4B_3_5B_3_6B_3_13B_3_14B_4_1B_4_2B_4_12B_4_4B_4_5B_4_6B_4_13B_4_14B_5_1B_5_2B_5_12B_5_4B_5_5B_5_6B_5_13B_5_14-C_1_2-C_9_2+C_1_4+C_9_4-C_1_3-C_9_3+C_1_5+C_9_5-C_2_2-C_10_2+C_2_4+C_10_4-C_2_3-C_10_3+C_2_5+C_10_5-C_12_2-C_11_2+C_12_4+C_11_4-C_12_3-C_11_3+C_12_5+C_11_5-C_4_2-C_3_2+C_4_4+C_3_4-C_4_3-C_3_3+C_4_5+C_3_5-C_5_2-C_7_2+C_5_4+C_7_4-C_5_3-C_7_3+C_5_5+C_7_5-C_6_2-C_8_2+C_6_4+C_8_4-C_6_3-C_8_3+C_6_5+C_8_5-C_13_2-C_15_2+C_13_4+C_15_4-C_13_3-C_15_3+C_13_5+C_15_5-C_14_2-C_16_2+C_14_4+C_16_4-C_14_3-C_16_3+C_14_5+C_16_5+TraceMulA_1_1-A_1_3+A_1_2-A_1_4+A_1_6-A_1_5+A_1_7A_2_1-A_2_3-A_4_3+A_2_2-A_2_4-A_4_4+A_2_6-A_2_5-A_4_5+A_2_7A_3_1-A_3_3-A_5_3+A_3_2-A_3_4-A_5_4+A_3_6-A_3_5-A_5_5+A_3_7B_1_9B_1_10B_1_11B_1_3B_1_7B_1_8B_1_15B_1_16-B_3_1+B_3_9+B_2_9-B_3_2+B_3_10+B_2_10-B_3_12+B_3_11+B_2_11B_3_3-B_3_4+B_2_3-B_3_5+B_3_7+B_2_7B_3_8-B_3_6+B_2_8-B_3_13+B_3_15+B_2_15-B_3_14+B_3_16+B_2_16-B_4_1+B_4_9+B_6_9-B_4_2+B_4_10+B_6_10-B_4_12+B_4_11+B_6_11B_4_3-B_4_4+B_6_3B_4_7-B_4_5+B_6_7B_4_8-B_4_6+B_6_8-B_4_13+B_4_15+B_6_15-B_4_14+B_4_16+B_6_16-B_5_1+B_5_9+B_7_9-B_5_2+B_5_10+B_7_10-B_5_12+B_5_11+B_7_11B_5_3-B_5_4+B_7_3B_5_7-B_5_5+B_7_7B_5_8-B_5_6+B_7_8-B_5_13+B_5_15+B_7_15-B_5_14+B_5_16+B_7_16C_1_1+C_9_1C_1_2+C_9_2-C_9_4C_1_3+C_9_3-C_9_5C_2_1+C_10_1C_2_2+C_10_2-C_10_4C_2_3+C_10_3-C_10_5C_12_1+C_11_1C_12_2+C_11_2-C_11_4C_12_3+C_11_3-C_11_5C_3_1+C_4_1C_3_2+C_4_2-C_3_4C_4_3+C_3_3-C_3_5C_7_1+C_5_1C_7_2+C_5_2-C_7_4C_5_3+C_7_3-C_7_5C_8_1+C_6_1C_8_2+C_6_2-C_8_4C_6_3+C_8_3-C_8_5C_13_1+C_15_1C_13_2+C_15_2-C_15_4C_13_3+C_15_3-C_15_5C_14_1+C_16_1C_14_2+C_16_2-C_16_4C_14_3+C_16_3-C_16_5TraceMulA_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_7A_5_1A_5_2A_5_3A_5_4A_5_5A_5_6A_5_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_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_16C_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_5C_16_1C_16_2C_16_3C_16_4C_16_5TraceMulA_1_3A_1_2A_1_4A_1_6A_1_5A_1_7A_2_3A_2_2A_2_4A_2_6A_2_5A_2_7A_3_3A_3_2A_3_4A_3_6A_3_5A_3_7B_3_1B_3_9B_3_2B_3_10B_3_12B_3_11B_3_3B_3_4B_3_5B_3_7B_3_8B_3_6B_3_13B_3_15B_3_14B_3_16B_4_1B_4_9B_4_2B_4_10B_4_12B_4_11B_4_3B_4_4B_4_7B_4_5B_4_8B_4_6B_4_13B_4_15B_4_14B_4_16B_5_1B_5_9B_5_2B_5_10B_5_12B_5_11B_5_3B_5_4B_5_7B_5_5B_5_8B_5_6B_5_13B_5_15B_5_14B_5_16C_9_1C_9_2C_9_4C_9_3C_9_5C_10_1C_10_2C_10_4C_10_3C_10_5C_11_1C_11_2C_11_4C_11_3C_11_5C_3_1C_3_2C_3_4C_3_3C_3_5C_7_1C_7_2C_7_4C_7_3C_7_5C_8_1C_8_2C_8_4C_8_3C_8_5C_15_1C_15_2C_15_4C_15_3C_15_5C_16_1C_16_2C_16_4C_16_3C_16_5TraceMulA_1_3A_1_4A_1_5A_2_3A_4_3A_2_4A_4_4A_2_5A_4_5A_3_3A_5_3A_3_4A_5_4A_3_5A_5_5B_3_9B_2_9B_3_10B_2_10B_3_11B_2_11B_3_3B_2_3B_3_7B_2_7B_3_8B_2_8B_3_15B_2_15B_3_16B_2_16B_4_9B_6_9B_4_10B_6_10B_4_11B_6_11B_4_3B_6_3B_4_7B_6_7B_4_8B_6_8B_4_15B_6_15B_4_16B_6_16B_5_9B_7_9B_5_10B_7_10B_5_11B_7_11B_5_3B_7_3B_5_7B_7_7B_5_8B_7_8B_5_15B_7_15B_5_16B_7_16C_1_1C_9_1C_1_2C_9_2C_1_3C_9_3C_2_1C_10_1C_2_2C_10_2C_2_3C_10_3C_12_1C_11_1C_12_2C_11_2C_12_3C_11_3C_3_1C_4_1C_3_2C_4_2C_4_3C_3_3C_7_1C_5_1C_7_2C_5_2C_5_3C_7_3C_8_1C_6_1C_8_2C_6_2C_6_3C_8_3C_13_1C_15_1C_13_2C_15_2C_13_3C_15_3C_14_1C_16_1C_14_2C_16_2C_14_3C_16_3TraceMulA_4_1A_4_2A_4_6A_4_7A_5_1A_5_2A_5_6A_5_7B_1_1B_1_2B_1_12B_1_4B_1_5B_1_6B_1_13B_1_14B_2_1B_2_2B_2_12B_2_4B_2_5B_2_6B_2_13B_2_14B_6_1B_6_2B_6_12B_6_4B_6_5B_6_6B_6_13B_6_14B_7_1B_7_2B_7_12B_7_4B_7_5B_7_6B_7_13B_7_14C_1_4C_1_5C_2_4C_2_5C_12_4C_12_5C_4_4C_4_5C_5_4C_5_5C_6_4C_6_5C_13_4C_13_5C_14_4C_14_5TraceMulA_1_1A_1_2A_1_6A_1_7A_2_1A_2_2A_2_6A_2_7A_3_1A_3_2A_3_6A_3_7B_1_1B_1_9B_1_2B_1_10B_1_12B_1_11B_1_4B_1_3B_1_5B_1_7B_1_6B_1_8B_1_13B_1_15B_1_14B_1_16B_3_1B_2_1B_3_9B_2_9B_3_2B_2_2B_3_10B_2_10B_3_12B_2_12B_3_11B_2_11B_3_4B_2_4B_3_3B_2_3B_3_5B_2_5B_3_7B_2_7B_3_6B_2_6B_3_8B_2_8B_3_13B_2_13B_3_15B_2_15B_3_14B_2_14B_3_16B_2_16B_4_1B_6_1B_4_9B_6_9B_4_2B_6_2B_4_10B_6_10B_4_12B_6_12B_4_11B_6_11B_4_4B_6_4B_4_3B_6_3B_4_5B_6_5B_4_7B_6_7B_4_6B_6_6B_4_8B_6_8B_4_13B_6_13B_4_15B_6_15B_4_14B_6_14B_4_16B_6_16B_5_1B_7_1B_5_9B_7_9B_5_2B_7_2B_5_10B_7_10B_5_12B_7_12B_5_11B_7_11B_5_4B_7_4B_5_3B_7_3B_5_5B_7_5B_5_7B_7_7B_5_6B_7_6B_5_8B_7_8B_5_13B_7_13B_5_15B_7_15B_5_14B_7_14B_5_16B_7_16C_1_1C_1_2C_1_3C_2_1C_2_2C_2_3C_12_1C_12_2C_12_3C_4_1C_4_2C_4_3C_5_1C_5_2C_5_3C_6_1C_6_2C_6_3C_13_1C_13_2C_13_3C_14_1C_14_2C_14_3TraceMulA_2_1A_4_1A_2_3A_4_3A_2_2A_4_2A_2_4A_4_4A_2_6A_4_6A_2_5A_4_5A_2_7A_4_7A_3_1A_5_1A_3_3A_5_3A_3_2A_5_2A_3_4A_5_4A_3_6A_5_6A_3_5A_5_5A_3_7A_5_7B_1_9B_1_10B_1_11B_1_3B_1_7B_1_8B_1_15B_1_16B_2_9B_2_10B_2_11B_2_3B_2_7B_2_8B_2_15B_2_16B_6_9B_6_10B_6_11B_6_3B_6_7B_6_8B_6_15B_6_16B_7_9B_7_10B_7_11B_7_3B_7_7B_7_8B_7_15B_7_16C_9_4C_9_5C_10_4C_10_5C_11_4C_11_5C_3_4C_3_5C_7_4C_7_5C_8_4C_8_5C_15_4C_15_5C_16_4C_16_5TraceMulA_4_3A_4_4A_4_5A_5_3A_5_4A_5_5B_3_1B_3_2B_3_12B_3_4B_3_5B_3_6B_3_13B_3_14B_4_1B_4_2B_4_12B_4_4B_4_5B_4_6B_4_13B_4_14B_5_1B_5_2B_5_12B_5_4B_5_5B_5_6B_5_13B_5_14C_1_2C_9_2C_1_4C_9_4C_1_3C_9_3C_1_5C_9_5C_2_2C_10_2C_2_4C_10_4C_2_3C_10_3C_2_5C_10_5C_12_2C_11_2C_12_4C_11_4C_12_3C_11_3C_12_5C_11_5C_4_2C_3_2C_4_4C_3_4C_4_3C_3_3C_4_5C_3_5C_5_2C_7_2C_5_4C_7_4C_5_3C_7_3C_5_5C_7_5C_6_2C_8_2C_6_4C_8_4C_6_3C_8_3C_6_5C_8_5C_13_2C_15_2C_13_4C_15_4C_13_3C_15_3C_13_5C_15_5C_14_2C_16_2C_14_4C_16_4C_14_3C_16_3C_14_5C_16_5TraceMulA_1_1A_1_3A_1_2A_1_4A_1_6A_1_5A_1_7A_2_1A_2_3A_4_3A_2_2A_2_4A_4_4A_2_6A_2_5A_4_5A_2_7A_3_1A_3_3A_5_3A_3_2A_3_4A_5_4A_3_6A_3_5A_5_5A_3_7B_1_9B_1_10B_1_11B_1_3B_1_7B_1_8B_1_15B_1_16B_3_1B_3_9B_2_9B_3_2B_3_10B_2_10B_3_12B_3_11B_2_11B_3_3B_3_4B_2_3B_3_5B_3_7B_2_7B_3_8B_3_6B_2_8B_3_13B_3_15B_2_15B_3_14B_3_16B_2_16B_4_1B_4_9B_6_9B_4_2B_4_10B_6_10B_4_12B_4_11B_6_11B_4_3B_4_4B_6_3B_4_7B_4_5B_6_7B_4_8B_4_6B_6_8B_4_13B_4_15B_6_15B_4_14B_4_16B_6_16B_5_1B_5_9B_7_9B_5_2B_5_10B_7_10B_5_12B_5_11B_7_11B_5_3B_5_4B_7_3B_5_7B_5_5B_7_7B_5_8B_5_6B_7_8B_5_13B_5_15B_7_15B_5_14B_5_16B_7_16C_1_1C_9_1C_1_2C_9_2C_9_4C_1_3C_9_3C_9_5C_2_1C_10_1C_2_2C_10_2C_10_4C_2_3C_10_3C_10_5C_12_1C_11_1C_12_2C_11_2C_11_4C_12_3C_11_3C_11_5C_3_1C_4_1C_3_2C_4_2C_3_4C_4_3C_3_3C_3_5C_7_1C_5_1C_7_2C_5_2C_7_4C_5_3C_7_3C_7_5C_8_1C_6_1C_8_2C_6_2C_8_4C_6_3C_8_3C_8_5C_13_1C_15_1C_13_2C_15_2C_15_4C_13_3C_15_3C_15_5C_14_1C_16_1C_14_2C_16_2C_16_4C_14_3C_16_3C_16_5Trace(Mul(Matrix(5, 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],[A_5_1,A_5_2,A_5_3,A_5_4,A_5_5,A_5_6,A_5_7]]),Matrix(7, 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]]),Matrix(16, 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],[C_16_1,C_16_2,C_16_3,C_16_4,C_16_5]]))) = Trace(Mul(Matrix(3, 3, [[A_1_3-A_1_2,A_1_4-A_1_6,A_1_5-A_1_7],[A_2_3-A_2_2,A_2_4-A_2_6,A_2_5-A_2_7],[A_3_3-A_3_2,A_3_4-A_3_6,A_3_5-A_3_7]]),Matrix(3, 8, [[-B_3_1+B_3_9,-B_3_2+B_3_10,-B_3_12+B_3_11,B_3_3-B_3_4,-B_3_5+B_3_7,B_3_8-B_3_6,-B_3_13+B_3_15,-B_3_14+B_3_16],[-B_4_1+B_4_9,-B_4_2+B_4_10,-B_4_12+B_4_11,B_4_3-B_4_4,B_4_7-B_4_5,B_4_8-B_4_6,-B_4_13+B_4_15,-B_4_14+B_4_16],[-B_5_1+B_5_9,-B_5_2+B_5_10,-B_5_12+B_5_11,B_5_3-B_5_4,B_5_7-B_5_5,B_5_8-B_5_6,-B_5_13+B_5_15,-B_5_14+B_5_16]]),Matrix(8, 3, [[C_9_1,C_9_2-C_9_4,C_9_3-C_9_5],[C_10_1,C_10_2-C_10_4,C_10_3-C_10_5],[C_11_1,C_11_2-C_11_4,C_11_3-C_11_5],[C_3_1,C_3_2-C_3_4,C_3_3-C_3_5],[C_7_1,C_7_2-C_7_4,C_7_3-C_7_5],[C_8_1,C_8_2-C_8_4,C_8_3-C_8_5],[C_15_1,C_15_2-C_15_4,C_15_3-C_15_5],[C_16_1,C_16_2-C_16_4,C_16_3-C_16_5]])))+Trace(Mul(Matrix(3, 3, [[A_1_3,A_1_4,A_1_5],[A_2_3+A_4_3,A_2_4+A_4_4,A_2_5+A_4_5],[A_3_3+A_5_3,A_3_4+A_5_4,A_3_5+A_5_5]]),Matrix(3, 8, [[B_3_9+B_2_9,B_3_10+B_2_10,B_3_11+B_2_11,B_3_3+B_2_3,B_3_7+B_2_7,B_3_8+B_2_8,B_3_15+B_2_15,B_3_16+B_2_16],[B_4_9+B_6_9,B_4_10+B_6_10,B_4_11+B_6_11,B_4_3+B_6_3,B_4_7+B_6_7,B_4_8+B_6_8,B_4_15+B_6_15,B_4_16+B_6_16],[B_5_9+B_7_9,B_5_10+B_7_10,B_5_11+B_7_11,B_5_3+B_7_3,B_5_7+B_7_7,B_5_8+B_7_8,B_5_15+B_7_15,B_5_16+B_7_16]]),Matrix(8, 3, [[C_1_1+C_9_1,C_1_2+C_9_2,C_1_3+C_9_3],[C_2_1+C_10_1,C_2_2+C_10_2,C_2_3+C_10_3],[C_12_1+C_11_1,C_12_2+C_11_2,C_12_3+C_11_3],[C_3_1+C_4_1,C_3_2+C_4_2,C_4_3+C_3_3],[C_7_1+C_5_1,C_7_2+C_5_2,C_5_3+C_7_3],[C_8_1+C_6_1,C_8_2+C_6_2,C_6_3+C_8_3],[C_13_1+C_15_1,C_13_2+C_15_2,C_13_3+C_15_3],[C_14_1+C_16_1,C_14_2+C_16_2,C_14_3+C_16_3]])))+Trace(Mul(Matrix(2, 4, [[A_4_1,A_4_2,A_4_6,A_4_7],[A_5_1,A_5_2,A_5_6,A_5_7]]),Matrix(4, 8, [[B_1_1,B_1_2,B_1_12,B_1_4,B_1_5,B_1_6,B_1_13,B_1_14],[B_2_1,B_2_2,B_2_12,B_2_4,B_2_5,B_2_6,B_2_13,B_2_14],[B_6_1,B_6_2,B_6_12,B_6_4,B_6_5,B_6_6,B_6_13,B_6_14],[B_7_1,B_7_2,B_7_12,B_7_4,B_7_5,B_7_6,B_7_13,B_7_14]]),Matrix(8, 2, [[C_1_4,C_1_5],[C_2_4,C_2_5],[C_12_4,C_12_5],[C_4_4,C_4_5],[C_5_4,C_5_5],[C_6_4,C_6_5],[C_13_4,C_13_5],[C_14_4,C_14_5]])))+Trace(Mul(Matrix(3, 4, [[A_1_1,A_1_2,A_1_6,A_1_7],[A_2_1,A_2_2,A_2_6,A_2_7],[A_3_1,A_3_2,A_3_6,A_3_7]]),Matrix(4, 8, [[B_1_1-B_1_9,B_1_2-B_1_10,B_1_12-B_1_11,B_1_4-B_1_3,B_1_5-B_1_7,B_1_6-B_1_8,B_1_13-B_1_15,B_1_14-B_1_16],[B_3_1+B_2_1-B_3_9-B_2_9,B_3_2+B_2_2-B_3_10-B_2_10,B_3_12+B_2_12-B_3_11-B_2_11,B_3_4+B_2_4-B_3_3-B_2_3,B_3_5+B_2_5-B_3_7-B_2_7,B_3_6+B_2_6-B_3_8-B_2_8,B_3_13+B_2_13-B_3_15-B_2_15,B_3_14+B_2_14-B_3_16-B_2_16],[B_4_1+B_6_1-B_4_9-B_6_9,B_4_2+B_6_2-B_4_10-B_6_10,B_4_12+B_6_12-B_4_11-B_6_11,B_4_4+B_6_4-B_4_3-B_6_3,B_4_5+B_6_5-B_4_7-B_6_7,B_4_6+B_6_6-B_4_8-B_6_8,B_4_13+B_6_13-B_4_15-B_6_15,B_4_14+B_6_14-B_4_16-B_6_16],[B_5_1+B_7_1-B_5_9-B_7_9,B_5_2+B_7_2-B_5_10-B_7_10,B_5_12+B_7_12-B_5_11-B_7_11,B_5_4+B_7_4-B_5_3-B_7_3,B_5_5+B_7_5-B_5_7-B_7_7,B_5_6+B_7_6-B_5_8-B_7_8,B_5_13+B_7_13-B_5_15-B_7_15,B_5_14+B_7_14-B_5_16-B_7_16]]),Matrix(8, 3, [[C_1_1,C_1_2,C_1_3],[C_2_1,C_2_2,C_2_3],[C_12_1,C_12_2,C_12_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_13_1,C_13_2,C_13_3],[C_14_1,C_14_2,C_14_3]])))+Trace(Mul(Matrix(2, 4, [[A_2_1+A_4_1,-A_2_3-A_4_3+A_2_2+A_4_2,-A_2_4-A_4_4+A_2_6+A_4_6,-A_2_5-A_4_5+A_2_7+A_4_7],[A_3_1+A_5_1,-A_3_3-A_5_3+A_3_2+A_5_2,-A_3_4-A_5_4+A_3_6+A_5_6,-A_3_5-A_5_5+A_3_7+A_5_7]]),Matrix(4, 8, [[B_1_9,B_1_10,B_1_11,B_1_3,B_1_7,B_1_8,B_1_15,B_1_16],[B_2_9,B_2_10,B_2_11,B_2_3,B_2_7,B_2_8,B_2_15,B_2_16],[B_6_9,B_6_10,B_6_11,B_6_3,B_6_7,B_6_8,B_6_15,B_6_16],[B_7_9,B_7_10,B_7_11,B_7_3,B_7_7,B_7_8,B_7_15,B_7_16]]),Matrix(8, 2, [[C_9_4,C_9_5],[C_10_4,C_10_5],[C_11_4,C_11_5],[C_3_4,C_3_5],[C_7_4,C_7_5],[C_8_4,C_8_5],[C_15_4,C_15_5],[C_16_4,C_16_5]])))+Trace(Mul(Matrix(2, 3, [[A_4_3,A_4_4,A_4_5],[A_5_3,A_5_4,A_5_5]]),Matrix(3, 8, [[B_3_1,B_3_2,B_3_12,B_3_4,B_3_5,B_3_6,B_3_13,B_3_14],[B_4_1,B_4_2,B_4_12,B_4_4,B_4_5,B_4_6,B_4_13,B_4_14],[B_5_1,B_5_2,B_5_12,B_5_4,B_5_5,B_5_6,B_5_13,B_5_14]]),Matrix(8, 2, [[-C_1_2-C_9_2+C_1_4+C_9_4,-C_1_3-C_9_3+C_1_5+C_9_5],[-C_2_2-C_10_2+C_2_4+C_10_4,-C_2_3-C_10_3+C_2_5+C_10_5],[-C_12_2-C_11_2+C_12_4+C_11_4,-C_12_3-C_11_3+C_12_5+C_11_5],[-C_4_2-C_3_2+C_4_4+C_3_4,-C_4_3-C_3_3+C_4_5+C_3_5],[-C_5_2-C_7_2+C_5_4+C_7_4,-C_5_3-C_7_3+C_5_5+C_7_5],[-C_6_2-C_8_2+C_6_4+C_8_4,-C_6_3-C_8_3+C_6_5+C_8_5],[-C_13_2-C_15_2+C_13_4+C_15_4,-C_13_3-C_15_3+C_13_5+C_15_5],[-C_14_2-C_16_2+C_14_4+C_16_4,-C_14_3-C_16_3+C_14_5+C_16_5]])))+Trace(Mul(Matrix(3, 4, [[A_1_1,-A_1_3+A_1_2,-A_1_4+A_1_6,-A_1_5+A_1_7],[A_2_1,-A_2_3-A_4_3+A_2_2,-A_2_4-A_4_4+A_2_6,-A_2_5-A_4_5+A_2_7],[A_3_1,-A_3_3-A_5_3+A_3_2,-A_3_4-A_5_4+A_3_6,-A_3_5-A_5_5+A_3_7]]),Matrix(4, 8, [[B_1_9,B_1_10,B_1_11,B_1_3,B_1_7,B_1_8,B_1_15,B_1_16],[-B_3_1+B_3_9+B_2_9,-B_3_2+B_3_10+B_2_10,-B_3_12+B_3_11+B_2_11,B_3_3-B_3_4+B_2_3,-B_3_5+B_3_7+B_2_7,B_3_8-B_3_6+B_2_8,-B_3_13+B_3_15+B_2_15,-B_3_14+B_3_16+B_2_16],[-B_4_1+B_4_9+B_6_9,-B_4_2+B_4_10+B_6_10,-B_4_12+B_4_11+B_6_11,B_4_3-B_4_4+B_6_3,B_4_7-B_4_5+B_6_7,B_4_8-B_4_6+B_6_8,-B_4_13+B_4_15+B_6_15,-B_4_14+B_4_16+B_6_16],[-B_5_1+B_5_9+B_7_9,-B_5_2+B_5_10+B_7_10,-B_5_12+B_5_11+B_7_11,B_5_3-B_5_4+B_7_3,B_5_7-B_5_5+B_7_7,B_5_8-B_5_6+B_7_8,-B_5_13+B_5_15+B_7_15,-B_5_14+B_5_16+B_7_16]]),Matrix(8, 3, [[C_1_1+C_9_1,C_1_2+C_9_2-C_9_4,C_1_3+C_9_3-C_9_5],[C_2_1+C_10_1,C_2_2+C_10_2-C_10_4,C_2_3+C_10_3-C_10_5],[C_12_1+C_11_1,C_12_2+C_11_2-C_11_4,C_12_3+C_11_3-C_11_5],[C_3_1+C_4_1,C_3_2+C_4_2-C_3_4,C_4_3+C_3_3-C_3_5],[C_7_1+C_5_1,C_7_2+C_5_2-C_7_4,C_5_3+C_7_3-C_7_5],[C_8_1+C_6_1,C_8_2+C_6_2-C_8_4,C_6_3+C_8_3-C_8_5],[C_13_1+C_15_1,C_13_2+C_15_2-C_15_4,C_13_3+C_15_3-C_15_5],[C_14_1+C_16_1,C_14_2+C_16_2-C_16_4,C_14_3+C_16_3-C_16_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.


Back to main table