Description of fast matrix multiplication algorithm: ⟨5×15×25:1270⟩

Algorithm type

2XY12Z+25X4Y4Z4+5X2Y8Z2+5X5Y4Z2+5X2Y8Z+5XY9Z+5X4Y4Z2+X3Y6Z+60X3Y4Z3+32X2Y6Z2+5X2Y4Z4+6XY8Z+XY6Z3+5XY4Z5+15X5Y2Z2+2X2Y6Z+20X2Y2Z5+2XY6Z2+10X5Y2Z+8X3Y4Z+131X2Y4Z2+38XY6Z+8XY4Z3+5XY2Z5+5X3Y3Z+5X3Y2Z2+6X2Y4Z+11XY4Z2+5XY3Z3+21X3Y2Z+X2Y3Z+187X2Y2Z2+91XY4Z+XY3Z2+26XY2Z3+30X3YZ+20X2Y2Z+53XY3Z+20XY2Z2+30XYZ3+6X2YZ+207XY2Z+6XYZ2+138XYZ2XY12Z25X4Y4Z45X2Y8Z25X5Y4Z25X2Y8Z5XY9Z5X4Y4Z2X3Y6Z60X3Y4Z332X2Y6Z25X2Y4Z46XY8ZXY6Z35XY4Z515X5Y2Z22X2Y6Z20X2Y2Z52XY6Z210X5Y2Z8X3Y4Z131X2Y4Z238XY6Z8XY4Z35XY2Z55X3Y3Z5X3Y2Z26X2Y4Z11XY4Z25XY3Z321X3Y2ZX2Y3Z187X2Y2Z291XY4ZXY3Z226XY2Z330X3YZ20X2Y2Z53XY3Z20XY2Z230XYZ36X2YZ207XY2Z6XYZ2138XYZ2*X*Y^12*Z+25*X^4*Y^4*Z^4+5*X^2*Y^8*Z^2+5*X^5*Y^4*Z^2+5*X^2*Y^8*Z+5*X*Y^9*Z+5*X^4*Y^4*Z^2+X^3*Y^6*Z+60*X^3*Y^4*Z^3+32*X^2*Y^6*Z^2+5*X^2*Y^4*Z^4+6*X*Y^8*Z+X*Y^6*Z^3+5*X*Y^4*Z^5+15*X^5*Y^2*Z^2+2*X^2*Y^6*Z+20*X^2*Y^2*Z^5+2*X*Y^6*Z^2+10*X^5*Y^2*Z+8*X^3*Y^4*Z+131*X^2*Y^4*Z^2+38*X*Y^6*Z+8*X*Y^4*Z^3+5*X*Y^2*Z^5+5*X^3*Y^3*Z+5*X^3*Y^2*Z^2+6*X^2*Y^4*Z+11*X*Y^4*Z^2+5*X*Y^3*Z^3+21*X^3*Y^2*Z+X^2*Y^3*Z+187*X^2*Y^2*Z^2+91*X*Y^4*Z+X*Y^3*Z^2+26*X*Y^2*Z^3+30*X^3*Y*Z+20*X^2*Y^2*Z+53*X*Y^3*Z+20*X*Y^2*Z^2+30*X*Y*Z^3+6*X^2*Y*Z+207*X*Y^2*Z+6*X*Y*Z^2+138*X*Y*Z

Algorithm definition

The algorithm ⟨5×15×25:1270⟩ could be constructed using the following decomposition:

⟨5×15×25:1270⟩ = ⟨3×5×5:58⟩ + ⟨2×5×5:40⟩ + ⟨3×5×5:58⟩ + ⟨2×5×5:40⟩ + ⟨2×5×5:40⟩ + ⟨3×5×5:58⟩ + ⟨3×5×5:58⟩ + ⟨3×5×5:58⟩ + ⟨3×5×5:58⟩ + ⟨2×5×5:40⟩ + ⟨2×5×5:40⟩ + ⟨3×5×5:58⟩ + ⟨2×5×5:40⟩ + ⟨2×5×5:40⟩ + ⟨3×5×5:58⟩ + ⟨2×5×5:40⟩ + ⟨2×5×5:40⟩ + ⟨3×5×5:58⟩ + ⟨3×5×5:58⟩ + ⟨3×5×5:58⟩ + ⟨3×5×5:58⟩ + ⟨3×5×5:58⟩ + ⟨3×5×5:58⟩ + ⟨2×5×5:40⟩ + ⟨3×5×5:58⟩.

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_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_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_4_1A_4_2A_4_3A_4_4A_4_5A_4_6A_4_7A_4_8A_4_9A_4_10A_4_11A_4_12A_4_13A_4_14A_4_15A_5_1A_5_2A_5_3A_5_4A_5_5A_5_6A_5_7A_5_8A_5_9A_5_10A_5_11A_5_12A_5_13A_5_14A_5_15B_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_15_17B_15_18B_15_19B_15_20B_15_21B_15_22B_15_23B_15_24B_15_25C_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_5C_17_1C_17_2C_17_3C_17_4C_17_5C_18_1C_18_2C_18_3C_18_4C_18_5C_19_1C_19_2C_19_3C_19_4C_19_5C_20_1C_20_2C_20_3C_20_4C_20_5C_21_1C_21_2C_21_3C_21_4C_21_5C_22_1C_22_2C_22_3C_22_4C_22_5C_23_1C_23_2C_23_3C_23_4C_23_5C_24_1C_24_2C_24_3C_24_4C_24_5C_25_1C_25_2C_25_3C_25_4C_25_5=TraceMulA_3_11A_3_12A_3_13A_3_14A_3_15A_4_11A_4_12A_4_13A_4_14A_4_15A_5_11A_5_12A_5_13A_5_14A_5_15-a21+a31+1B_1_11a11-B_6_11+B_11_11-a21+a31+1B_1_12a11-B_6_12+B_11_12-a21+a31+1B_1_13a11-B_6_13+B_11_13-a21+a31+1B_1_14a11-B_6_14+B_11_14-a21+a31+1B_1_15a11-B_6_15+B_11_15-a21+a31+1B_2_11a11-B_7_11+B_12_11-a21+a31+1B_2_12a11-B_7_12+B_12_12-a21+a31+1B_2_13a11-B_7_13+B_12_13-a21+a31+1B_2_14a11-B_7_14+B_12_14-a21+a31+1B_2_15a11-B_7_15+B_12_15-a21+a31+1B_3_11a11-B_8_11+B_13_11-a21+a31+1B_3_12a11-B_8_12+B_13_12-a21+a31+1B_3_13a11-B_8_13+B_13_13-a21+a31+1B_3_14a11-B_8_14+B_13_14-a21+a31+1B_3_15a11-B_8_15+B_13_15-a21+a31+1B_4_11a11-B_9_11+B_14_11-a21+a31+1B_4_12a11-B_9_12+B_14_12-a21+a31+1B_4_13a11-B_9_13+B_14_13-a21+a31+1B_4_14a11-B_9_14+B_14_14-a21+a31+1B_4_15a11-B_9_15+B_14_15-a21+a31+1B_5_11a11-B_10_11+B_15_11-a21+a31+1B_5_12a11-B_10_12+B_15_12-a21+a31+1B_5_13a11-B_10_13+B_15_13-a21+a31+1B_5_14a11-B_10_14+B_15_14-a21+a31+1B_5_15a11-B_10_15+B_15_15C_11_3C_11_1+C_11_4C_11_2+C_11_5C_12_3C_12_1+C_12_4C_12_2+C_12_5C_13_3C_13_1+C_13_4C_13_2+C_13_5C_14_3C_14_1+C_14_4C_14_2+C_14_5C_15_3C_15_1+C_15_4C_15_2+C_15_5+TraceMulA_1_1+A_1_6A_1_2+A_1_7A_1_3+A_1_8A_1_4+A_1_9A_1_5+A_1_10A_2_1+A_2_6A_2_2+A_2_7A_2_3+A_2_8A_2_4+A_2_9A_2_5+A_2_10B_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_10B_4_6B_4_7B_4_8B_4_9B_4_10B_5_6B_5_7B_5_8B_5_9B_5_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_4_1A_1_2-A_4_2A_1_3-A_4_3A_1_4-A_4_4A_1_5-A_4_5A_2_1-A_5_1A_2_2-A_5_2A_2_3-A_5_3A_2_4-A_5_4A_2_5-A_5_5B_1_1-a12-1B_11_1a32-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