Description of fast matrix multiplication algorithm: ⟨4×23×27:1639⟩

Algorithm type

3X2Y14Z2+10X2Y12Z2+4XY14Z+2X2Y11Z2+27X4Y6Z4+10X2Y10Z2+18XY12Z+2X2Y9Z2+2XY11Z+56X4Y4Z4+20X2Y8Z2+30XY10Z+8X3Y4Z4+2XY9Z+2XY8Z2+X4Y2Z4+2X3Y3Z4+74X2Y6Z2+40XY8Z+2X4YZ4+4XY7Z+2XY6Z2+172X2Y4Z2+62XY6Z+60X2Y3Z2+32XY5Z+242X2Y2Z2+134XY4Z+8X2YZ2+42XY3Z+328XY2Z+238XYZ3X2Y14Z210X2Y12Z24XY14Z2X2Y11Z227X4Y6Z410X2Y10Z218XY12Z2X2Y9Z22XY11Z56X4Y4Z420X2Y8Z230XY10Z8X3Y4Z42XY9Z2XY8Z2X4Y2Z42X3Y3Z474X2Y6Z240XY8Z2X4YZ44XY7Z2XY6Z2172X2Y4Z262XY6Z60X2Y3Z232XY5Z242X2Y2Z2134XY4Z8X2YZ242XY3Z328XY2Z238XYZ3*X^2*Y^14*Z^2+10*X^2*Y^12*Z^2+4*X*Y^14*Z+2*X^2*Y^11*Z^2+27*X^4*Y^6*Z^4+10*X^2*Y^10*Z^2+18*X*Y^12*Z+2*X^2*Y^9*Z^2+2*X*Y^11*Z+56*X^4*Y^4*Z^4+20*X^2*Y^8*Z^2+30*X*Y^10*Z+8*X^3*Y^4*Z^4+2*X*Y^9*Z+2*X*Y^8*Z^2+X^4*Y^2*Z^4+2*X^3*Y^3*Z^4+74*X^2*Y^6*Z^2+40*X*Y^8*Z+2*X^4*Y*Z^4+4*X*Y^7*Z+2*X*Y^6*Z^2+172*X^2*Y^4*Z^2+62*X*Y^6*Z+60*X^2*Y^3*Z^2+32*X*Y^5*Z+242*X^2*Y^2*Z^2+134*X*Y^4*Z+8*X^2*Y*Z^2+42*X*Y^3*Z+328*X*Y^2*Z+238*X*Y*Z

Algorithm definition

The algorithm ⟨4×23×27:1639⟩ could be constructed using the following decomposition:

⟨4×23×27:1639⟩ = ⟨2×7×9:99⟩ + ⟨2×8×9:113⟩ + ⟨2×8×9:113⟩ + ⟨2×8×9:113⟩ + ⟨2×7×9:99⟩ + ⟨2×7×9:99⟩ + ⟨2×7×9:99⟩ + ⟨2×8×9:113⟩ + ⟨2×8×9:113⟩ + ⟨2×8×9:113⟩ + ⟨2×8×9:113⟩ + ⟨2×8×9:113⟩ + ⟨2×8×9:113⟩ + ⟨2×8×9:113⟩ + ⟨2×8×9:113⟩.

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_1_17A_1_18A_1_19A_1_20A_1_21A_1_22A_1_23A_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_2_17A_2_18A_2_19A_2_20A_2_21A_2_22A_2_23A_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_16A_3_17A_3_18A_3_19A_3_20A_3_21A_3_22A_3_23A_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_4_16A_4_17A_4_18A_4_19A_4_20A_4_21A_4_22A_4_23B_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_1_26B_1_27B_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_2_26B_2_27B_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_3_26B_3_27B_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_4_26B_4_27B_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_5_26B_5_27B_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_6_26B_6_27B_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_7_26B_7_27B_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_8_26B_8_27B_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_9_26B_9_27B_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_10_26B_10_27B_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_11_26B_11_27B_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_12_26B_12_27B_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_13_26B_13_27B_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_14_26B_14_27B_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_25B_15_26B_15_27B_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_16B_16_17B_16_18B_16_19B_16_20B_16_21B_16_22B_16_23B_16_24B_16_25B_16_26B_16_27B_17_1B_17_2B_17_3B_17_4B_17_5B_17_6B_17_7B_17_8B_17_9B_17_10B_17_11B_17_12B_17_13B_17_14B_17_15B_17_16B_17_17B_17_18B_17_19B_17_20B_17_21B_17_22B_17_23B_17_24B_17_25B_17_26B_17_27B_18_1B_18_2B_18_3B_18_4B_18_5B_18_6B_18_7B_18_8B_18_9B_18_10B_18_11B_18_12B_18_13B_18_14B_18_15B_18_16B_18_17B_18_18B_18_19B_18_20B_18_21B_18_22B_18_23B_18_24B_18_25B_18_26B_18_27B_19_1B_19_2B_19_3B_19_4B_19_5B_19_6B_19_7B_19_8B_19_9B_19_10B_19_11B_19_12B_19_13B_19_14B_19_15B_19_16B_19_17B_19_18B_19_19B_19_20B_19_21B_19_22B_19_23B_19_24B_19_25B_19_26B_19_27B_20_1B_20_2B_20_3B_20_4B_20_5B_20_6B_20_7B_20_8B_20_9B_20_10B_20_11B_20_12B_20_13B_20_14B_20_15B_20_16B_20_17B_20_18B_20_19B_20_20B_20_21B_20_22B_20_23B_20_24B_20_25B_20_26B_20_27B_21_1B_21_2B_21_3B_21_4B_21_5B_21_6B_21_7B_21_8B_21_9B_21_10B_21_11B_21_12B_21_13B_21_14B_21_15B_21_16B_21_17B_21_18B_21_19B_21_20B_21_21B_21_22B_21_23B_21_24B_21_25B_21_26B_21_27B_22_1B_22_2B_22_3B_22_4B_22_5B_22_6B_22_7B_22_8B_22_9B_22_10B_22_11B_22_12B_22_13B_22_14B_22_15B_22_16B_22_17B_22_18B_22_19B_22_20B_22_21B_22_22B_22_23B_22_24B_22_25B_22_26B_22_27B_23_1B_23_2B_23_3B_23_4B_23_5B_23_6B_23_7B_23_8B_23_9B_23_10B_23_11B_23_12B_23_13B_23_14B_23_15B_23_16B_23_17B_23_18B_23_19B_23_20B_23_21B_23_22B_23_23B_23_24B_23_25B_23_26B_23_27C_1_1C_1_2C_1_3C_1_4C_2_1C_2_2C_2_3C_2_4C_3_1C_3_2C_3_3C_3_4C_4_1C_4_2C_4_3C_4_4C_5_1C_5_2C_5_3C_5_4C_6_1C_6_2C_6_3C_6_4C_7_1C_7_2C_7_3C_7_4C_8_1C_8_2C_8_3C_8_4C_9_1C_9_2C_9_3C_9_4C_10_1C_10_2C_10_3C_10_4C_11_1C_11_2C_11_3C_11_4C_12_1C_12_2C_12_3C_12_4C_13_1C_13_2C_13_3C_13_4C_14_1C_14_2C_14_3C_14_4C_15_1C_15_2C_15_3C_15_4C_16_1C_16_2C_16_3C_16_4C_17_1C_17_2C_17_3C_17_4C_18_1C_18_2C_18_3C_18_4C_19_1C_19_2C_19_3C_19_4C_20_1C_20_2C_20_3C_20_4C_21_1C_21_2C_21_3C_21_4C_22_1C_22_2C_22_3C_22_4C_23_1C_23_2C_23_3C_23_4C_24_1C_24_2C_24_3C_24_4C_25_1C_25_2C_25_3C_25_4C_26_1C_26_2C_2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23, [[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_1_17,A_1_18,A_1_19,A_1_20,A_1_21,A_1_22,A_1_23],[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_2_17,A_2_18,A_2_19,A_2_20,A_2_21,A_2_22,A_2_23],[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,A_3_17,A_3_18,A_3_19,A_3_20,A_3_21,A_3_22,A_3_23],[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_4_10,A_4_11,A_4_12,A_4_13,A_4_14,A_4_15,A_4_16,A_4_17,A_4_18,A_4_19,A_4_20,A_4_21,A_4_22,A_4_23]]),Matrix(23, 27, [[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_1_17,B_1_18,B_1_19,B_1_20,B_1_21,B_1_22,B_1_23,B_1_24,B_1_25,B_1_26,B_1_27],[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_2_17,B_2_18,B_2_19,B_2_20,B_2_21,B_2_22,B_2_23,B_2_24,B_2_25,B_2_26,B_2_27],[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_3_17,B_3_18,B_3_19,B_3_20,B_3_21,B_3_22,B_3_23,B_3_24,B_3_25,B_3_26,B_3_27],[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_4_17,B_4_18,B_4_19,B_4_20,B_4_21,B_4_22,B_4_23,B_4_24,B_4_25,B_4_26,B_4_27],[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_5_17,B_5_18,B_5_19,B_5_20,B_5_21,B_5_22,B_5_23,B_5_24,B_5_25,B_5_26,B_5_27],[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_6_17,B_6_18,B_6_19,B_6_20,B_6_21,B_6_22,B_6_23,B_6_24,B_6_25,B_6_26,B_6_27],[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_7_17,B_7_18,B_7_19,B_7_20,B_7_21,B_7_22,B_7_23,B_7_24,B_7_25,B_7_26,B_7_27],[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_8_17,B_8_18,B_8_19,B_8_20,B_8_21,B_8_22,B_8_23,B_8_24,B_8_25,B_8_26,B_8_27],[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_9_17,B_9_18,B_9_19,B_9_20,B_9_21,B_9_22,B_9_23,B_9_24,B_9_25,B_9_26,B_9_27],[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_10_17,B_10_18,B_10_19,B_10_20,B_10_21,B_10_22,B_10_23,B_10_24,B_10_25,B_10_26,B_10_27],[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_11_17,B_11_18,B_11_19,B_11_20,B_11_21,B_11_22,B_11_23,B_11_24,B_11_25,B_11_26,B_11_27],[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_12_17,B_12_18,B_12_19,B_12_20,B_12_21,B_12_22,B_12_23,B_12_24,B_12_25,B_12_26,B_12_27],[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_13_17,B_13_18,B_13_19,B_13_20,B_13_21,B_13_22,B_13_23,B_13_24,B_13_25,B_13_26,B_13_27],[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_14_17,B_14_18,B_14_19,B_14_20,B_14_21,B_14_22,B_14_23,B_14_24,B_14_25,B_14_26,B_14_27],[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_15_17,B_15_18,B_15_19,B_15_20,B_15_21,B_15_22,B_15_23,B_15_24,B_15_25,B_15_26,B_15_27],[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,B_16_17,B_16_18,B_16_19,B_16_20,B_16_21,B_16_22,B_16_23,B_16_24,B_16_25,B_16_26,B_16_27],[B_17_1,B_17_2,B_17_3,B_17_4,B_17_5,B_17_6,B_17_7,B_17_8,B_17_9,B_17_10,B_17_11,B_17_12,B_17_13,B_17_14,B_17_15,B_17_16,B_17_17,B_17_18,B_17_19,B_17_20,B_17_21,B_17_22,B_17_23,B_17_24,B_17_25,B_17_26,B_17_27],[B_18_1,B_18_2,B_18_3,B_18_4,B_18_5,B_18_6,B_18_7,B_18_8,B_18_9,B_18_10,B_18_11,B_18_12,B_18_13,B_18_14,B_18_15,B_18_16,B_18_17,B_18_18,B_18_19,B_18_20,B_18_21,B_18_22,B_18_23,B_18_24,B_18_25,B_18_26,B_18_27],[B_19_1,B_19_2,B_19_3,B_19_4,B_19_5,B_19_6,B_19_7,B_19_8,B_19_9,B_19_10,B_19_11,B_19_12,B_19_13,B_19_14,B_19_15,B_19_16,B_19_17,B_19_18,B_19_19,B_19_20,B_19_21,B_19_22,B_19_23,B_19_24,B_19_25,B_19_26,B_19_27],[B_20_1,B_20_2,B_20_3,B_20_4,B_20_5,B_20_6,B_20_7,B_20_8,B_20_9,B_20_10,B_20_11,B_20_12,B_20_13,B_20_14,B_20_15,B_20_16,B_20_17,B_20_18,B_20_19,B_20_20,B_20_21,B_20_22,B_20_23,B_20_24,B_20_25,B_20_26,B_20_27],[B_21_1,B_21_2,B_21_3,B_21_4,B_21_5,B_21_6,B_21_7,B_21_8,B_21_9,B_21_10,B_21_11,B_21_12,B_21_13,B_21_14,B_21_15,B_21_16,B_21_17,B_21_18,B_21_19,B_21_20,B_21_21,B_21_22,B_21_23,B_21_24,B_21_25,B_21_26,B_21_27],[B_22_1,B_22_2,B_22_3,B_22_4,B_22_5,B_22_6,B_22_7,B_22_8,B_22_9,B_22_10,B_22_11,B_22_12,B_22_13,B_22_14,B_22_15,B_22_16,B_22_17,B_22_18,B_22_19,B_22_20,B_22_21,B_22_22,B_22_23,B_22_24,B_22_25,B_22_26,B_22_27],[B_23_1,B_23_2,B_23_3,B_23_4,B_23_5,B_23_6,B_23_7,B_23_8,B_23_9,B_23_10,B_23_11,B_23_12,B_23_13,B_23_14,B_23_15,B_23_16,B_23_17,B_23_18,B_23_19,B_23_20,B_23_21,B_23_22,B_23_23,B_23_24,B_23_25,B_23_26,B_23_27]]),Matrix(27, 4, [[C_1_1,C_1_2,C_1_3,C_1_4],[C_2_1,C_2_2,C_2_3,C_2_4],[C_3_1,C_3_2,C_3_3,C_3_4],[C_4_1,C_4_2,C_4_3,C_4_4],[C_5_1,C_5_2,C_5_3,C_5_4],[C_6_1,C_6_2,C_6_3,C_6_4],[C_7_1,C_7_2,C_7_3,C_7_4],[C_8_1,C_8_2,C_8_3,C_8_4],[C_9_1,C_9_2,C_9_3,C_9_4],[C_10_1,C_10_2,C_10_3,C_10_4],[C_11_1,C_11_2,C_11_3,C_11_4],[C_12_1,C_12_2,C_12_3,C_12_4],[C_13_1,C_13_2,C_13_3,C_13_4],[C_14_1,C_14_2,C_14_3,C_14_4],[C_15_1,C_15_2,C_15_3,C_15_4],[C_16_1,C_16_2,C_16_3,C_16_4],[C_17_1,C_17_2,C_17_3,C_17_4],[C_18_1,C_18_2,C_18_3,C_18_4],[C_19_1,C_19_2,C_19_3,C_19_4],[C_20_1,C_20_2,C_20_3,C_20_4],[C_21_1,C_21_2,C_21_3,C_21_4],[C_22_1,C_22_2,C_22_3,C_22_4],[C_23_1,C_23_2,C_23_3,C_23_4],[C_24_1,C_24_2,C_24_3,C_24_4],[C_25_1,C_25_2,C_25_3,C_25_4],[C_26_1,C_26_2,C_26_3,C_26_4],[C_27_1,C_27_2,C_27_3,C_27_4]]))) = Trace(Mul(Matrix(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]]),Matrix(7, 9, [[B_1_1-B_9_1-B_1_19,B_1_2-B_9_2-B_1_20,B_1_3-B_9_3-B_1_21,B_1_4-B_9_4-B_1_22,B_1_5-B_9_5-B_1_23,B_1_6-B_9_6-B_1_24,B_1_7-B_9_7-B_1_25,B_1_8-B_9_8-B_1_26,B_1_9-B_9_9-B_1_27],[B_2_1-B_10_1-B_2_19,B_2_2-B_10_2-B_2_20,B_2_3-B_10_3-B_2_21,B_2_4-B_10_4-B_2_22,B_2_5-B_10_5-B_2_23,B_2_6-B_10_6-B_2_24,B_2_7-B_10_7-B_2_25,B_2_8-B_10_8-B_2_26,B_2_9-B_10_9-B_2_27],[B_3_1-B_11_1-B_3_19,B_3_2-B_11_2-B_3_20,B_3_3-B_11_3-B_3_21,B_3_4-B_11_4-B_3_22,B_3_5-B_11_5-B_3_23,B_3_6-B_11_6-B_3_24,B_3_7-B_11_7-B_3_25,B_3_8-B_11_8-B_3_26,B_3_9-B_11_9-B_3_27],[B_4_1-B_12_1-B_4_19,B_4_2-B_12_2-B_4_20,B_4_3-B_12_3-B_4_21,B_4_4-B_12_4-B_4_22,B_4_5-B_12_5-B_4_23,B_4_6-B_12_6-B_4_24,B_4_7-B_12_7-B_4_25,B_4_8-B_12_8-B_4_26,B_4_9-B_12_9-B_4_27],[B_5_1-B_13_1-B_5_19,B_5_2-B_13_2-B_5_20,B_5_3-B_13_3-B_5_21,B_5_4-B_13_4-B_5_22,B_5_5-B_13_5-B_5_23,B_5_6-B_13_6-B_5_24,B_5_7-B_13_7-B_5_25,B_5_8-B_13_8-B_5_26,B_5_9-B_13_9-B_5_27],[B_6_1-B_14_1-B_6_19,B_6_2-B_14_2-B_6_20,B_6_3-B_14_3-B_6_21,B_6_4-B_14_4-B_6_22,B_6_5-B_14_5-B_6_23,B_6_6-B_14_6-B_6_24,B_6_7-B_14_7-B_6_25,B_6_8-B_14_8-B_6_26,B_6_9-B_14_9-B_6_27],[B_7_1-B_15_1-B_7_19,B_7_2-B_15_2-B_7_20,B_7_3-B_15_3-B_7_21,B_7_4-B_15_4-B_7_22,B_7_5-B_15_5-B_7_23,B_7_6-B_15_6-B_7_24,B_7_7-B_15_7-B_7_25,B_7_8-B_15_8-B_7_26,B_7_9-B_15_9-B_7_27]]),Matrix(9, 2, [[C_1_1+C_1_3,C_1_2+C_1_4],[C_2_1+C_2_3,C_2_2+C_2_4],[C_3_1+C_3_3,C_3_2+C_3_4],[C_4_1+C_4_3,C_4_2+C_4_4],[C_5_1+C_5_3,C_5_4+C_5_2],[C_6_1+C_6_3,C_6_2+C_6_4],[C_7_1+C_7_3,C_7_2+C_7_4],[C_8_1+C_8_3,C_8_2+C_8_4],[C_9_1+C_9_3,C_9_2+C_9_4]])))+Trace(Mul(Matrix(2, 8, [[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_2_8,A_2_9,A_2_10,A_2_11,A_2_12,A_2_13,A_2_14,A_2_15]]),Matrix(8, 9, [[B_8_10-B_8_19,B_8_11-B_8_20,B_8_12-B_8_21,B_8_13-B_8_22,B_8_14-B_8_23,B_8_15-B_8_24,B_8_16-B_8_25,B_8_17-B_8_26,B_8_18-B_8_27],[-B_1_10+B_9_10-B_9_19,-B_1_11+B_9_11-B_9_20,-B_1_12+B_9_12-B_9_21,-B_1_13+B_9_13-B_9_22,-B_1_14+B_9_14-B_9_23,-B_1_15+B_9_15-B_9_24,-B_1_16+B_9_16-B_9_25,-B_1_17+B_9_17-B_9_26,-B_1_18+B_9_18-B_9_27],[-B_2_10+B_10_10-B_10_19,-B_2_11+B_10_11-B_10_20,-B_2_12+B_10_12-B_10_21,-B_2_13+B_10_13-B_10_22,-B_2_14+B_10_14-B_10_23,-B_2_15+B_10_15-B_10_24,-B_2_16+B_10_16-B_10_25,-B_2_17+B_10_17-B_10_26,-B_2_18+B_10_18-B_10_27],[-B_3_10+B_11_10-B_11_19,-B_3_11+B_11_11-B_11_20,-B_3_12+B_11_12-B_11_21,-B_3_13+B_11_13-B_11_22,-B_3_14+B_11_14-B_11_23,-B_3_15+B_11_15-B_11_24,-B_3_16+B_11_16-B_11_25,-B_3_17+B_11_17-B_11_26,-B_3_18+B_11_18-B_11_27],[-B_4_10+B_12_10-B_12_19,-B_4_11+B_12_11-B_12_20,-B_4_12+B_12_12-B_12_21,-B_4_13+B_12_13-B_12_22,-B_4_14+B_12_14-B_12_23,-B_4_15+B_12_15-B_12_24,-B_4_16+B_12_16-B_12_25,-B_4_17+B_12_17-B_12_26,-B_4_18+B_12_18-B_12_27],[-B_5_10+B_13_10-B_13_19,-B_5_11+B_13_11-B_13_20,-B_5_12+B_13_12-B_13_21,-B_5_13+B_13_13-B_13_22,-B_5_14+B_13_14-B_13_23,-B_5_15+B_13_15-B_13_24,-B_5_16+B_13_16-B_13_25,-B_5_17+B_13_17-B_13_26,-B_5_18+B_13_18-B_13_27],[-B_6_10+B_14_10-B_14_19,-B_6_11+B_14_11-B_14_20,-B_6_12+B_14_12-B_14_21,-B_6_13+B_14_13-B_14_22,-B_6_14+B_14_14-B_14_23,-B_6_15+B_14_15-B_14_24,-B_6_16+B_14_16-B_14_25,-B_6_17+B_14_17-B_14_26,-B_6_18+B_14_18-B_14_27],[-B_7_10+B_15_10-B_15_19,-B_7_11+B_15_11-B_15_20,-B_7_12+B_15_12-B_15_21,-B_7_13+B_15_13-B_15_22,-B_7_14+B_15_14-B_15_23,-B_7_15+B_15_15-B_15_24,-B_7_16+B_15_16-B_15_25,-B_7_17+B_15_17-B_15_26,-B_7_18+B_15_18-B_15_27]]),Matrix(9, 2, [[C_10_1+C_10_3,C_10_2+C_10_4],[C_11_1+C_11_3,C_11_2+C_11_4],[C_12_1+C_12_3,C_12_2+C_12_4],[C_13_1+C_13_3,C_13_2+C_13_4],[C_14_1+C_14_3,C_14_2+C_14_4],[C_15_1+C_15_3,C_15_2+C_15_4],[C_16_1+C_16_3,C_16_2+C_16_4],[C_17_1+C_17_3,C_17_2+C_17_4],[C_18_1+C_18_3,C_18_2+C_18_4]])))+Trace(Mul(Matrix(2, 8, [[A_1_16,A_1_17,A_1_18,A_1_19,A_1_20,A_1_21,A_1_22,A_1_23],[A_2_16,A_2_17,A_2_18,A_2_19,A_2_20,A_2_21,A_2_22,A_2_23]]),Matrix(8, 9, [[-B_16_10+B_16_19,-B_16_11+B_16_20,-B_16_12+B_16_21,-B_16_13+B_16_22,-B_16_14+B_16_23,-B_16_15+B_16_24,-B_16_16+B_16_25,-B_16_17+B_16_26,-B_16_18+B_16_27],[-B_17_10-B_1_19+B_17_19,-B_17_11-B_1_20+B_17_20,-B_17_12-B_1_21+B_17_21,-B_17_13-B_1_22+B_17_22,-B_17_14-B_1_23+B_17_23,-B_17_15-B_1_24+B_17_24,-B_17_16-B_1_25+B_17_25,-B_17_17-B_1_26+B_17_26,-B_17_18-B_1_27+B_17_27],[-B_18_10-B_2_19+B_18_19,-B_18_11-B_2_20+B_18_20,-B_18_12-B_2_21+B_18_21,-B_18_13-B_2_22+B_18_22,-B_18_14-B_2_23+B_18_23,-B_18_15-B_2_24+B_18_24,-B_18_16-B_2_25+B_18_25,-B_18_17-B_2_26+B_18_26,-B_18_18-B_2_27+B_18_27],[-B_19_10-B_3_19+B_19_19,-B_19_11-B_3_20+B_19_20,-B_19_12-B_3_21+B_19_21,-B_19_13-B_3_22+B_19_22,-B_19_14-B_3_23+B_19_23,-B_19_15-B_3_24+B_19_24,-B_19_16-B_3_25+B_19_25,-B_19_17-B_3_26+B_19_26,-B_19_18-B_3_27+B_19_27],[-B_20_10-B_4_19+B_20_19,-B_20_11-B_4_20+B_20_20,-B_20_12-B_4_21+B_20_21,-B_20_13-B_4_22+B_20_22,-B_20_14-B_4_23+B_20_23,-B_20_15-B_4_24+B_20_24,-B_20_16-B_4_25+B_20_25,-B_20_17-B_4_26+B_20_26,-B_20_18-B_4_27+B_20_27],[-B_21_10-B_5_19+B_21_19,-B_21_11-B_5_20+B_21_20,-B_21_12-B_5_21+B_21_21,-B_21_13-B_5_22+B_21_22,-B_21_14-B_5_23+B_21_23,-B_21_15-B_5_24+B_21_24,-B_21_16-B_5_25+B_21_25,-B_21_17-B_5_26+B_21_26,-B_21_18-B_5_27+B_21_27],[-B_22_10-B_6_19+B_22_19,-B_22_11-B_6_20+B_22_20,-B_22_12-B_6_21+B_22_21,-B_22_13-B_6_22+B_22_22,-B_22_14-B_6_23+B_22_23,-B_22_15-B_6_24+B_22_24,-B_22_16-B_6_25+B_22_25,-B_22_17-B_6_26+B_22_26,-B_22_18-B_6_27+B_22_27],[-B_23_10-B_7_19+B_23_19,-B_23_11-B_7_20+B_23_20,-B_23_12-B_7_21+B_23_21,-B_23_13-B_7_22+B_23_22,-B_23_14-B_7_23+B_23_23,-B_23_15-B_7_24+B_23_24,-B_23_16-B_7_25+B_23_25,-B_23_17-B_7_26+B_23_26,-B_23_18-B_7_27+B_23_27]]),Matrix(9, 2, [[C_19_1,C_19_2],[C_20_1,C_20_2],[C_21_1,C_21_2],[C_22_1,C_22_2],[C_23_1,C_23_2],[C_24_1,C_24_2],[C_25_1,C_25_2],[C_26_1,C_26_2],[C_27_1,C_27_2]])))+Trace(Mul(Matrix(2, 8, [[A_3_16,A_3_17,A_3_18,A_3_19,A_3_20,A_3_21,A_3_22,A_3_23],[A_4_16,A_4_17,A_4_18,A_4_19,A_4_20,A_4_21,A_4_22,A_4_23]]),Matrix(8, 9, [[-B_16_1-B_8_19+B_16_19,-B_16_2-B_8_20+B_16_20,-B_16_3-B_8_21+B_16_21,-B_16_4-B_8_22+B_16_22,-B_16_5-B_8_23+B_16_23,-B_16_6-B_8_24+B_16_24,-B_16_7-B_8_25+B_16_25,-B_16_8-B_8_26+B_16_26,-B_16_9-B_8_27+B_16_27],[-B_17_1-B_9_19+B_17_19,-B_17_2-B_9_20+B_17_20,-B_17_3-B_9_21+B_17_21,-B_17_4-B_9_22+B_17_22,-B_17_5-B_9_23+B_17_23,-B_17_6-B_9_24+B_17_24,-B_17_7-B_9_25+B_17_25,-B_17_8-B_9_26+B_17_26,-B_17_9-B_9_27+B_17_27],[-B_18_1-B_10_19+B_18_19,-B_18_2-B_10_20+B_18_20,-B_18_3-B_10_21+B_18_21,-B_18_4-B_10_22+B_18_22,-B_18_5-B_10_23+B_18_23,-B_18_6-B_10_24+B_18_24,-B_18_7-B_10_25+B_18_25,-B_18_8-B_10_26+B_18_26,-B_18_9-B_10_27+B_18_27],[-B_19_1-B_11_19+B_19_19,-B_19_2-B_11_20+B_19_20,-B_19_3-B_11_21+B_19_21,-B_19_4-B_11_22+B_19_22,-B_19_5-B_11_23+B_19_23,-B_19_6-B_11_24+B_19_24,-B_19_7-B_11_25+B_19_25,-B_19_8-B_11_26+B_19_26,-B_19_9-B_11_27+B_19_27],[-B_20_1-B_12_19+B_20_19,-B_20_2-B_12_20+B_20_20,-B_20_3-B_12_21+B_20_21,-B_20_4-B_12_22+B_20_22,-B_20_5-B_12_23+B_20_23,-B_20_6-B_12_24+B_20_24,-B_20_7-B_12_25+B_20_25,-B_20_8-B_12_26+B_20_26,-B_20_9-B_12_27+B_20_27],[-B_21_1-B_13_19+B_21_19,-B_21_2-B_13_20+B_21_20,-B_21_3-B_13_21+B_21_21,-B_21_4-B_13_22+B_21_22,-B_21_5-B_13_23+B_21_23,-B_21_6-B_13_24+B_21_24,-B_21_7-B_13_25+B_21_25,-B_21_8-B_13_26+B_21_26,-B_21_9-B_13_27+B_21_27],[-B_22_1-B_14_19+B_22_19,-B_22_2-B_14_20+B_22_20,-B_22_3-B_14_21+B_22_21,-B_22_4-B_14_22+B_22_22,-B_22_5-B_14_23+B_22_23,-B_22_6-B_14_24+B_22_24,-B_22_7-B_14_25+B_22_25,-B_22_8-B_14_26+B_22_26,-B_22_9-B_14_27+B_22_27],[-B_23_1-B_15_19+B_23_19,-B_23_2-B_15_20+B_23_20,-B_23_3-B_15_21+B_23_21,-B_23_4-B_15_22+B_23_22,-B_23_5-B_15_23+B_23_23,-B_23_6-B_15_24+B_23_24,-B_23_7-B_15_25+B_23_25,-B_23_8-B_15_26+B_23_26,-B_23_9-B_15_27+B_23_27]]),Matrix(9, 2, [[C_19_3,C_19_4],[C_20_3,C_20_4],[C_21_3,C_21_4],[C_22_3,C_22_4],[C_23_3,C_23_4],[C_24_3,C_24_4],[C_25_3,C_25_4],[C_26_3,C_26_4],[C_27_3,C_27_4]])))+Trace(Mul(Matrix(2, 7, [[A_1_1+A_1_9,A_1_2+A_1_10,A_1_3+A_1_11,A_1_4+A_1_12,A_1_5+A_1_13,A_1_6+A_1_14,A_1_7+A_1_15],[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_2_5+A_2_13,A_2_6+A_2_14,A_2_7+A_2_15]]),Matrix(7, 9, [[B_1_10,B_1_11,B_1_12,B_1_13,B_1_14,B_1_15,B_1_16,B_1_17,B_1_18],[B_2_10,B_2_11,B_2_12,B_2_13,B_2_14,B_2_15,B_2_16,B_2_17,B_2_18],[B_3_10,B_3_11,B_3_12,B_3_13,B_3_14,B_3_15,B_3_16,B_3_17,B_3_18],[B_4_10,B_4_11,B_4_12,B_4_13,B_4_14,B_4_15,B_4_16,B_4_17,B_4_18],[B_5_10,B_5_11,B_5_12,B_5_13,B_5_14,B_5_15,B_5_16,B_5_17,B_5_18],[B_6_10,B_6_11,B_6_12,B_6_13,B_6_14,B_6_15,B_6_16,B_6_17,B_6_18],[B_7_10,B_7_11,B_7_12,B_7_13,B_7_14,B_7_15,B_7_16,B_7_17,B_7_18]]),Matrix(9, 2, [[C_1_1+C_10_1,C_1_2+C_10_2],[C_2_1+C_11_1,C_2_2+C_11_2],[C_3_1+C_12_1,C_3_2+C_12_2],[C_4_1+C_13_1,C_4_2+C_13_2],[C_5_1+C_14_1,C_5_2+C_14_2],[C_6_1+C_15_1,C_6_2+C_15_2],[C_7_1+C_16_1,C_7_2+C_16_2],[C_8_1+C_17_1,C_8_2+C_17_2],[C_9_1+C_18_1,C_9_2+C_18_2]])))+Trace(Mul(Matrix(2, 7, [[A_1_1+A_1_17,A_1_2+A_1_18,A_1_3+A_1_19,A_1_4+A_1_20,A_1_5+A_1_21,A_1_6+A_1_22,A_1_7+A_1_23],[A_2_1+A_2_17,A_2_2+A_2_18,A_2_3+A_2_19,A_2_4+A_2_20,A_2_5+A_2_21,A_2_6+A_2_22,A_2_7+A_2_23]]),Matrix(7, 9, [[B_1_19,B_1_20,B_1_21,B_1_22,B_1_23,B_1_24,B_1_25,B_1_26,B_1_27],[B_2_19,B_2_20,B_2_21,B_2_22,B_2_23,B_2_24,B_2_25,B_2_26,B_2_27],[B_3_19,B_3_20,B_3_21,B_3_22,B_3_23,B_3_24,B_3_25,B_3_26,B_3_27],[B_4_19,B_4_20,B_4_21,B_4_22,B_4_23,B_4_24,B_4_25,B_4_26,B_4_27],[B_5_19,B_5_20,B_5_21,B_5_22,B_5_23,B_5_24,B_5_25,B_5_26,B_5_27],[B_6_19,B_6_20,B_6_21,B_6_22,B_6_23,B_6_24,B_6_25,B_6_26,B_6_27],[B_7_19,B_7_20,B_7_21,B_7_22,B_7_23,B_7_24,B_7_25,B_7_26,B_7_27]]),Matrix(9, 2, [[C_1_1+C_19_1+C_1_3+C_19_3,C_1_2+C_19_2+C_1_4+C_19_4],[C_2_1+C_20_1+C_2_3+C_20_3,C_2_2+C_20_2+C_2_4+C_20_4],[C_3_1+C_21_1+C_3_3+C_21_3,C_3_2+C_21_2+C_3_4+C_21_4],[C_4_1+C_22_1+C_4_3+C_22_3,C_4_2+C_22_2+C_4_4+C_22_4],[C_5_1+C_23_1+C_5_3+C_23_3,C_5_2+C_23_2+C_5_4+C_23_4],[C_6_1+C_24_1+C_6_3+C_24_3,C_6_2+C_24_2+C_6_4+C_24_4],[C_7_1+C_25_1+C_7_3+C_25_3,C_7_2+C_25_2+C_7_4+C_25_4],[C_8_1+C_26_1+C_8_3+C_26_3,C_8_2+C_26_2+C_8_4+C_26_4],[C_9_1+C_27_1+C_9_3+C_27_3,C_9_2+C_27_2+C_9_4+C_27_4]])))+Trace(Mul(Matrix(2, 7, [[A_1_1-A_3_1,A_1_2-A_3_2,A_1_3-A_3_3,A_1_4-A_3_4,A_1_5-A_3_5,A_1_6-A_3_6,A_1_7-A_3_7],[A_2_1-A_4_1,A_2_2-A_4_2,A_2_3-A_4_3,A_2_4-A_4_4,-A_4_5+A_2_5,-A_4_6+A_2_6,A_2_7-A_4_7]]),Matrix(7, 9, [[B_1_1-B_17_1-B_1_10,B_1_2-B_17_2-B_1_11,B_1_3-B_17_3-B_1_12,B_1_4-B_17_4-B_1_13,B_1_5-B_17_5-B_1_14,B_1_6-B_17_6-B_1_15,B_1_7-B_17_7-B_1_16,B_1_8-B_17_8-B_1_17,B_1_9-B_17_9-B_1_18],[B_2_1-B_18_1-B_2_10,B_2_2-B_18_2-B_2_11,B_2_3-B_18_3-B_2_12,B_2_4-B_18_4-B_2_13,B_2_5-B_18_5-B_2_14,B_2_6-B_18_6-B_2_15,B_2_7-B_18_7-B_2_16,B_2_8-B_18_8-B_2_17,B_2_9-B_18_9-B_2_18],[B_3_1-B_19_1-B_3_10,B_3_2-B_19_2-B_3_11,B_3_3-B_19_3-B_3_12,B_3_4-B_19_4-B_3_13,B_3_5-B_19_5-B_3_14,B_3_6-B_19_6-B_3_15,B_3_7-B_19_7-B_3_16,B_3_8-B_19_8-B_3_17,B_3_9-B_19_9-B_3_18],[B_4_1-B_20_1-B_4_10,B_4_2-B_20_2-B_4_11,B_4_3-B_20_3-B_4_12,B_4_4-B_20_4-B_4_13,B_4_5-B_20_5-B_4_14,B_4_6-B_20_6-B_4_15,B_4_7-B_20_7-B_4_16,B_4_8-B_20_8-B_4_17,B_4_9-B_20_9-B_4_18],[B_5_1-B_21_1-B_5_10,B_5_2-B_21_2-B_5_11,B_5_3-B_21_3-B_5_12,B_5_4-B_21_4-B_5_13,B_5_5-B_21_5-B_5_14,B_5_6-B_21_6-B_5_15,B_5_7-B_21_7-B_5_16,B_5_8-B_21_8-B_5_17,B_5_9-B_21_9-B_5_18],[B_6_1-B_22_1-B_6_10,B_6_2-B_22_2-B_6_11,B_6_3-B_22_3-B_6_12,B_6_4-B_22_4-B_6_13,B_6_5-B_22_5-B_6_14,B_6_6-B_22_6-B_6_15,B_6_7-B_22_7-B_6_16,B_6_8-B_22_8-B_6_17,B_6_9-B_22_9-B_6_18],[B_7_1-B_23_1-B_7_10,B_7_2-B_23_2-B_7_11,B_7_3-B_23_3-B_7_12,B_7_4-B_23_4-B_7_13,B_7_5-B_23_5-B_7_14,B_7_6-B_23_6-B_7_15,B_7_7-B_23_7-B_7_16,B_7_8-B_23_8-B_7_17,B_7_9-B_23_9-B_7_18]]),Matrix(9, 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],[C_6_1,C_6_2],[C_7_1,C_7_2],[C_8_1,C_8_2],[C_9_1,C_9_2]])))+Trace(Mul(Matrix(2, 8, [[A_1_8,A_3_1+A_1_9,A_3_2+A_1_10,A_3_3+A_1_11,A_3_4+A_1_12,A_3_5+A_1_13,A_3_6+A_1_14,A_3_7+A_1_15],[A_2_8,A_4_1+A_2_9,A_4_2+A_2_10,A_4_3+A_2_11,A_4_4+A_2_12,A_4_5+A_2_13,A_4_6+A_2_14,A_4_7+A_2_15]]),Matrix(8, 9, [[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_9_1-B_1_10,B_9_2-B_1_11,B_9_3-B_1_12,B_9_4-B_1_13,B_9_5-B_1_14,B_9_6-B_1_15,B_9_7-B_1_16,B_9_8-B_1_17,B_9_9-B_1_18],[B_10_1-B_2_10,B_10_2-B_2_11,B_10_3-B_2_12,B_10_4-B_2_13,B_10_5-B_2_14,B_10_6-B_2_15,B_10_7-B_2_16,B_10_8-B_2_17,B_10_9-B_2_18],[B_11_1-B_3_10,B_11_2-B_3_11,B_11_3-B_3_12,B_11_4-B_3_13,B_11_5-B_3_14,B_11_6-B_3_15,B_11_7-B_3_16,B_11_8-B_3_17,B_11_9-B_3_18],[B_12_1-B_4_10,B_12_2-B_4_11,B_12_3-B_4_12,B_12_4-B_4_13,B_12_5-B_4_14,B_12_6-B_4_15,B_12_7-B_4_16,B_12_8-B_4_17,B_12_9-B_4_18],[B_13_1-B_5_10,B_13_2-B_5_11,B_13_3-B_5_12,B_13_4-B_5_13,B_13_5-B_5_14,B_13_6-B_5_15,B_13_7-B_5_16,B_13_8-B_5_17,B_13_9-B_5_18],[B_14_1-B_6_10,B_14_2-B_6_11,B_14_3-B_6_12,B_14_4-B_6_13,B_14_5-B_6_14,B_14_6-B_6_15,B_14_7-B_6_16,B_14_8-B_6_17,B_14_9-B_6_18],[B_15_1-B_7_10,B_15_2-B_7_11,B_15_3-B_7_12,B_15_4-B_7_13,B_15_5-B_7_14,B_15_6-B_7_15,B_15_7-B_7_16,B_15_8-B_7_17,B_15_9-B_7_18]]),Matrix(9, 2, [[C_1_1-C_10_3,C_1_2-C_10_4],[C_2_1-C_11_3,C_2_2-C_11_4],[C_3_1-C_12_3,C_3_2-C_12_4],[C_4_1-C_13_3,C_4_2-C_13_4],[C_5_1-C_14_3,C_5_2-C_14_4],[C_6_1-C_15_3,C_6_2-C_15_4],[C_7_1-C_16_3,C_7_2-C_16_4],[C_8_1-C_17_3,C_8_2-C_17_4],[C_9_1-C_18_3,C_9_2-C_18_4]])))+Trace(Mul(Matrix(2, 8, [[A_3_8,A_3_1+A_3_9,A_3_2+A_3_10,A_3_3+A_3_11,A_3_4+A_3_12,A_3_5+A_3_13,A_3_6+A_3_14,A_3_7+A_3_15],[A_4_8,A_4_1+A_4_9,A_4_2+A_4_10,A_4_3+A_4_11,A_4_4+A_4_12,A_4_5+A_4_13,A_4_6+A_4_14,A_4_7+A_4_15]]),Matrix(8, 9, [[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_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_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_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_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_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_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_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]]),Matrix(9, 2, [[C_1_3+C_10_3,C_1_4+C_10_4],[C_2_3+C_11_3,C_2_4+C_11_4],[C_3_3+C_12_3,C_3_4+C_12_4],[C_4_3+C_13_3,C_4_4+C_13_4],[C_5_3+C_14_3,C_5_4+C_14_4],[C_6_3+C_15_3,C_6_4+C_15_4],[C_7_3+C_16_3,C_7_4+C_16_4],[C_8_3+C_17_3,C_8_4+C_17_4],[C_9_3+C_18_3,C_9_4+C_18_4]])))+Trace(Mul(Matrix(2, 8, [[A_1_8-A_3_8,A_1_9-A_3_9,A_1_10-A_3_10,A_1_11-A_3_11,A_1_12-A_3_12,A_1_13-A_3_13,A_1_14-A_3_14,A_1_15-A_3_15],[A_2_8-A_4_8,A_2_9-A_4_9,A_2_10-A_4_10,A_2_11-A_4_11,A_2_12-A_4_12,A_2_13-A_4_13,A_2_14-A_4_14,A_2_15-A_4_15]]),Matrix(8, 9, [[B_8_1-B_8_10+B_16_10,B_8_2-B_8_11+B_16_11,B_8_3-B_8_12+B_16_12,B_8_4-B_8_13+B_16_13,B_8_5-B_8_14+B_16_14,B_8_6-B_8_15+B_16_15,B_8_7-B_8_16+B_16_16,B_8_8-B_8_17+B_16_17,B_8_9-B_8_18+B_16_18],[B_9_1-B_9_10+B_17_10,B_9_2-B_9_11+B_17_11,B_9_3-B_9_12+B_17_12,B_9_4-B_9_13+B_17_13,B_9_5-B_9_14+B_17_14,B_9_6-B_9_15+B_17_15,B_9_7-B_9_16+B_17_16,B_9_8-B_9_17+B_17_17,B_9_9-B_9_18+B_17_18],[B_10_1-B_10_10+B_18_10,B_10_2-B_10_11+B_18_11,B_10_3-B_10_12+B_18_12,B_10_4-B_10_13+B_18_13,B_10_5-B_10_14+B_18_14,B_10_6-B_10_15+B_18_15,B_10_7-B_10_16+B_18_16,B_10_8-B_10_17+B_18_17,B_10_9-B_10_18+B_18_18],[B_11_1-B_11_10+B_19_10,B_11_2-B_11_11+B_19_11,B_11_3-B_11_12+B_19_12,B_11_4-B_11_13+B_19_13,B_11_5-B_11_14+B_19_14,B_11_6-B_11_15+B_19_15,B_11_7-B_11_16+B_19_16,B_11_8-B_11_17+B_19_17,B_11_9-B_11_18+B_19_18],[B_12_1-B_12_10+B_20_10,B_12_2-B_12_11+B_20_11,B_12_3-B_12_12+B_20_12,B_12_4-B_12_13+B_20_13,B_12_5-B_12_14+B_20_14,B_12_6-B_12_15+B_20_15,B_12_7-B_12_16+B_20_16,B_12_8-B_12_17+B_20_17,B_12_9-B_12_18+B_20_18],[B_13_1-B_13_10+B_21_10,B_13_2-B_13_11+B_21_11,B_13_3-B_13_12+B_21_12,B_13_4-B_13_13+B_21_13,B_13_5-B_13_14+B_21_14,B_13_6-B_13_15+B_21_15,B_13_7-B_13_16+B_21_16,B_13_8-B_13_17+B_21_17,B_13_9-B_13_18+B_21_18],[B_14_1-B_14_10+B_22_10,B_14_2-B_14_11+B_22_11,B_14_3-B_14_12+B_22_12,B_14_4-B_14_13+B_22_13,B_14_5-B_14_14+B_22_14,B_14_6-B_14_15+B_22_15,B_14_7-B_14_16+B_22_16,B_14_8-B_14_17+B_22_17,B_14_9-B_14_18+B_22_18],[B_15_1-B_15_10+B_23_10,B_15_2-B_15_11+B_23_11,B_15_3-B_15_12+B_23_12,B_15_4-B_15_13+B_23_13,B_15_5-B_15_14+B_23_14,B_15_6-B_15_15+B_23_15,B_15_7-B_15_16+B_23_16,B_15_8-B_15_17+B_23_17,B_15_9-B_15_18+B_23_18]]),Matrix(9, 2, [[C_10_3,C_10_4],[C_11_3,C_11_4],[C_12_3,C_12_4],[C_13_3,C_13_4],[C_14_3,C_14_4],[C_15_3,C_15_4],[C_16_3,C_16_4],[C_17_3,C_17_4],[C_18_3,C_18_4]])))+Trace(Mul(Matrix(2, 8, [[A_3_8+A_3_16,A_3_9+A_3_17,A_3_10+A_3_18,A_3_11+A_3_19,A_3_12+A_3_20,A_3_13+A_3_21,A_3_14+A_3_22,A_3_15+A_3_23],[A_4_8+A_4_16,A_4_9+A_4_17,A_4_10+A_4_18,A_4_11+A_4_19,A_4_12+A_4_20,A_4_13+A_4_21,A_4_14+A_4_22,A_4_15+A_4_23]]),Matrix(8, 9, [[B_8_19,B_8_20,B_8_21,B_8_22,B_8_23,B_8_24,B_8_25,B_8_26,B_8_27],[B_9_19,B_9_20,B_9_21,B_9_22,B_9_23,B_9_24,B_9_25,B_9_26,B_9_27],[B_10_19,B_10_20,B_10_21,B_10_22,B_10_23,B_10_24,B_10_25,B_10_26,B_10_27],[B_11_19,B_11_20,B_11_21,B_11_22,B_11_23,B_11_24,B_11_25,B_11_26,B_11_27],[B_12_19,B_12_20,B_12_21,B_12_22,B_12_23,B_12_24,B_12_25,B_12_26,B_12_27],[B_13_19,B_13_20,B_13_21,B_13_22,B_13_23,B_13_24,B_13_25,B_13_26,B_13_27],[B_14_19,B_14_20,B_14_21,B_14_22,B_14_23,B_14_24,B_14_25,B_14_26,B_14_27],[B_15_19,B_15_20,B_15_21,B_15_22,B_15_23,B_15_24,B_15_25,B_15_26,B_15_27]]),Matrix(9, 2, [[C_10_1+C_19_1+C_10_3+C_19_3,C_10_2+C_19_2+C_10_4+C_19_4],[C_11_1+C_20_1+C_11_3+C_20_3,C_11_2+C_20_2+C_11_4+C_20_4],[C_12_1+C_21_1+C_12_3+C_21_3,C_12_2+C_21_2+C_12_4+C_21_4],[C_13_1+C_22_1+C_13_3+C_22_3,C_13_2+C_22_2+C_13_4+C_22_4],[C_14_1+C_23_1+C_14_3+C_23_3,C_14_2+C_23_2+C_14_4+C_23_4],[C_15_1+C_24_1+C_15_3+C_24_3,C_15_2+C_24_2+C_15_4+C_24_4],[C_16_1+C_25_1+C_16_3+C_25_3,C_16_2+C_25_2+C_16_4+C_25_4],[C_17_1+C_26_1+C_17_3+C_26_3,C_17_2+C_26_2+C_17_4+C_26_4],[C_18_1+C_27_1+C_18_3+C_27_3,C_18_2+C_27_2+C_18_4+C_27_4]])))+Trace(Mul(Matrix(2, 8, [[A_1_16,A_1_1-A_3_1+A_1_17,A_1_2-A_3_2+A_1_18,A_1_3-A_3_3+A_1_19,A_1_4-A_3_4+A_1_20,A_1_5-A_3_5+A_1_21,A_1_6-A_3_6+A_1_22,A_1_7-A_3_7+A_1_23],[A_2_16,A_2_1-A_4_1+A_2_17,A_2_2-A_4_2+A_2_18,A_2_3-A_4_3+A_2_19,A_2_4-A_4_4+A_2_20,-A_4_5+A_2_5+A_2_21,-A_4_6+A_2_6+A_2_22,A_2_7-A_4_7+A_2_23]]),Matrix(8, 9, [[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_17_1-B_1_19,B_17_2-B_1_20,B_17_3-B_1_21,B_17_4-B_1_22,B_17_5-B_1_23,B_17_6-B_1_24,B_17_7-B_1_25,B_17_8-B_1_26,B_17_9-B_1_27],[B_18_1-B_2_19,B_18_2-B_2_20,B_18_3-B_2_21,B_18_4-B_2_22,B_18_5-B_2_23,B_18_6-B_2_24,B_18_7-B_2_25,B_18_8-B_2_26,B_18_9-B_2_27],[B_19_1-B_3_19,B_19_2-B_3_20,B_19_3-B_3_21,B_19_4-B_3_22,B_19_5-B_3_23,B_19_6-B_3_24,B_19_7-B_3_25,B_19_8-B_3_26,B_19_9-B_3_27],[B_20_1-B_4_19,B_20_2-B_4_20,B_20_3-B_4_21,B_20_4-B_4_22,B_20_5-B_4_23,B_20_6-B_4_24,B_20_7-B_4_25,B_20_8-B_4_26,B_20_9-B_4_27],[B_21_1-B_5_19,B_21_2-B_5_20,B_21_3-B_5_21,B_21_4-B_5_22,B_21_5-B_5_23,B_21_6-B_5_24,B_21_7-B_5_25,B_21_8-B_5_26,B_21_9-B_5_27],[B_22_1-B_6_19,B_22_2-B_6_20,B_22_3-B_6_21,B_22_4-B_6_22,B_22_5-B_6_23,B_22_6-B_6_24,B_22_7-B_6_25,B_22_8-B_6_26,B_22_9-B_6_27],[B_23_1-B_7_19,B_23_2-B_7_20,B_23_3-B_7_21,B_23_4-B_7_22,B_23_5-B_7_23,B_23_6-B_7_24,B_23_7-B_7_25,B_23_8-B_7_26,B_23_9-B_7_27]]),Matrix(9, 2, [[C_1_1+C_1_3+C_19_3,C_1_2+C_1_4+C_19_4],[C_2_1+C_2_3+C_20_3,C_2_2+C_2_4+C_20_4],[C_3_1+C_3_3+C_21_3,C_3_2+C_3_4+C_21_4],[C_4_1+C_4_3+C_22_3,C_4_2+C_4_4+C_22_4],[C_5_1+C_5_3+C_23_3,C_5_4+C_5_2+C_23_4],[C_6_1+C_6_3+C_24_3,C_6_2+C_6_4+C_24_4],[C_7_1+C_7_3+C_25_3,C_7_2+C_7_4+C_25_4],[C_8_1+C_8_3+C_26_3,C_8_2+C_8_4+C_26_4],[C_9_1+C_9_3+C_27_3,C_9_2+C_9_4+C_27_4]])))+Trace(Mul(Matrix(2, 8, [[A_1_8-A_3_8-A_3_16,A_1_9-A_3_9-A_3_17,A_1_10-A_3_10-A_3_18,A_1_11-A_3_11-A_3_19,A_1_12-A_3_12-A_3_20,A_1_13-A_3_13-A_3_21,A_1_14-A_3_14-A_3_22,A_1_15-A_3_15-A_3_23],[A_2_8-A_4_8-A_4_16,A_2_9-A_4_9-A_4_17,A_2_10-A_4_10-A_4_18,A_2_11-A_4_11-A_4_19,A_2_12-A_4_12-A_4_20,A_2_13-A_4_13-A_4_21,A_2_14-A_4_14-A_4_22,A_2_15-A_4_15-A_4_23]]),Matrix(8, 9, [[-B_16_10+B_8_19,-B_16_11+B_8_20,-B_16_12+B_8_21,-B_16_13+B_8_22,-B_16_14+B_8_23,-B_16_15+B_8_24,-B_16_16+B_8_25,-B_16_17+B_8_26,-B_16_18+B_8_27],[-B_17_10+B_9_19,-B_17_11+B_9_20,-B_17_12+B_9_21,-B_17_13+B_9_22,-B_17_14+B_9_23,-B_17_15+B_9_24,-B_17_16+B_9_25,-B_17_17+B_9_26,-B_17_18+B_9_27],[-B_18_10+B_10_19,-B_18_11+B_10_20,-B_18_12+B_10_21,-B_18_13+B_10_22,-B_18_14+B_10_23,-B_18_15+B_10_24,-B_18_16+B_10_25,-B_18_17+B_10_26,-B_18_18+B_10_27],[-B_19_10+B_11_19,-B_19_11+B_11_20,-B_19_12+B_11_21,-B_19_13+B_11_22,-B_19_14+B_11_23,-B_19_15+B_11_24,-B_19_16+B_11_25,-B_19_17+B_11_26,-B_19_18+B_11_27],[-B_20_10+B_12_19,-B_20_11+B_12_20,-B_20_12+B_12_21,-B_20_13+B_12_22,-B_20_14+B_12_23,-B_20_15+B_12_24,-B_20_16+B_12_25,-B_20_17+B_12_26,-B_20_18+B_12_27],[-B_21_10+B_13_19,-B_21_11+B_13_20,-B_21_12+B_13_21,-B_21_13+B_13_22,-B_21_14+B_13_23,-B_21_15+B_13_24,-B_21_16+B_13_25,-B_21_17+B_13_26,-B_21_18+B_13_27],[-B_22_10+B_14_19,-B_22_11+B_14_20,-B_22_12+B_14_21,-B_22_13+B_14_22,-B_22_14+B_14_23,-B_22_15+B_14_24,-B_22_16+B_14_25,-B_22_17+B_14_26,-B_22_18+B_14_27],[-B_23_10+B_15_19,-B_23_11+B_15_20,-B_23_12+B_15_21,-B_23_13+B_15_22,-B_23_14+B_15_23,-B_23_15+B_15_24,-B_23_16+B_15_25,-B_23_17+B_15_26,-B_23_18+B_15_27]]),Matrix(9, 2, [[C_10_1+C_19_1+C_10_3,C_10_2+C_19_2+C_10_4],[C_11_1+C_20_1+C_11_3,C_11_2+C_20_2+C_11_4],[C_12_1+C_21_1+C_12_3,C_12_2+C_21_2+C_12_4],[C_13_1+C_22_1+C_13_3,C_13_2+C_22_2+C_13_4],[C_14_1+C_23_1+C_14_3,C_14_2+C_23_2+C_14_4],[C_15_1+C_24_1+C_15_3,C_15_2+C_24_2+C_15_4],[C_16_1+C_25_1+C_16_3,C_16_2+C_25_2+C_16_4],[C_17_1+C_26_1+C_17_3,C_17_2+C_26_2+C_17_4],[C_18_1+C_27_1+C_18_3,C_18_2+C_27_2+C_18_4]])))+Trace(Mul(Matrix(2, 8, [[A_1_8-A_3_8+A_1_16-A_3_16,A_1_9-A_3_9+A_1_17-A_3_17,A_1_10-A_3_10+A_1_18-A_3_18,A_1_11-A_3_11+A_1_19-A_3_19,A_1_12-A_3_12+A_1_20-A_3_20,A_1_13-A_3_13+A_1_21-A_3_21,A_1_14-A_3_14+A_1_22-A_3_22,A_1_15-A_3_15+A_1_23-A_3_23],[A_2_8-A_4_8+A_2_16-A_4_16,A_2_9-A_4_9+A_2_17-A_4_17,A_2_10-A_4_10+A_2_18-A_4_18,A_2_11-A_4_11+A_2_19-A_4_19,A_2_12-A_4_12+A_2_20-A_4_20,A_2_13-A_4_13+A_2_21-A_4_21,A_2_14-A_4_14+A_2_22-A_4_22,A_2_15-A_4_15+A_2_23-A_4_23]]),Matrix(8, 9, [[B_16_10,B_16_11,B_16_12,B_16_13,B_16_14,B_16_15,B_16_16,B_16_17,B_16_18],[B_17_10,B_17_11,B_17_12,B_17_13,B_17_14,B_17_15,B_17_16,B_17_17,B_17_18],[B_18_10,B_18_11,B_18_12,B_18_13,B_18_14,B_18_15,B_18_16,B_18_17,B_18_18],[B_19_10,B_19_11,B_19_12,B_19_13,B_19_14,B_19_15,B_19_16,B_19_17,B_19_18],[B_20_10,B_20_11,B_20_12,B_20_13,B_20_14,B_20_15,B_20_16,B_20_17,B_20_18],[B_21_10,B_21_11,B_21_12,B_21_13,B_21_14,B_21_15,B_21_16,B_21_17,B_21_18],[B_22_10,B_22_11,B_22_12,B_22_13,B_22_14,B_22_15,B_22_16,B_22_17,B_22_18],[B_23_10,B_23_11,B_23_12,B_23_13,B_23_14,B_23_15,B_23_16,B_23_17,B_23_18]]),Matrix(9, 2, [[C_10_1+C_19_1,C_10_2+C_19_2],[C_11_1+C_20_1,C_11_2+C_20_2],[C_12_1+C_21_1,C_12_2+C_21_2],[C_13_1+C_22_1,C_13_2+C_22_2],[C_14_1+C_23_1,C_14_2+C_23_2],[C_15_1+C_24_1,C_15_2+C_24_2],[C_16_1+C_25_1,C_16_2+C_25_2],[C_17_1+C_26_1,C_17_2+C_26_2],[C_18_1+C_27_1,C_18_2+C_27_2]])))+Trace(Mul(Matrix(2, 8, [[-A_1_16+A_3_16,-A_1_1+A_3_1-A_1_17+A_3_17,-A_1_2+A_3_2-A_1_18+A_3_18,-A_1_3+A_3_3-A_1_19+A_3_19,-A_1_4+A_3_4-A_1_20+A_3_20,-A_1_5+A_3_5-A_1_21+A_3_21,-A_1_6+A_3_6-A_1_22+A_3_22,-A_1_7+A_3_7-A_1_23+A_3_23],[-A_2_16+A_4_16,-A_2_1+A_4_1-A_2_17+A_4_17,-A_2_2+A_4_2-A_2_18+A_4_18,A_4_3-A_2_3-A_2_19+A_4_19,-A_2_4+A_4_4-A_2_20+A_4_20,A_4_5-A_2_5-A_2_21+A_4_21,-A_2_6+A_4_6-A_2_22+A_4_22,-A_2_7+A_4_7-A_2_23+A_4_23]]),Matrix(8, 9, [[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_17_1,B_17_2,B_17_3,B_17_4,B_17_5,B_17_6,B_17_7,B_17_8,B_17_9],[B_18_1,B_18_2,B_18_3,B_18_4,B_18_5,B_18_6,B_18_7,B_18_8,B_18_9],[B_19_1,B_19_2,B_19_3,B_19_4,B_19_5,B_19_6,B_19_7,B_19_8,B_19_9],[B_20_1,B_20_2,B_20_3,B_20_4,B_20_5,B_20_6,B_20_7,B_20_8,B_20_9],[B_21_1,B_21_2,B_21_3,B_21_4,B_21_5,B_21_6,B_21_7,B_21_8,B_21_9],[B_22_1,B_22_2,B_22_3,B_22_4,B_22_5,B_22_6,B_22_7,B_22_8,B_22_9],[B_23_1,B_23_2,B_23_3,B_23_4,B_23_5,B_23_6,B_23_7,B_23_8,B_23_9]]),Matrix(9, 2, [[C_1_3+C_19_3,C_1_4+C_19_4],[C_2_3+C_20_3,C_2_4+C_20_4],[C_3_3+C_21_3,C_3_4+C_21_4],[C_4_3+C_22_3,C_4_4+C_22_4],[C_5_3+C_23_3,C_5_4+C_23_4],[C_6_3+C_24_3,C_6_4+C_24_4],[C_7_3+C_25_3,C_7_4+C_25_4],[C_8_3+C_26_3,C_8_4+C_26_4],[C_9_3+C_27_3,C_9_4+C_27_4]])))

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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