Description of fast matrix multiplication algorithm: ⟨6×9×28:956⟩

Algorithm type

80X4Y6Z6+32X2Y9Z3+120X2Y6Z6+48XY9Z3+64X2Y6Z3+96XY6Z3+14X2Y4Z2+160X2Y3Z3+4XY6Z+4XY4Z2+240XY3Z3+2X3Y2Z+14X2Y2Z2+8XY4Z+4XY2Z3+2X3YZ+4XY3Z+12XY2Z2+4XYZ3+22XY2Z+8XYZ2+14XYZ80X4Y6Z632X2Y9Z3120X2Y6Z648XY9Z364X2Y6Z396XY6Z314X2Y4Z2160X2Y3Z34XY6Z4XY4Z2240XY3Z32X3Y2Z14X2Y2Z28XY4Z4XY2Z32X3YZ4XY3Z12XY2Z24XYZ322XY2Z8XYZ214XYZ80*X^4*Y^6*Z^6+32*X^2*Y^9*Z^3+120*X^2*Y^6*Z^6+48*X*Y^9*Z^3+64*X^2*Y^6*Z^3+96*X*Y^6*Z^3+14*X^2*Y^4*Z^2+160*X^2*Y^3*Z^3+4*X*Y^6*Z+4*X*Y^4*Z^2+240*X*Y^3*Z^3+2*X^3*Y^2*Z+14*X^2*Y^2*Z^2+8*X*Y^4*Z+4*X*Y^2*Z^3+2*X^3*Y*Z+4*X*Y^3*Z+12*X*Y^2*Z^2+4*X*Y*Z^3+22*X*Y^2*Z+8*X*Y*Z^2+14*X*Y*Z

Algorithm definition

The algorithm ⟨6×9×28:956⟩ could be constructed using the following decomposition:

⟨6×9×28:956⟩ = ⟨3×3×4:29⟩ + ⟨3×3×6:40⟩ + ⟨3×3×6:40⟩ + ⟨3×3×6:40⟩ + ⟨3×3×4:29⟩ + ⟨3×3×6:40⟩ + ⟨3×3×4:29⟩ + ⟨3×3×6:40⟩ + ⟨3×3×6:40⟩ + ⟨3×3×6:40⟩ + ⟨3×3×6:40⟩ + ⟨3×3×6:40⟩ + ⟨3×3×6:40⟩ + ⟨3×3×6:40⟩ + ⟨3×3×6:40⟩ + ⟨3×3×6:40⟩ + ⟨3×3×6:40⟩ + ⟨3×3×6:40⟩ + ⟨3×3×6:40⟩ + ⟨3×3×6:40⟩ + ⟨3×3×6:40⟩ + ⟨3×3×6:40⟩ + ⟨3×3×6:40⟩ + ⟨3×3×6:40⟩ + ⟨3×3×4:29⟩.

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_9A_6_1A_6_2A_6_3A_6_4A_6_5A_6_6A_6_7A_6_8A_6_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_1_16B_1_17B_1_18B_1_19B_1_20B_1_21B_1_22B_1_23B_1_24B_1_25B_1_26B_1_27B_1_28B_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_2_28B_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_3_28B_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_4_28B_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_5_28B_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_6_28B_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_7_28B_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_8_28B_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_9_28C_1_1C_1_2C_1_3C_1_4C_1_5C_1_6C_2_1C_2_2C_2_3C_2_4C_2_5C_2_6C_3_1C_3_2C_3_3C_3_4C_3_5C_3_6C_4_1C_4_2C_4_3C_4_4C_4_5C_4_6C_5_1C_5_2C_5_3C_5_4C_5_5C_5_6C_6_1C_6_2C_6_3C_6_4C_6_5C_6_6C_7_1C_7_2C_7_3C_7_4C_7_5C_7_6C_8_1C_8_2C_8_3C_8_4C_8_5C_8_6C_9_1C_9_2C_9_3C_9_4C_9_5C_9_6C_10_1C_10_2C_10_3C_10_4C_10_5C_10_6C_11_1C_11_2C_11_3C_11_4C_11_5C_11_6C_12_1C_12_2C_12_3C_12_4C_12_5C_12_6C_13_1C_13_2C_13_3C_13_4C_13_5C_13_6C_14_1C_14_2C_14_3C_14_4C_14_5C_14_6C_15_1C_15_2C_15_3C_15_4C_15_5C_15_6C_16_1C_16_2C_16_3C_16_4C_16_5C_16_6C_17_1C_17_2C_17_3C_17_4C_17_5C_17_6C_18_1C_18_2C_18_3C_18_4C_18_5C_18_6C_19_1C_19_2C_19_3C_19_4C_19_5C_19_6C_20_1C_20_2C_20_3C_20_4C_20_5C_20_6C_21_1C_21_2C_21_3C_21_4C_21_5C_21_6C_22_1C_22_2C_22_3C_22_4C_22_5C_22_6C_23_1C_23_2C_23_3C_23_4C_23_5C_23_6C_24_1C_24_2C_24_3C_24_4C_24_5C_24_6C_25_1C_25_2C_25_3C_25_4C_25_5C_25_6C_26_1C_26_2C_26_3C_26_4C_26_5C_26_6C_27_1C_27_2C_27_3C_27_4C_27_5C_27_6C_28_1C_28_2C_28_3C_28_4C_28_5C_28_6=TraceMulA_4_1A_4_2A_4_3A_5_1A_5_2A_5_3A_6_1A_6_2A_6_3B_1_1-B_4_1-a12+a22B_7_1a32-B_1_25-a32a11-a31a12B_4_25a21a32-a22a31-a22a11-a21a12B_7_25a21a32-a22a31B_1_2-B_4_2-a12+a22B_7_2a32-B_1_26-a32a11-a31a12B_4_26a21a32-a22a31-a22a11-a21a12B_7_26a21a32-a22a31B_1_3-B_4_3-a12+a22B_7_3a32-B_1_27-a32a11-a31a12B_4_27a21a32-a22a31-a22a11-a21a12B_7_27a21a32-a22a31B_1_4-B_4_4-a12+a22B_7_4a32-B_1_28-a32a11-a31a12B_4_28a21a32-a22a31-a22a11-a21a12B_7_28a21a32-a22a31B_2_1-B_5_1-a12+a22B_8_1a32-B_2_25-a32a11-a31a12B_5_25a21a32-a22a31-a22a11-a21a12B_8_25a21a32-a22a31B_2_2-B_5_2-a12+a22B_8_2a32-B_2_26-a32a11-a31a12B_5_26a21a32-a22a31-a22a11-a21a12B_8_26a21a32-a22a31B_2_3-B_5_3-a12+a22B_8_3a32-B_2_27-a32a11-a31a12B_5_27a21a32-a22a31-a22a11-a21a12B_8_27a21a32-a22a31B_2_4-B_5_4-a12+a22B_8_4a32-B_2_28-a32a11-a31a12B_5_28a21a32-a22a31-a22a11-a21a12B_8_28a21a32-a22a31-B_6_1+B_3_1-a12+a22B_9_1a32-B_3_25-a32a11-a31a12B_6_25a21a32-a22a31-a22a11-a21a12B_9_25a21a32-a22a31B_3_2-B_6_2-a12+a22B_9_2a32-B_3_26-a32a11-a31a12B_6_26a21a32-a22a31-a22a11-a21a12B_9_26a21a32-a22a31B_3_3-B_6_3-a12+a22B_9_3a32-B_3_27-a32a11-a31a12B_6_27a21a32-a22a31-a22a11-a21a12B_9_27a21a32-a22a31B_3_4-B_6_4-a12+a22B_9_4a32-B_3_28-a32a11-a31a12B_6_28a21a32-a22a31-a22a11-a21a12B_9_28a21a32-a22a31C_1_1+C_1_4C_1_2+C_1_5C_1_3+C_1_6C_2_1+C_2_4C_2_2+C_2_5C_2_3+C_2_6C_3_1+C_3_4C_3_2+C_3_5C_3_3+C_3_6C_4_1+C_4_4C_4_2+C_4_5C_4_3+C_4_6+TraceMulA_1_4A_1_5A_1_6A_2_4A_2_5A_2_6A_3_4A_3_5A_3_6-B_1_5+B_4_5-B_7_5-B_1_6+B_4_6-B_7_6-B_1_7+B_4_7-B_7_7-B_1_8+B_4_8-B_7_8-B_1_9+B_4_9-B_7_9-B_1_10+B_4_10-B_7_10-B_2_5+B_5_5-B_8_5-B_2_6+B_5_6-B_8_6-B_2_7+B_5_7-B_8_7-B_2_8+B_5_8-B_8_8-B_2_9+B_5_9-B_8_9-B_2_10+B_5_10-B_8_10-B_3_5+B_6_5-B_9_5-B_3_6+B_6_6-B_9_6-B_3_7+B_6_7-B_9_7-B_3_8+B_6_8-B_9_8-B_3_9+B_6_9-B_9_9-B_3_10+B_6_10-B_9_10C_5_1+C_5_4C_5_2+C_5_5C_5_3+C_5_6C_6_1+C_6_4C_6_2+C_6_5C_6_3+C_6_6C_7_1+C_7_4C_7_2+C_7_5C_7_3+C_7_6C_8_1+C_8_4C_8_2+C_8_5C_8_3+C_8_6C_9_1+C_9_4C_9_2+C_9_5C_9_3+C_9_6C_10_1+C_10_4C_10_2+C_10_5C_10_3+C_10_6+TraceMulA_4_7A_4_8A_4_9A_5_7A_5_8A_5_9A_6_7A_6_8A_6_9-a21+a31+1B_1_11a11-B_4_11+B_7_11-a21+a31+1B_1_12a11-B_4_12+B_7_12-a21+a31+1B_1_13a11-B_4_13+B_7_13-a21+a31+1B_1_14a11-B_4_14+B_7_14-a21+a31+1B_1_15a11-B_4_15+B_7_15-a21+a31+1B_1_16a11-B_4_16+B_7_16-a21+a31+1B_2_11a11-B_5_11+B_8_11-a21+a31+1B_2_12a11-B_5_12+B_8_12-a21+a31+1B_2_13a11-B_5_13+B_8_13-a21+a31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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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