Description of fast matrix multiplication algorithm: ⟨9×11×17:1104⟩

Algorithm type

16 X^{2} Y^{3} Z^{9} + 24 X Y^{3} Z^{9} + 52 X^{4} Y^{4} Z^{4} + 2 X^{4} Y^{4} Z^{3} + X^{4} Y^{3} Z^{4} + 7 X^{3} Y^{4} Z^{4} + 3 X^{8} Y Z + 19 X^{6} Y^{2} Z^{2} + 7 X^{4} Y^{3} Z^{3} + X^{3} Y^{3} Z^{4} + 26 X^{2} Y^{6} Z^{2} + 13 X^{2} Y^{2} Z^{6} + X^{6} Y^{2} Z + 12 X^{5} Y^{2} Z^{2} + X^{3} Y^{3} Z^{3} + X^{3} Y^{2} Z^{4} + 9 X^{2} Y^{5} Z^{2} + 5 X^{2} Y^{2} Z^{5} + X^{6} Y Z + 59 X^{2} Y^{4} Z^{2} + 48 X^{2} Y^{3} Z^{3} + 59 X^{2} Y^{2} Z^{4} + 28 X Y^{6} Z + 14 X Y Z^{6} + 2 X^{5} Y Z + X^{3} Y^{3} Z + X^{3} Y^{2} Z^{2} + 4 X^{3} Y Z^{3} + 2 X^{2} Y^{4} Z + 6 X^{2} Y^{3} Z^{2} + 6 X^{2} Y^{2} Z^{3} + X^{2} Y Z^{4} + 8 X Y^{5} Z + 4 X Y^{4} Z^{2} + 72 X Y^{3} Z^{3} + 3 X Y^{2} Z^{4} + 4 X Y Z^{5} + 23 X^{3} Y^{2} Z + 21 X^{3} Y Z^{2} + 3 X^{2} Y^{3} Z + 139 X^{2} Y^{2} Z^{2} + 2 X^{2} Y Z^{3} + 32 X Y^{3} Z^{2} + 16 X Y^{2} Z^{3} + X Y Z^{4} + 14 X^{3} Y Z + 6 X^{2} Y^{2} Z + 3 X^{2} Y Z^{2} + 6 X Y^{3} Z + 12 X Y^{2} Z^{2} + 18 X Y Z^{3} + 9 X^{2} Y Z + 114 X Y^{2} Z + 108 X Y Z^{2} + 54 X Y Z

Algorithm definition

The algorithm ⟨9×11×17:1104⟩ could be constructed using the following decomposition:

⟨9×11×17:1104⟩ = ⟨3×4×6:54⟩ + ⟨3×4×6:54⟩ + ⟨3×4×6:54⟩ + ⟨3×4×6:54⟩ + ⟨3×4×6:54⟩ + ⟨3×4×6:54⟩ + ⟨3×4×6:54⟩ + ⟨3×3×6:40⟩ + ⟨3×3×6:40⟩ + ⟨3×4×6:54⟩ + ⟨3×3×6:40⟩ + ⟨3×4×6:54⟩ + ⟨3×3×5:36⟩ + ⟨3×3×5:36⟩ + ⟨3×4×5:47⟩ + ⟨3×4×6:54⟩ + ⟨3×4×6:54⟩ + ⟨3×4×5:47⟩ + ⟨3×4×5:47⟩ + ⟨3×4×5:47⟩ + ⟨3×3×6:40⟩ + ⟨3×3×5:36⟩ + ⟨3×4×6:54⟩.

This decomposition is defined by the following equality:

Trace (Mul ((\begin{matrix} 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_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_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_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_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 & A_5_10 & A_5_11 \\ A_6_1 & A_6_2 & A_6_3 & A_6_4 & A_6_5 & A_6_6 & A_6_7 & A_6_8 & A_6_9 & A_6_10 & A_6_11 \\ A_7_1 & A_7_2 & A_7_3 & A_7_4 & A_7_5 & A_7_6 & A_7_7 & A_7_8 & A_7_9 & A_7_10 & A_7_11 \\ A_8_1 & A_8_2 & A_8_3 & A_8_4 & A_8_5 & A_8_6 & A_8_7 & A_8_8 & A_8_9 & A_8_10 & A_8_11 \\ A_9_1 & A_9_2 & A_9_3 & A_9_4 & A_9_5 & A_9_6 & A_9_7 & A_9_8 & A_9_9 & A_9_10 & A_9_11 \end{matrix}), (\begin{matrix} 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_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_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_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_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_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_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_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_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_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_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 \end{matrix}), (\begin{matrix} C_1_1 & C_1_2 & C_1_3 & C_1_4 & C_1_5 & C_1_6 & C_1_7 & C_1_8 & C_1_9 \\ C_2_1 & C_2_2 & C_2_3 & C_2_4 & C_2_5 & C_2_6 & C_2_7 & C_2_8 & C_2_9 \\ C_3_1 & C_3_2 & C_3_3 & C_3_4 & C_3_5 & C_3_6 & C_3_7 & C_3_8 & C_3_9 \\ C_4_1 & C_4_2 & C_4_3 & C_4_4 & C_4_5 & C_4_6 & C_4_7 & C_4_8 & C_4_9 \\ C_5_1 & C_5_2 & C_5_3 & C_5_4 & C_5_5 & C_5_6 & C_5_7 & C_5_8 & C_5_9 \\ C_6_1 & C_6_2 & C_6_3 & C_6_4 & C_6_5 & C_6_6 & C_6_7 & C_6_8 & C_6_9 \\ C_7_1 & C_7_2 & C_7_3 & C_7_4 & C_7_5 & C_7_6 & C_7_7 & C_7_8 & C_7_9 \\ C_8_1 & C_8_2 & C_8_3 & C_8_4 & C_8_5 & C_8_6 & C_8_7 & C_8_8 & C_8_9 \\ C_9_1 & C_9_2 & C_9_3 & C_9_4 & C_9_5 & C_9_6 & C_9_7 & C_9_8 & C_9_9 \\ C_10_1 & C_10_2 & C_10_3 & C_10_4 & C_10_5 & C_10_6 & C_10_7 & C_10_8 & C_10_9 \\ C_11_1 & C_11_2 & C_11_3 & C_11_4 & C_11_5 & C_11_6 & C_11_7 & C_11_8 & C_11_9 \\ C_12_1 & C_12_2 & C_12_3 & C_12_4 & C_12_5 & C_12_6 & C_12_7 & C_12_8 & C_12_9 \\ C_13_1 & C_13_2 & C_13_3 & C_13_4 & C_13_5 & C_13_6 & C_13_7 & C_13_8 & C_13_9 \\ C_14_1 & C_14_2 & C_14_3 & C_14_4 & C_14_5 & C_14_6 & C_14_7 & C_14_8 & C_14_9 \\ C_15_1 & C_15_2 & C_15_3 & C_15_4 & C_15_5 & C_15_6 & C_15_7 & C_15_8 & C_15_9 \\ C_16_1 & C_16_2 & C_16_3 & C_16_4 & C_16_5 & C_16_6 & C_16_7 & C_16_8 & C_16_9 \\ C_17_1 & C_17_2 & C_17_3 & C_17_4 & C_17_5 & C_17_6 & C_17_7 & C_17_8 & C_17_9 \end{matrix}))) = Trace (Mul ((\begin{matrix} - A_4_8 l + A_4_4 & - A_4_9 l + A_4_5 & - A_4_10 l + A_4_6 & - A_4_11 l + A_4_7 \\ - A_5_8 l + A_5_4 & - A_5_9 l + A_5_5 & - A_5_10 l + A_5_6 & - A_5_11 l + A_5_7 \\ - A_6_8 l + A_6_4 & - A_6_9 l + A_6_5 & - A_6_10 l + A_6_6 & - A_6_11 l + A_6_7 \end{matrix}), (\begin{matrix} - B_4_12 l + B_4_6 & - B_4_13 l + B_4_7 & - B_4_14 l + B_4_8 & - B_4_15 l + B_4_9 & - B_4_16 l + B_4_10 & - B_4_17 l + B_4_11 \\ - B_5_12 l + B_5_6 & - B_5_13 l + B_5_7 & - B_5_14 l + B_5_8 & - B_5_15 l + B_5_9 & - B_5_16 l + B_5_10 & - B_5_17 l + B_5_11 \\ - B_6_12 l + B_6_6 & - B_6_13 l + B_6_7 & - B_6_14 l + B_6_8 & - B_6_15 l + B_6_9 & - B_6_16 l + B_6_10 & - B_6_17 l + B_6_11 \\ - B_7_12 l + B_7_6 & - B_7_13 l + B_7_7 & - B_7_14 l + B_7_8 & - B_7_15 l + B_7_9 & - B_7_16 l + B_7_10 & - B_7_17 l + B_7_11 \end{matrix}), (\begin{matrix} - C_6_7 l + C_6_4 & - C_6_8 l + C_6_5 & - C_6_9 l + C_6_6 \\ - C_7_7 l + C_7_4 & - C_7_8 l + C_7_5 & - C_7_9 l + C_7_6 \\ - C_8_7 l + C_8_4 & - C_8_8 l + C_8_5 & - C_8_9 l + C_8_6 \\ - C_9_7 l + C_9_4 & - C_9_8 l + C_9_5 & - C_9_9 l + C_9_6 \\ - C_10_7 l + C_10_4 & - C_10_8 l + C_10_5 & - C_10_9 l + C_10_6 \\ - C_11_7 l + C_11_4 & - C_11_8 l + C_11_5 & - C_11_9 l + C_11_6 \end{matrix}))) + Trace (Mul ((\begin{matrix} A_4_4 + \frac{A_7_4}{l} & A_4_5 + \frac{A_7_5}{l} & A_4_6 + \frac{A_7_6}{l} & A_4_7 + \frac{A_7_7}{l} \\ A_5_4 + \frac{A_8_4}{l} & A_5_5 + \frac{A_8_5}{l} & A_5_6 + \frac{A_8_6}{l} & A_5_7 + \frac{A_8_7}{l} \\ A_6_4 + \frac{A_9_4}{l} & A_6_5 + \frac{A_9_5}{l} & A_6_6 + \frac{A_9_6}{l} & A_6_7 + \frac{A_9_7}{l} \end{matrix}), (\begin{matrix} B_4_6 + \frac{B_8_6}{l} & B_4_7 + \frac{B_8_7}{l} & B_4_8 + \frac{B_8_8}{l} & B_4_9 + \frac{B_8_9}{l} & B_4_10 + \frac{B_8_10}{l} & B_4_11 + \frac{B_8_11}{l} \\ B_5_6 + \frac{B_9_6}{l} & B_5_7 + \frac{B_9_7}{l} & B_5_8 + \frac{B_9_8}{l} & B_5_9 + \frac{B_9_9}{l} & B_5_10 + \frac{B_9_10}{l} & B_5_11 + \frac{B_9_11}{l} \\ B_6_6 + \frac{B_10_6}{l} & B_6_7 + \frac{B_10_7}{l} & B_6_8 + \frac{B_10_8}{l} & B_6_9 + \frac{B_10_9}{l} & B_6_10 + \frac{B_10_10}{l} & B_6_11 + \frac{B_10_11}{l} \\ B_7_6 + \frac{B_11_6}{l} & B_7_7 + \frac{B_11_7}{l} & B_7_8 + \frac{B_11_8}{l} & B_7_9 + \frac{B_11_9}{l} & B_7_10 + \frac{B_11_10}{l} & B_7_11 + \frac{B_11_11}{l} \end{matrix}), (\begin{matrix} C_6_4 + \frac{C_12_4}{l} & C_6_5 + \frac{C_12_5}{l} & C_6_6 + \frac{C_12_6}{l} \\ C_7_4 + \frac{C_13_4}{l} & C_7_5 + \frac{C_13_5}{l} & C_7_6 + \frac{C_13_6}{l} \\ C_8_4 + \frac{C_14_4}{l} & C_8_5 + \frac{C_14_5}{l} & C_8_6 + \frac{C_14_6}{l} \\ C_9_4 + \frac{C_15_4}{l} & C_9_5 + \frac{C_15_5}{l} & C_9_6 + \frac{C_15_6}{l} \\ C_10_4 + \frac{C_16_4}{l} & C_10_5 + \frac{C_16_5}{l} & C_10_6 + \frac{C_16_6}{l} \\ C_11_4 + \frac{C_17_4}{l} & C_11_5 + \frac{C_17_5}{l} & C_11_6 + \frac{C_17_6}{l} \end{matrix}))) + Trace (Mul ((\begin{matrix} A_7_8 & A_7_9 & A_7_10 & A_7_11 \\ A_8_8 & A_8_9 & A_8_10 & A_8_11 \\ A_9_8 & A_9_9 & A_9_10 & A_9_11 \end{matrix}), (\begin{matrix} B_8_12 & B_8_13 & B_8_14 & B_8_15 & B_8_16 & B_8_17 \\ B_9_12 & B_9_13 & B_9_14 & B_9_15 & B_9_16 & B_9_17 \\ B_10_12 & B_10_13 & B_10_14 & B_10_15 & B_10_16 & B_10_17 \\ B_11_12 & B_11_13 & B_11_14 & B_11_15 & B_11_16 & B_11_17 \end{matrix}), (\begin{matrix} C_12_7 & C_12_8 & C_12_9 \\ C_13_7 & C_13_8 & C_13_9 \\ C_14_7 & C_14_8 & C_14_9 \\ C_15_7 & C_15_8 & C_15_9 \\ C_16_7 & C_16_8 & C_16_9 \\ C_17_7 & C_17_8 & C_17_9 \end{matrix}))) + Trace (Mul ((\begin{matrix} - A_4_8 & - A_4_9 & - A_4_10 & - A_4_11 \\ - A_5_8 & - A_5_9 & - A_5_10 & - A_5_11 \\ - A_6_8 & - A_6_9 & - A_6_10 & - A_6_11 \end{matrix}), (\begin{matrix} B_4_6 + \frac{B_8_6}{l} - B_4_12 l - B_8_12 & \frac{B_8_1}{l} + B_4_7 + \frac{B_8_7}{l} - B_4_13 l - B_8_13 & \frac{B_8_2}{l} + B_4_8 + \frac{B_8_8}{l} - B_4_14 l - B_8_14 & \frac{B_8_3}{l} + B_4_9 + \frac{B_8_9}{l} - B_4_15 l - B_8_15 & \frac{B_8_4}{l} + B_4_10 + \frac{B_8_10}{l} - B_4_16 l - B_8_16 & \frac{B_8_5}{l} + B_4_11 + \frac{B_8_11}{l} - B_4_17 l - B_8_17 \\ B_5_6 + \frac{B_9_6}{l} - l B_1_12 - B_5_12 l - B_9_12 & B_1_1 + \frac{B_9_1}{l} + B_5_7 + \frac{B_9_7}{l} - l B_1_13 - B_5_13 l - B_9_13 & B_1_2 + \frac{B_9_2}{l} + B_5_8 + \frac{B_9_8}{l} - l B_1_14 - B_5_14 l - B_9_14 & B_1_3 + \frac{B_9_3}{l} + B_5_9 + \frac{B_9_9}{l} - l B_1_15 - B_5_15 l - B_9_15 & B_1_4 + \frac{B_9_4}{l} + B_5_10 + \frac{B_9_10}{l} - l B_1_16 - B_5_16 l - B_9_16 & B_1_5 + \frac{B_9_5}{l} + B_5_11 + \frac{B_9_11}{l} - l B_1_17 - B_5_17 l - B_9_17 \\ B_6_6 + \frac{B_10_6}{l} - l B_2_12 - B_6_12 l - B_10_12 & B_2_1 + \frac{B_10_1}{l} + B_6_7 + \frac{B_10_7}{l} - l B_2_13 - B_6_13 l - B_10_13 & B_2_2 + \frac{B_10_2}{l} + B_6_8 + \frac{B_10_8}{l} - l B_2_14 - B_6_14 l - B_10_14 & B_2_3 + \frac{B_10_3}{l} + B_6_9 + \frac{B_10_9}{l} - l B_2_15 - B_6_15 l - B_10_15 & B_2_4 + \frac{B_10_4}{l} + B_6_10 + \frac{B_10_10}{l} - l B_2_16 - B_6_16 l - B_10_16 & B_2_5 + \frac{B_10_5}{l} + B_6_11 + \frac{B_10_11}{l} - l B_2_17 - B_6_17 l - B_10_17 \\ B_7_6 + \frac{B_11_6}{l} - l B_3_12 - B_7_12 l - B_11_12 & B_3_1 + \frac{B_11_1}{l} + B_7_7 + \frac{B_11_7}{l} - l B_3_13 - B_7_13 l - B_11_13 & B_3_2 + \frac{B_11_2}{l} + B_7_8 + \frac{B_11_8}{l} - l B_3_14 - B_7_14 l - B_11_14 & B_3_3 + \frac{B_11_3}{l} + B_7_9 + \frac{B_11_9}{l} - l B_3_15 - B_7_15 l - B_11_15 & B_3_4 + \frac{B_11_4}{l} + B_7_10 + \frac{B_11_10}{l} - l B_3_16 - B_7_16 l - B_11_16 & B_3_5 + \frac{B_11_5}{l} + B_7_11 + \frac{B_11_11}{l} - l B_3_17 - B_7_17 l - B_11_17 \end{matrix}), (\begin{matrix} C_12_4 & C_12_5 & C_12_6 \\ C_13_4 & C_13_5 & C_13_6 \\ C_14_4 & C_14_5 & C_14_6 \\ C_15_4 & C_15_5 & C_15_6 \\ C_16_4 & C_16_5 & C_16_6 \\ C_17_4 & C_17_5 & C_17_6 \end{matrix}))) + Trace (Mul ((\begin{matrix} - A_4_4 - \frac{A_7_4}{l} + l A_1_8 + A_4_8 l + A_7_8 & - A_1_1 - \frac{A_7_1}{l} - A_4_5 - \frac{A_7_5}{l} + l A_1_9 + A_4_9 l + A_7_9 & - A_1_2 - \frac{A_7_2}{l} - A_4_6 - \frac{A_7_6}{l} + l A_1_10 + A_4_10 l + A_7_10 & - A_1_3 - \frac{A_7_3}{l} - A_4_7 - \frac{A_7_7}{l} + l A_1_11 + A_4_11 l + A_7_11 \\ - A_5_4 - \frac{A_8_4}{l} + l A_2_8 + A_5_8 l + A_8_8 & - A_2_1 - \frac{A_8_1}{l} - A_5_5 - \frac{A_8_5}{l} + l A_2_9 + A_5_9 l + A_8_9 & - A_2_2 - \frac{A_8_2}{l} - A_5_6 - \frac{A_8_6}{l} + l A_2_10 + A_5_10 l + A_8_10 & - A_2_3 - \frac{A_8_3}{l} - A_5_7 - \frac{A_8_7}{l} + l A_2_11 + A_5_11 l + A_8_11 \\ - A_6_4 - \frac{A_9_4}{l} + l A_3_8 + A_6_8 l + A_9_8 & - A_3_1 - \frac{A_9_1}{l} - A_6_5 - \frac{A_9_5}{l} + l A_3_9 + A_6_9 l + A_9_9 & - A_3_2 - \frac{A_9_2}{l} - A_6_6 - \frac{A_9_6}{l} + l A_3_10 + A_6_10 l + A_9_10 & - A_3_3 - \frac{A_9_3}{l} - A_6_7 - \frac{A_9_7}{l} + l A_3_11 + A_6_11 l + A_9_11 \end{matrix}), (\begin{matrix} B_8_6 & B_8_7 & B_8_8 & B_8_9 & B_8_10 & B_8_11 \\ B_9_6 & B_9_7 & B_9_8 & B_9_9 & B_9_10 & B_9_11 \\ B_10_6 & B_10_7 & B_10_8 & B_10_9 & B_10_10 & B_10_11 \\ B_11_6 & B_11_7 & B_11_8 & B_11_9 & B_11_10 & B_11_11 \end{matrix}), (\begin{matrix} C_6_7 & C_6_8 & C_6_9 \\ C_7_7 & C_7_8 & C_7_9 \\ C_8_7 & C_8_8 & C_8_9 \\ C_9_7 & C_9_8 & C_9_9 \\ C_10_7 & C_10_8 & C_10_9 \\ C_11_7 & C_11_8 & C_11_9 \end{matrix}))) + Trace (Mul ((\begin{matrix} A_7_4 & A_7_5 & A_7_6 & A_7_7 \\ A_8_4 & A_8_5 & A_8_6 & A_8_7 \\ A_9_4 & A_9_5 & A_9_6 & A_9_7 \end{matrix}), (\begin{matrix} B_4_12 & B_4_13 & B_4_14 & B_4_15 & B_4_16 & B_4_17 \\ B_5_12 & B_5_13 & B_5_14 & B_5_15 & B_5_16 & B_5_17 \\ B_6_12 & B_6_13 & B_6_14 & B_6_15 & B_6_16 & B_6_17 \\ B_7_12 & B_7_13 & B_7_14 & B_7_15 & B_7_16 & B_7_17 \end{matrix}), (\begin{matrix} - \frac{C_12_1}{l} - C_6_4 - \frac{C_12_4}{l} + C_6_7 l + C_12_7 & - \frac{C_12_2}{l} - C_6_5 - \frac{C_12_5}{l} + C_6_8 l + C_12_8 & - \frac{C_12_3}{l} - C_6_6 - \frac{C_12_6}{l} + C_6_9 l + C_12_9 \\ - C_1_1 - \frac{C_13_1}{l} - C_7_4 - \frac{C_13_4}{l} + l C_1_7 + C_7_7 l + C_13_7 & - C_1_2 - \frac{C_13_2}{l} - C_7_5 - \frac{C_13_5}{l} + l C_1_8 + C_7_8 l + C_13_8 & - C_1_3 - \frac{C_13_3}{l} - C_7_6 - \frac{C_13_6}{l} + l C_1_9 + C_7_9 l + C_13_9 \\ - C_2_1 - \frac{C_14_1}{l} - C_8_4 - \frac{C_14_4}{l} + l C_2_7 + C_8_7 l + C_14_7 & - C_2_2 - \frac{C_14_2}{l} - C_8_5 - \frac{C_14_5}{l} + l C_2_8 + C_8_8 l + C_14_8 & - C_2_3 - \frac{C_14_3}{l} - C_8_6 - \frac{C_14_6}{l} + l C_2_9 + C_8_9 l + C_14_9 \\ - C_3_1 - \frac{C_15_1}{l} - C_9_4 - \frac{C_15_4}{l} + l C_3_7 + C_9_7 l + C_15_7 & - C_3_2 - \frac{C_15_2}{l} - C_9_5 - \frac{C_15_5}{l} + l C_3_8 + C_9_8 l + C_15_8 & - C_3_3 - \frac{C_15_3}{l} - C_9_6 - \frac{C_15_6}{l} + l C_3_9 + C_9_9 l + C_15_9 \\ - C_4_1 - \frac{C_16_1}{l} - C_10_4 - \frac{C_16_4}{l} + l C_4_7 + C_10_7 l + C_16_7 & - C_4_2 - \frac{C_16_2}{l} - C_10_5 - \frac{C_16_5}{l} + l C_4_8 + C_10_8 l + C_16_8 & - C_4_3 - \frac{C_16_3}{l} - C_10_6 - \frac{C_16_6}{l} + l C_4_9 + C_10_9 l + C_16_9 \\ - C_5_1 - \frac{C_17_1}{l} - C_11_4 - \frac{C_17_4}{l} + l C_5_7 + C_11_7 l + C_17_7 & - C_5_2 - \frac{C_17_2}{l} - C_11_5 - \frac{C_17_5}{l} + l C_5_8 + C_11_8 l + C_17_8 & - C_5_3 - \frac{C_17_3}{l} - C_11_6 - \frac{C_17_6}{l} + l C_5_9 + C_11_9 l + C_17_9 \end{matrix}))) + Trace (Mul ((\begin{matrix} - A_4_4 - \frac{A_7_4}{l} + A_4_8 l & - A_4_5 - \frac{A_7_5}{l} + A_4_9 l & - A_4_6 - \frac{A_7_6}{l} + A_4_10 l & - A_4_7 - \frac{A_7_7}{l} + A_4_11 l \\ - A_5_4 - \frac{A_8_4}{l} + A_5_8 l & - A_5_5 - \frac{A_8_5}{l} + A_5_9 l & - A_5_6 - \frac{A_8_6}{l} + A_5_10 l & - A_5_7 - \frac{A_8_7}{l} + A_5_11 l \\ - A_6_4 - \frac{A_9_4}{l} + A_6_8 l & - A_6_5 - \frac{A_9_5}{l} + A_6_9 l & - A_6_6 - \frac{A_9_6}{l} + A_6_10 l & - A_6_7 - \frac{A_9_7}{l} + A_6_11 l \end{matrix}), (\begin{matrix} B_4_6 + \frac{B_8_6}{l} - B_4_12 l & B_4_7 + \frac{B_8_7}{l} - B_4_13 l & B_4_8 + \frac{B_8_8}{l} - B_4_14 l & B_4_9 + \frac{B_8_9}{l} - B_4_15 l & B_4_10 + \frac{B_8_10}{l} - B_4_16 l & B_4_11 + \frac{B_8_11}{l} - B_4_17 l \\ B_5_6 + \frac{B_9_6}{l} - B_5_12 l & B_5_7 + \frac{B_9_7}{l} - B_5_13 l & B_5_8 + \frac{B_9_8}{l} - B_5_14 l & B_5_9 + \frac{B_9_9}{l} - B_5_15 l & B_5_10 + \frac{B_9_10}{l} - B_5_16 l & B_5_11 + \frac{B_9_11}{l} - B_5_17 l \\ B_6_6 + \frac{B_10_6}{l} - B_6_12 l & B_6_7 + \frac{B_10_7}{l} - B_6_13 l & B_6_8 + \frac{B_10_8}{l} - B_6_14 l & B_6_9 + \frac{B_10_9}{l} - B_6_15 l & B_6_10 + \frac{B_10_10}{l} - B_6_16 l & B_6_11 + \frac{B_10_11}{l} - B_6_17 l \\ B_7_6 + \frac{B_11_6}{l} - B_7_12 l & B_7_7 + \frac{B_11_7}{l} - B_7_13 l & B_7_8 + \frac{B_11_8}{l} - B_7_14 l & B_7_9 + \frac{B_11_9}{l} - B_7_15 l & B_7_10 + \frac{B_11_10}{l} - B_7_16 l & B_7_11 + \frac{B_11_11}{l} - B_7_17 l \end{matrix}), (\begin{matrix} C_6_4 + \frac{C_12_4}{l} - C_6_7 l & C_6_5 + \frac{C_12_5}{l} - C_6_8 l & C_6_6 + \frac{C_12_6}{l} - C_6_9 l \\ C_7_4 + \frac{C_13_4}{l} - C_7_7 l & C_7_5 + \frac{C_13_5}{l} - C_7_8 l & C_7_6 + \frac{C_13_6}{l} - C_7_9 l \\ C_8_4 + \frac{C_14_4}{l} - C_8_7 l & C_8_5 + \frac{C_14_5}{l} - C_8_8 l & C_8_6 + \frac{C_14_6}{l} - C_8_9 l \\ C_9_4 + \frac{C_15_4}{l} - C_9_7 l & C_9_5 + \frac{C_15_5}{l} - C_9_8 l & C_9_6 + \frac{C_15_6}{l} - C_9_9 l \\ C_10_4 + \frac{C_16_4}{l} - C_10_7 l & C_10_5 + \frac{C_16_5}{l} - C_10_8 l & C_10_6 + \frac{C_16_6}{l} - C_10_9 l \\ C_11_4 + \frac{C_17_4}{l} - C_11_7 l & C_11_5 + \frac{C_17_5}{l} - C_11_8 l & C_11_6 + \frac{C_17_6}{l} - C_11_9 l \end{matrix}))) + Trace (Mul ((\begin{matrix} - l A_1_9 + A_1_1 & - l A_1_10 + A_1_2 & - l A_1_11 + A_1_3 \\ - l A_2_9 + A_2_1 & - l A_2_10 + A_2_2 & - l A_2_11 + A_2_3 \\ - l A_3_9 + A_3_1 & - l A_3_10 + A_3_2 & - l A_3_11 + A_3_3 \end{matrix}), (\begin{matrix} - l B_1_12 + B_1_6 & - l B_1_13 + B_1_7 & - l B_1_14 + B_1_8 & - l B_1_15 + B_1_9 & - l B_1_16 + B_1_10 & - l B_1_17 + B_1_11 \\ - l B_2_12 + B_2_6 & - l B_2_13 + B_2_7 & - l B_2_14 + B_2_8 & - l B_2_15 + B_2_9 & - l B_2_16 + B_2_10 & - l B_2_17 + B_2_11 \\ - l B_3_12 + B_3_6 & - l B_3_13 + B_3_7 & - l B_3_14 + B_3_8 & - l B_3_15 + B_3_9 & - l B_3_16 + B_3_10 & - l B_3_17 + B_3_11 \end{matrix}), (\begin{matrix} - C_6_7 l + C_6_1 & - C_6_8 l + C_6_2 & - C_6_9 l + C_6_3 \\ - C_7_7 l + C_7_1 & - C_7_8 l + C_7_2 & - C_7_9 l + C_7_3 \\ - C_8_7 l + C_8_1 & - C_8_8 l + C_8_2 & - C_8_9 l + C_8_3 \\ - C_9_7 l + C_9_1 & - C_9_8 l + C_9_2 & - C_9_9 l + C_9_3 \\ - C_10_7 l + C_10_1 & - C_10_8 l + C_10_2 & - C_10_9 l + C_10_3 \\ - C_11_7 l + C_11_1 & - C_11_8 l + C_11_2 & - C_11_9 l + C_11_3 \end{matrix}))) + Trace (Mul ((\begin{matrix} A_1_1 + \frac{A_7_1}{l} & A_1_2 + \frac{A_7_2}{l} & A_1_3 + \frac{A_7_3}{l} \\ A_2_1 + \frac{A_8_1}{l} & A_2_2 + \frac{A_8_2}{l} & A_2_3 + \frac{A_8_3}{l} \\ A_3_1 + \frac{A_9_1}{l} & A_3_2 + \frac{A_9_2}{l} & A_3_3 + \frac{A_9_3}{l} \end{matrix}), (\begin{matrix} B_1_6 + \frac{B_9_6}{l} & B_1_7 + \frac{B_9_7}{l} & B_1_8 + \frac{B_9_8}{l} & B_1_9 + \frac{B_9_9}{l} & B_1_10 + \frac{B_9_10}{l} & B_1_11 + \frac{B_9_11}{l} \\ B_2_6 + \frac{B_10_6}{l} & B_2_7 + \frac{B_10_7}{l} & B_2_8 + \frac{B_10_8}{l} & B_2_9 + \frac{B_10_9}{l} & B_2_10 + \frac{B_10_10}{l} & B_2_11 + \frac{B_10_11}{l} \\ B_3_6 + \frac{B_11_6}{l} & B_3_7 + \frac{B_11_7}{l} & B_3_8 + \frac{B_11_8}{l} & B_3_9 + \frac{B_11_9}{l} & B_3_10 + \frac{B_11_10}{l} & B_3_11 + \frac{B_11_11}{l} \end{matrix}), (\begin{matrix} C_6_1 + \frac{C_12_1}{l} & C_6_2 + \frac{C_12_2}{l} & C_6_3 + \frac{C_12_3}{l} \\ C_7_1 + \frac{C_13_1}{l} & C_7_2 + \frac{C_13_2}{l} & C_7_3 + \frac{C_13_3}{l} \\ C_8_1 + \frac{C_14_1}{l} & C_8_2 + \frac{C_14_2}{l} & C_8_3 + \frac{C_14_3}{l} \\ C_9_1 + \frac{C_15_1}{l} & C_9_2 + \frac{C_15_2}{l} & C_9_3 + \frac{C_15_3}{l} \\ C_10_1 + \frac{C_16_1}{l} & C_10_2 + \frac{C_16_2}{l} & C_10_3 + \frac{C_16_3}{l} \\ C_11_1 + \frac{C_17_1}{l} & C_11_2 + \frac{C_17_2}{l} & C_11_3 + \frac{C_17_3}{l} \end{matrix}))) + Trace (Mul ((\begin{matrix} A_1_8 & A_1_9 & A_1_10 & A_1_11 \\ A_2_8 & A_2_9 & A_2_10 & A_2_11 \\ A_3_8 & A_3_9 & A_3_10 & A_3_11 \end{matrix}), (\begin{matrix} - \frac{B_8_6}{l} + B_4_12 l + B_8_12 & - B_4_1 - \frac{B_8_1}{l} - \frac{B_8_7}{l} + B_4_13 l + B_8_13 & - B_4_2 - \frac{B_8_2}{l} - \frac{B_8_8}{l} + B_4_14 l + B_8_14 & - B_4_3 - \frac{B_8_3}{l} - \frac{B_8_9}{l} + B_4_15 l + B_8_15 & - B_4_4 - \frac{B_8_4}{l} - \frac{B_8_10}{l} + B_4_16 l + B_8_16 & - B_4_5 - \frac{B_8_5}{l} - \frac{B_8_11}{l} + B_4_17 l + B_8_17 \\ - B_1_6 - \frac{B_9_6}{l} + l B_1_12 + B_5_12 l + B_9_12 & - B_5_1 - \frac{B_9_1}{l} - B_1_7 - \frac{B_9_7}{l} + l B_1_13 + B_5_13 l + B_9_13 & - B_5_2 - \frac{B_9_2}{l} - B_1_8 - \frac{B_9_8}{l} + l B_1_14 + B_5_14 l + B_9_14 & - B_5_3 - \frac{B_9_3}{l} - B_1_9 - \frac{B_9_9}{l} + l B_1_15 + B_5_15 l + B_9_15 & - B_5_4 - \frac{B_9_4}{l} - B_1_10 - \frac{B_9_10}{l} + l B_1_16 + B_5_16 l + B_9_16 & - B_5_5 - \frac{B_9_5}{l} - B_1_11 - \frac{B_9_11}{l} + l B_1_17 + B_5_17 l + B_9_17 \\ - B_2_6 - \frac{B_10_6}{l} + l B_2_12 + B_6_12 l + B_10_12 & - B_6_1 - \frac{B_10_1}{l} - B_2_7 - \frac{B_10_7}{l} + l B_2_13 + B_6_13 l + B_10_13 & - B_6_2 - \frac{B_10_2}{l} - B_2_8 - \frac{B_10_8}{l} + l B_2_14 + B_6_14 l + B_10_14 & - B_6_3 - \frac{B_10_3}{l} - B_2_9 - \frac{B_10_9}{l} + l B_2_15 + B_6_15 l + B_10_15 & - B_6_4 - \frac{B_10_4}{l} - B_2_10 - \frac{B_10_10}{l} + l B_2_16 + B_6_16 l + B_10_16 & - B_6_5 - \frac{B_10_5}{l} - B_2_11 - \frac{B_10_11}{l} + l B_2_17 + B_6_17 l + B_10_17 \\ - B_3_6 - \frac{B_11_6}{l} + l B_3_12 + B_7_12 l + B_11_12 & - B_7_1 - \frac{B_11_1}{l} - B_3_7 - \frac{B_11_7}{l} + l B_3_13 + B_7_13 l + B_11_13 & - B_7_2 - \frac{B_11_2}{l} - B_3_8 - \frac{B_11_8}{l} + l B_3_14 + B_7_14 l + B_11_14 & - B_7_3 - \frac{B_11_3}{l} - B_3_9 - \frac{B_11_9}{l} + l B_3_15 + B_7_15 l + B_11_15 & - B_7_4 - \frac{B_11_4}{l} - B_3_10 - \frac{B_11_10}{l} + l B_3_16 + B_7_16 l + B_11_16 & - B_7_5 - \frac{B_11_5}{l} - B_3_11 - \frac{B_11_11}{l} + l B_3_17 + B_7_17 l + B_11_17 \end{matrix}), (\begin{matrix} C_12_1 & C_12_2 & C_12_3 \\ C_13_1 & C_13_2 & C_13_3 \\ C_14_1 & C_14_2 & C_14_3 \\ C_15_1 & C_15_2 & C_15_3 \\ C_16_1 & C_16_2 & C_16_3 \\ C_17_1 & C_17_2 & C_17_3 \end{matrix}))) + Trace (Mul ((\begin{matrix} - A_7_1 & - A_7_2 & - A_7_3 \\ - A_8_1 & - A_8_2 & - A_8_3 \\ - A_9_1 & - A_9_2 & - A_9_3 \end{matrix}), (\begin{matrix} B_1_12 & B_1_13 & B_1_14 & B_1_15 & B_1_16 & B_1_17 \\ B_2_12 & B_2_13 & B_2_14 & B_2_15 & B_2_16 & B_2_17 \\ B_3_12 & B_3_13 & B_3_14 & B_3_15 & B_3_16 & B_3_17 \end{matrix}), (\begin{matrix} C_6_1 + \frac{C_12_1}{l} + \frac{C_12_4}{l} - C_6_7 l - C_12_7 & C_6_2 + \frac{C_12_2}{l} + \frac{C_12_5}{l} - C_6_8 l - C_12_8 & C_6_3 + \frac{C_12_3}{l} + \frac{C_12_6}{l} - C_6_9 l - C_12_9 \\ C_7_1 + \frac{C_13_1}{l} + C_1_4 + \frac{C_13_4}{l} - l C_1_7 - C_7_7 l - C_13_7 & C_7_2 + \frac{C_13_2}{l} + C_1_5 + \frac{C_13_5}{l} - l C_1_8 - C_7_8 l - C_13_8 & C_7_3 + \frac{C_13_3}{l} + C_1_6 + \frac{C_13_6}{l} - l C_1_9 - C_7_9 l - C_13_9 \\ C_8_1 + \frac{C_14_1}{l} + C_2_4 + \frac{C_14_4}{l} - l C_2_7 - C_8_7 l - C_14_7 & C_8_2 + \frac{C_14_2}{l} + C_2_5 + \frac{C_14_5}{l} - l C_2_8 - C_8_8 l - C_14_8 & C_8_3 + \frac{C_14_3}{l} + C_2_6 + \frac{C_14_6}{l} - l C_2_9 - C_8_9 l - C_14_9 \\ C_9_1 + \frac{C_15_1}{l} + C_3_4 + \frac{C_15_4}{l} - l C_3_7 - C_9_7 l - C_15_7 & C_9_2 + \frac{C_15_2}{l} + C_3_5 + \frac{C_15_5}{l} - l C_3_8 - C_9_8 l - C_15_8 & C_9_3 + \frac{C_15_3}{l} + C_3_6 + \frac{C_15_6}{l} - l C_3_9 - C_9_9 l - C_15_9 \\ C_10_1 + \frac{C_16_1}{l} + C_4_4 + \frac{C_16_4}{l} - l C_4_7 - C_10_7 l - C_16_7 & C_10_2 + \frac{C_16_2}{l} + C_4_5 + \frac{C_16_5}{l} - l C_4_8 - C_10_8 l - C_16_8 & C_10_3 + \frac{C_16_3}{l} + C_4_6 + \frac{C_16_6}{l} - l C_4_9 - C_10_9 l - C_16_9 \\ C_11_1 + \frac{C_17_1}{l} + C_5_4 + \frac{C_17_4}{l} - l C_5_7 - C_11_7 l - C_17_7 & C_11_2 + \frac{C_17_2}{l} + C_5_5 + \frac{C_17_5}{l} - l C_5_8 - C_11_8 l - C_17_8 & C_11_3 + \frac{C_17_3}{l} + C_5_6 + \frac{C_17_6}{l} - l C_5_9 - C_11_9 l - C_17_9 \end{matrix}))) + Trace (Mul ((\begin{matrix} l A_1_8 & - A_1_1 - \frac{A_7_1}{l} + l A_1_9 & - A_1_2 - \frac{A_7_2}{l} + l A_1_10 & - A_1_3 - \frac{A_7_3}{l} + l A_1_11 \\ l A_2_8 & - A_2_1 - \frac{A_8_1}{l} + l A_2_9 & - A_2_2 - \frac{A_8_2}{l} + l A_2_10 & - A_2_3 - \frac{A_8_3}{l} + l A_2_11 \\ l A_3_8 & - A_3_1 - \frac{A_9_1}{l} + l A_3_9 & - A_3_2 - \frac{A_9_2}{l} + l A_3_10 & - A_3_3 - \frac{A_9_3}{l} + l A_3_11 \end{matrix}), (\begin{matrix} \frac{B_8_6}{l} & \frac{B_8_7}{l} & \frac{B_8_8}{l} & \frac{B_8_9}{l} & \frac{B_8_10}{l} & \frac{B_8_11}{l} \\ B_1_6 + \frac{B_9_6}{l} - l B_1_12 & B_1_7 + \frac{B_9_7}{l} - l B_1_13 & B_1_8 + \frac{B_9_8}{l} - l B_1_14 & B_1_9 + \frac{B_9_9}{l} - l B_1_15 & B_1_10 + \frac{B_9_10}{l} - l B_1_16 & B_1_11 + \frac{B_9_11}{l} - l B_1_17 \\ B_2_6 + \frac{B_10_6}{l} - l B_2_12 & B_2_7 + \frac{B_10_7}{l} - l B_2_13 & B_2_8 + \frac{B_10_8}{l} - l B_2_14 & B_2_9 + \frac{B_10_9}{l} - l B_2_15 & B_2_10 + \frac{B_10_10}{l} - l B_2_16 & B_2_11 + \frac{B_10_11}{l} - l B_2_17 \\ B_3_6 + \frac{B_11_6}{l} - l B_3_12 & B_3_7 + \frac{B_11_7}{l} - l B_3_13 & B_3_8 + \frac{B_11_8}{l} - l B_3_14 & B_3_9 + \frac{B_11_9}{l} - l B_3_15 & B_3_10 + \frac{B_11_10}{l} - l B_3_16 & B_3_11 + \frac{B_11_11}{l} - l B_3_17 \end{matrix}), (\begin{matrix} C_6_1 + \frac{C_12_1}{l} - C_6_7 l & C_6_2 + \frac{C_12_2}{l} - C_6_8 l & C_6_3 + \frac{C_12_3}{l} - C_6_9 l \\ C_7_1 + \frac{C_13_1}{l} - C_7_7 l & C_7_2 + \frac{C_13_2}{l} - C_7_8 l & C_7_3 + \frac{C_13_3}{l} - C_7_9 l \\ C_8_1 + \frac{C_14_1}{l} - C_8_7 l & C_8_2 + \frac{C_14_2}{l} - C_8_8 l & C_8_3 + \frac{C_14_3}{l} - C_8_9 l \\ C_9_1 + \frac{C_15_1}{l} - C_9_7 l & C_9_2 + \frac{C_15_2}{l} - C_9_8 l & C_9_3 + \frac{C_15_3}{l} - C_9_9 l \\ C_10_1 + \frac{C_16_1}{l} - C_10_7 l & C_10_2 + \frac{C_16_2}{l} - C_10_8 l & C_10_3 + \frac{C_16_3}{l} - C_10_9 l \\ C_11_1 + \frac{C_17_1}{l} - C_11_7 l & C_11_2 + \frac{C_17_2}{l} - C_11_8 l & C_11_3 + \frac{C_17_3}{l} - C_11_9 l \end{matrix}))) + Trace (Mul ((\begin{matrix} - A_4_9 l + A_4_1 & - A_4_10 l + A_4_2 & - A_4_11 l + A_4_3 \\ - A_5_9 l + A_5_1 & - A_5_10 l + A_5_2 & - A_5_11 l + A_5_3 \\ - A_6_9 l + A_6_1 & - A_6_10 l + A_6_2 & - A_6_11 l + A_6_3 \end{matrix}), (\begin{matrix} - l B_1_13 + B_1_1 & - l B_1_14 + B_1_2 & - l B_1_15 + B_1_3 & - l B_1_16 + B_1_4 & - l B_1_17 + B_1_5 \\ - l B_2_13 + B_2_1 & - l B_2_14 + B_2_2 & - l B_2_15 + B_2_3 & - l B_2_16 + B_2_4 & - l B_2_17 + B_2_5 \\ - l B_3_13 + B_3_1 & - l B_3_14 + B_3_2 & - l B_3_15 + B_3_3 & - l B_3_16 + B_3_4 & - l B_3_17 + B_3_5 \end{matrix}), (\begin{matrix} - l C_1_7 + C_1_4 & - l C_1_8 + C_1_5 & - l C_1_9 + C_1_6 \\ - l C_2_7 + C_2_4 & - l C_2_8 + C_2_5 & - l C_2_9 + C_2_6 \\ - l C_3_7 + C_3_4 & - l C_3_8 + C_3_5 & - l C_3_9 + C_3_6 \\ - l C_4_7 + C_4_4 & - l C_4_8 + C_4_5 & - l C_4_9 + C_4_6 \\ - l C_5_7 + C_5_4 & - l C_5_8 + C_5_5 & - l C_5_9 + C_5_6 \end{matrix}))) + Trace (Mul ((\begin{matrix} A_4_1 + \frac{A_7_1}{l} & A_4_2 + \frac{A_7_2}{l} & A_4_3 + \frac{A_7_3}{l} \\ A_5_1 + \frac{A_8_1}{l} & A_5_2 + \frac{A_8_2}{l} & A_5_3 + \frac{A_8_3}{l} \\ A_6_1 + \frac{A_9_1}{l} & A_6_2 + \frac{A_9_2}{l} & A_6_3 + \frac{A_9_3}{l} \end{matrix}), (\begin{matrix} B_1_1 + \frac{B_9_1}{l} & B_1_2 + \frac{B_9_2}{l} & B_1_3 + \frac{B_9_3}{l} & B_1_4 + \frac{B_9_4}{l} & B_1_5 + \frac{B_9_5}{l} \\ B_2_1 + \frac{B_10_1}{l} & B_2_2 + \frac{B_10_2}{l} & B_2_3 + \frac{B_10_3}{l} & B_2_4 + \frac{B_10_4}{l} & B_2_5 + \frac{B_10_5}{l} \\ B_3_1 + \frac{B_11_1}{l} & B_3_2 + \frac{B_11_2}{l} & B_3_3 + \frac{B_11_3}{l} & B_3_4 + \frac{B_11_4}{l} & B_3_5 + \frac{B_11_5}{l} \end{matrix}), (\begin{matrix} C_1_4 + \frac{C_13_4}{l} & C_1_5 + \frac{C_13_5}{l} & C_1_6 + \frac{C_13_6}{l} \\ C_2_4 + \frac{C_14_4}{l} & C_2_5 + \frac{C_14_5}{l} & C_2_6 + \frac{C_14_6}{l} \\ C_3_4 + \frac{C_15_4}{l} & C_3_5 + \frac{C_15_5}{l} & C_3_6 + \frac{C_15_6}{l} \\ C_4_4 + \frac{C_16_4}{l} & C_4_5 + \frac{C_16_5}{l} & C_4_6 + \frac{C_16_6}{l} \\ C_5_4 + \frac{C_17_4}{l} & C_5_5 + \frac{C_17_5}{l} & C_5_6 + \frac{C_17_6}{l} \end{matrix}))) + Trace (Mul ((\begin{matrix} - A_1_4 - \frac{A_7_4}{l} + l A_1_8 + A_4_8 l + A_7_8 & - A_4_1 - \frac{A_7_1}{l} - A_1_5 - \frac{A_7_5}{l} + l A_1_9 + A_4_9 l + A_7_9 & - A_4_2 - \frac{A_7_2}{l} - A_1_6 - \frac{A_7_6}{l} + l A_1_10 + A_4_10 l + A_7_10 & - A_4_3 - \frac{A_7_3}{l} - A_1_7 - \frac{A_7_7}{l} + l A_1_11 + A_4_11 l + A_7_11 \\ - A_2_4 - \frac{A_8_4}{l} + l A_2_8 + A_5_8 l + A_8_8 & - A_5_1 - \frac{A_8_1}{l} - A_2_5 - \frac{A_8_5}{l} + l A_2_9 + A_5_9 l + A_8_9 & - A_5_2 - \frac{A_8_2}{l} - A_2_6 - \frac{A_8_6}{l} + l A_2_10 + A_5_10 l + A_8_10 & - A_5_3 - \frac{A_8_3}{l} - A_2_7 - \frac{A_8_7}{l} + l A_2_11 + A_5_11 l + A_8_11 \\ - A_3_4 - \frac{A_9_4}{l} + l A_3_8 + A_6_8 l + A_9_8 & - A_6_1 - \frac{A_9_1}{l} - A_3_5 - \frac{A_9_5}{l} + l A_3_9 + A_6_9 l + A_9_9 & - A_6_2 - \frac{A_9_2}{l} - A_3_6 - \frac{A_9_6}{l} + l A_3_10 + A_6_10 l + A_9_10 & - A_6_3 - \frac{A_9_3}{l} - A_3_7 - \frac{A_9_7}{l} + l A_3_11 + A_6_11 l + A_9_11 \end{matrix}), (\begin{matrix} B_8_1 & B_8_2 & B_8_3 & B_8_4 & B_8_5 \\ B_9_1 & B_9_2 & B_9_3 & B_9_4 & B_9_5 \\ B_10_1 & B_10_2 & B_10_3 & B_10_4 & B_10_5 \\ B_11_1 & B_11_2 & B_11_3 & B_11_4 & B_11_5 \end{matrix}), (\begin{matrix} C_1_7 & C_1_8 & C_1_9 \\ C_2_7 & C_2_8 & C_2_9 \\ C_3_7 & C_3_8 & C_3_9 \\ C_4_7 & C_4_8 & C_4_9 \\ C_5_7 & C_5_8 & C_5_9 \end{matrix}))) + Trace (Mul ((\begin{matrix} A_4_8 l & - A_4_1 - \frac{A_7_1}{l} + A_4_9 l & - A_4_2 - \frac{A_7_2}{l} + A_4_10 l & - A_4_3 - \frac{A_7_3}{l} + A_4_11 l \\ A_5_8 l & - A_5_1 - \frac{A_8_1}{l} + A_5_9 l & - A_5_2 - \frac{A_8_2}{l} + A_5_10 l & - A_5_3 - \frac{A_8_3}{l} + A_5_11 l \\ A_6_8 l & - A_6_1 - \frac{A_9_1}{l} + A_6_9 l & - A_6_2 - \frac{A_9_2}{l} + A_6_10 l & - A_6_3 - \frac{A_9_3}{l} + A_6_11 l \end{matrix}), (\begin{matrix} 0 & \frac{B_8_1}{l} & \frac{B_8_2}{l} & \frac{B_8_3}{l} & \frac{B_8_4}{l} & \frac{B_8_5}{l} \\ - l B_1_12 & B_1_1 + \frac{B_9_1}{l} - l B_1_13 & B_1_2 + \frac{B_9_2}{l} - l B_1_14 & B_1_3 + \frac{B_9_3}{l} - l B_1_15 & B_1_4 + \frac{B_9_4}{l} - l B_1_16 & B_1_5 + \frac{B_9_5}{l} - l B_1_17 \\ - l B_2_12 & B_2_1 + \frac{B_10_1}{l} - l B_2_13 & B_2_2 + \frac{B_10_2}{l} - l B_2_14 & B_2_3 + \frac{B_10_3}{l} - l B_2_15 & B_2_4 + \frac{B_10_4}{l} - l B_2_16 & B_2_5 + \frac{B_10_5}{l} - l B_2_17 \\ - l B_3_12 & B_3_1 + \frac{B_11_1}{l} - l B_3_13 & B_3_2 + \frac{B_11_2}{l} - l B_3_14 & B_3_3 + \frac{B_11_3}{l} - l B_3_15 & B_3_4 + \frac{B_11_4}{l} - l B_3_16 & B_3_5 + \frac{B_11_5}{l} - l B_3_17 \end{matrix}), (\begin{matrix} \frac{C_12_4}{l} & \frac{C_12_5}{l} & \frac{C_12_6}{l} \\ C_1_4 + \frac{C_13_4}{l} - l C_1_7 & C_1_5 + \frac{C_13_5}{l} - l C_1_8 & C_1_6 + \frac{C_13_6}{l} - l C_1_9 \\ C_2_4 + \frac{C_14_4}{l} - l C_2_7 & C_2_5 + \frac{C_14_5}{l} - l C_2_8 & C_2_6 + \frac{C_14_6}{l} - l C_2_9 \\ C_3_4 + \frac{C_15_4}{l} - l C_3_7 & C_3_5 + \frac{C_15_5}{l} - l C_3_8 & C_3_6 + \frac{C_15_6}{l} - l C_3_9 \\ C_4_4 + \frac{C_16_4}{l} - l C_4_7 & C_4_5 + \frac{C_16_5}{l} - l C_4_8 & C_4_6 + \frac{C_16_6}{l} - l C_4_9 \\ C_5_4 + \frac{C_17_4}{l} - l C_5_7 & C_5_5 + \frac{C_17_5}{l} - l C_5_8 & C_5_6 + \frac{C_17_6}{l} - l C_5_9 \end{matrix}))) + Trace (Mul ((\begin{matrix} - A_1_4 - \frac{A_7_4}{l} + l A_1_8 & - A_1_5 - \frac{A_7_5}{l} + l A_1_9 & - A_1_6 - \frac{A_7_6}{l} + l A_1_10 & - A_1_7 - \frac{A_7_7}{l} + l A_1_11 \\ - A_2_4 - \frac{A_8_4}{l} + l A_2_8 & - A_2_5 - \frac{A_8_5}{l} + l A_2_9 & - A_2_6 - \frac{A_8_6}{l} + l A_2_10 & - A_2_7 - \frac{A_8_7}{l} + l A_2_11 \\ - A_3_4 - \frac{A_9_4}{l} + l A_3_8 & - A_3_5 - \frac{A_9_5}{l} + l A_3_9 & - A_3_6 - \frac{A_9_6}{l} + l A_3_10 & - A_3_7 - \frac{A_9_7}{l} + l A_3_11 \end{matrix}), (\begin{matrix} - B_4_12 l & B_4_1 + \frac{B_8_1}{l} - B_4_13 l & B_4_2 + \frac{B_8_2}{l} - B_4_14 l & B_4_3 + \frac{B_8_3}{l} - B_4_15 l & B_4_4 + \frac{B_8_4}{l} - B_4_16 l & B_4_5 + \frac{B_8_5}{l} - B_4_17 l \\ - B_5_12 l & B_5_1 + \frac{B_9_1}{l} - B_5_13 l & B_5_2 + \frac{B_9_2}{l} - B_5_14 l & B_5_3 + \frac{B_9_3}{l} - B_5_15 l & B_5_4 + \frac{B_9_4}{l} - B_5_16 l & B_5_5 + \frac{B_9_5}{l} - B_5_17 l \\ - B_6_12 l & B_6_1 + \frac{B_10_1}{l} - B_6_13 l & B_6_2 + \frac{B_10_2}{l} - B_6_14 l & B_6_3 + \frac{B_10_3}{l} - B_6_15 l & B_6_4 + \frac{B_10_4}{l} - B_6_16 l & B_6_5 + \frac{B_10_5}{l} - B_6_17 l \\ - B_7_12 l & B_7_1 + \frac{B_11_1}{l} - B_7_13 l & B_7_2 + \frac{B_11_2}{l} - B_7_14 l & B_7_3 + \frac{B_11_3}{l} - B_7_15 l & B_7_4 + \frac{B_11_4}{l} - B_7_16 l & B_7_5 + \frac{B_11_5}{l} - B_7_17 l \end{matrix}), (\begin{matrix} \frac{C_12_1}{l} & \frac{C_12_2}{l} & \frac{C_12_3}{l} \\ C_1_1 + \frac{C_13_1}{l} - l C_1_7 & C_1_2 + \frac{C_13_2}{l} - l C_1_8 & C_1_3 + \frac{C_13_3}{l} - l C_1_9 \\ C_2_1 + \frac{C_14_1}{l} - l C_2_7 & C_2_2 + \frac{C_14_2}{l} - l C_2_8 & C_2_3 + \frac{C_14_3}{l} - l C_2_9 \\ C_3_1 + \frac{C_15_1}{l} - l C_3_7 & C_3_2 + \frac{C_15_2}{l} - l C_3_8 & C_3_3 + \frac{C_15_3}{l} - l C_3_9 \\ C_4_1 + \frac{C_16_1}{l} - l C_4_7 & C_4_2 + \frac{C_16_2}{l} - l C_4_8 & C_4_3 + \frac{C_16_3}{l} - l C_4_9 \\ C_5_1 + \frac{C_17_1}{l} - l C_5_7 & C_5_2 + \frac{C_17_2}{l} - l C_5_8 & C_5_3 + \frac{C_17_3}{l} - l C_5_9 \end{matrix}))) + Trace (Mul ((\begin{matrix} - l A_1_8 + A_1_4 & - l A_1_9 + A_1_5 & - l A_1_10 + A_1_6 & - l A_1_11 + A_1_7 \\ - l A_2_8 + A_2_4 & - l A_2_9 + A_2_5 & - l A_2_10 + A_2_6 & - l A_2_11 + A_2_7 \\ - l A_3_8 + A_3_4 & - l A_3_9 + A_3_5 & - l A_3_10 + A_3_6 & - l A_3_11 + A_3_7 \end{matrix}), (\begin{matrix} - B_4_13 l + B_4_1 & - B_4_14 l + B_4_2 & - B_4_15 l + B_4_3 & - B_4_16 l + B_4_4 & - B_4_17 l + B_4_5 \\ - B_5_13 l + B_5_1 & - B_5_14 l + B_5_2 & - B_5_15 l + B_5_3 & - B_5_16 l + B_5_4 & - B_5_17 l + B_5_5 \\ - B_6_13 l + B_6_1 & - B_6_14 l + B_6_2 & - B_6_15 l + B_6_3 & - B_6_16 l + B_6_4 & - B_6_17 l + B_6_5 \\ - B_7_13 l + B_7_1 & - B_7_14 l + B_7_2 & - B_7_15 l + B_7_3 & - B_7_16 l + B_7_4 & - B_7_17 l + B_7_5 \end{matrix}), (\begin{matrix} - l C_1_7 + C_1_1 & - l C_1_8 + C_1_2 & - l C_1_9 + C_1_3 \\ - l C_2_7 + C_2_1 & - l C_2_8 + C_2_2 & - l C_2_9 + C_2_3 \\ - l C_3_7 + C_3_1 & - l C_3_8 + C_3_2 & - l C_3_9 + C_3_3 \\ - l C_4_7 + C_4_1 & - l C_4_8 + C_4_2 & - l C_4_9 + C_4_3 \\ - l C_5_7 + C_5_1 & - l C_5_8 + C_5_2 & - l C_5_9 + C_5_3 \end{matrix}))) + Trace (Mul ((\begin{matrix} A_1_4 + \frac{A_7_4}{l} & A_1_5 + \frac{A_7_5}{l} & A_1_6 + \frac{A_7_6}{l} & A_1_7 + \frac{A_7_7}{l} \\ A_2_4 + \frac{A_8_4}{l} & A_2_5 + \frac{A_8_5}{l} & A_2_6 + \frac{A_8_6}{l} & A_2_7 + \frac{A_8_7}{l} \\ A_3_4 + \frac{A_9_4}{l} & A_3_5 + \frac{A_9_5}{l} & A_3_6 + \frac{A_9_6}{l} & A_3_7 + \frac{A_9_7}{l} \end{matrix}), (\begin{matrix} B_4_1 + \frac{B_8_1}{l} & B_4_2 + \frac{B_8_2}{l} & B_4_3 + \frac{B_8_3}{l} & B_4_4 + \frac{B_8_4}{l} & B_4_5 + \frac{B_8_5}{l} \\ B_5_1 + \frac{B_9_1}{l} & B_5_2 + \frac{B_9_2}{l} & B_5_3 + \frac{B_9_3}{l} & B_5_4 + \frac{B_9_4}{l} & B_5_5 + \frac{B_9_5}{l} \\ B_6_1 + \frac{B_10_1}{l} & B_6_2 + \frac{B_10_2}{l} & B_6_3 + \frac{B_10_3}{l} & B_6_4 + \frac{B_10_4}{l} & B_6_5 + \frac{B_10_5}{l} \\ B_7_1 + \frac{B_11_1}{l} & B_7_2 + \frac{B_11_2}{l} & B_7_3 + \frac{B_11_3}{l} & B_7_4 + \frac{B_11_4}{l} & B_7_5 + \frac{B_11_5}{l} \end{matrix}), (\begin{matrix} C_1_1 + \frac{C_13_1}{l} & C_1_2 + \frac{C_13_2}{l} & C_1_3 + \frac{C_13_3}{l} \\ C_2_1 + \frac{C_14_1}{l} & C_2_2 + \frac{C_14_2}{l} & C_2_3 + \frac{C_14_3}{l} \\ C_3_1 + \frac{C_15_1}{l} & C_3_2 + \frac{C_15_2}{l} & C_3_3 + \frac{C_15_3}{l} \\ C_4_1 + \frac{C_16_1}{l} & C_4_2 + \frac{C_16_2}{l} & C_4_3 + \frac{C_16_3}{l} \\ C_5_1 + \frac{C_17_1}{l} & C_5_2 + \frac{C_17_2}{l} & C_5_3 + \frac{C_17_3}{l} \end{matrix}))) + Trace (Mul ((\begin{matrix} A_4_4 & A_4_5 & A_4_6 & A_4_7 \\ A_5_4 & A_5_5 & A_5_6 & A_5_7 \\ A_6_4 & A_6_5 & A_6_6 & A_6_7 \end{matrix}), (\begin{matrix} B_4_1 & B_4_2 & B_4_3 & B_4_4 & B_4_5 \\ B_5_1 & B_5_2 & B_5_3 & B_5_4 & B_5_5 \\ B_6_1 & B_6_2 & B_6_3 & B_6_4 & B_6_5 \\ B_7_1 & B_7_2 & B_7_3 & B_7_4 & B_7_5 \end{matrix}), (\begin{matrix} C_1_4 & C_1_5 & C_1_6 \\ C_2_4 & C_2_5 & C_2_6 \\ C_3_4 & C_3_5 & C_3_6 \\ C_4_4 & C_4_5 & C_4_6 \\ C_5_4 & C_5_5 & C_5_6 \end{matrix}))) + Trace (Mul ((\begin{matrix} A_4_1 & A_4_2 & A_4_3 \\ A_5_1 & A_5_2 & A_5_3 \\ A_6_1 & A_6_2 & A_6_3 \end{matrix}), (\begin{matrix} B_1_6 & B_1_7 & B_1_8 & B_1_9 & B_1_10 & B_1_11 \\ B_2_6 & B_2_7 & B_2_8 & B_2_9 & B_2_10 & B_2_11 \\ B_3_6 & B_3_7 & B_3_8 & B_3_9 & B_3_10 & B_3_11 \end{matrix}), (\begin{matrix} C_6_4 & C_6_5 & C_6_6 \\ C_7_4 & C_7_5 & C_7_6 \\ C_8_4 & C_8_5 & C_8_6 \\ C_9_4 & C_9_5 & C_9_6 \\ C_10_4 & C_10_5 & C_10_6 \\ C_11_4 & C_11_5 & C_11_6 \end{matrix}))) + Trace (Mul ((\begin{matrix} A_1_1 & A_1_2 & A_1_3 \\ A_2_1 & A_2_2 & A_2_3 \\ A_3_1 & A_3_2 & A_3_3 \end{matrix}), (\begin{matrix} B_1_1 & B_1_2 & B_1_3 & B_1_4 & B_1_5 \\ B_2_1 & B_2_2 & B_2_3 & B_2_4 & B_2_5 \\ B_3_1 & B_3_2 & B_3_3 & B_3_4 & B_3_5 \end{matrix}), (\begin{matrix} C_1_1 & C_1_2 & C_1_3 \\ C_2_1 & C_2_2 & C_2_3 \\ C_3_1 & C_3_2 & C_3_3 \\ C_4_1 & C_4_2 & C_4_3 \\ C_5_1 & C_5_2 & C_5_3 \end{matrix}))) + Trace (Mul ((\begin{matrix} A_1_4 & A_1_5 & A_1_6 & A_1_7 \\ A_2_4 & A_2_5 & A_2_6 & A_2_7 \\ A_3_4 & A_3_5 & A_3_6 & A_3_7 \end{matrix}), (\begin{matrix} B_4_6 & B_4_7 & B_4_8 & B_4_9 & B_4_10 & B_4_11 \\ B_5_6 & B_5_7 & B_5_8 & B_5_9 & B_5_10 & B_5_11 \\ B_6_6 & B_6_7 & B_6_8 & B_6_9 & B_6_10 & B_6_11 \\ B_7_6 & B_7_7 & B_7_8 & B_7_9 & B_7_10 & B_7_11 \end{matrix}), (\begin{matrix} C_6_1 & C_6_2 & C_6_3 \\ C_7_1 & C_7_2 & C_7_3 \\ C_8_1 & C_8_2 & C_8_3 \\ C_9_1 & C_9_2 & C_9_3 \\ C_10_1 & C_10_2 & C_10_3 \\ C_11_1 & C_11_2 & C_11_3 \end{matrix})))

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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