Description of fast matrix multiplication algorithm: ⟨3×24×32:1696⟩

Algorithm type

X2⁢Y15⁢Z2+X⁢Y15⁢Z2+2⁢X⁢Y15⁢Z+6⁢X3⁢Y10⁢Z3+3⁢X2⁢Y12⁢Z2+3⁢X⁢Y12⁢Z2+18⁢X3⁢Y8⁢Z3+10⁢X2⁢Y10⁢Z2+14⁢X⁢Y12⁢Z+2⁢X2⁢Y10⁢Z+5⁢X2⁢Y9⁢Z2+30⁢X3⁢Y6⁢Z3+54⁢X2⁢Y8⁢Z2+10⁢X⁢Y10⁢Z+5⁢X⁢Y9⁢Z2+6⁢X2⁢Y8⁢Z+6⁢X⁢Y9⁢Z+66⁢X3⁢Y4⁢Z3+49⁢X2⁢Y6⁢Z2+62⁢X⁢Y8⁢Z+10⁢X2⁢Y6⁢Z+5⁢X2⁢Y5⁢Z2+11⁢X⁢Y6⁢Z2+X2⁢Y5⁢Z+77⁢X2⁢Y4⁢Z2+88⁢X⁢Y6⁢Z+3⁢X⁢Y5⁢Z2+25⁢X2⁢Y4⁢Z+25⁢X2⁢Y3⁢Z2+43⁢X⁢Y5⁢Z+9⁢X⁢Y4⁢Z2+5⁢X2⁢Y3⁢Z+139⁢X2⁢Y2⁢Z2+151⁢X⁢Y4⁢Z+15⁢X⁢Y3⁢Z2+11⁢X2⁢Y2⁢Z+143⁢X⁢Y3⁢Z+33⁢X⁢Y2⁢Z2+217⁢X⁢Y2⁢Z+332⁢X⁢Y⁢ZX2Y15Z2XY15Z22XY15Z6X3Y10Z33X2Y12Z23XY12Z218X3Y8Z310X2Y10Z214XY12Z2X2Y10Z5X2Y9Z230X3Y6Z354X2Y8Z210XY10Z5XY9Z26X2Y8Z6XY9Z66X3Y4Z349X2Y6Z262XY8Z10X2Y6Z5X2Y5Z211XY6Z2X2Y5Z77X2Y4Z288XY6Z3XY5Z225X2Y4Z25X2Y3Z243XY5Z9XY4Z25X2Y3Z139X2Y2Z2151XY4Z15XY3Z211X2Y2Z143XY3Z33XY2Z2217XY2Z332XYZX^2*Y^15*Z^2+X*Y^15*Z^2+2*X*Y^15*Z+6*X^3*Y^10*Z^3+3*X^2*Y^12*Z^2+3*X*Y^12*Z^2+18*X^3*Y^8*Z^3+10*X^2*Y^10*Z^2+14*X*Y^12*Z+2*X^2*Y^10*Z+5*X^2*Y^9*Z^2+30*X^3*Y^6*Z^3+54*X^2*Y^8*Z^2+10*X*Y^10*Z+5*X*Y^9*Z^2+6*X^2*Y^8*Z+6*X*Y^9*Z+66*X^3*Y^4*Z^3+49*X^2*Y^6*Z^2+62*X*Y^8*Z+10*X^2*Y^6*Z+5*X^2*Y^5*Z^2+11*X*Y^6*Z^2+X^2*Y^5*Z+77*X^2*Y^4*Z^2+88*X*Y^6*Z+3*X*Y^5*Z^2+25*X^2*Y^4*Z+25*X^2*Y^3*Z^2+43*X*Y^5*Z+9*X*Y^4*Z^2+5*X^2*Y^3*Z+139*X^2*Y^2*Z^2+151*X*Y^4*Z+15*X*Y^3*Z^2+11*X^2*Y^2*Z+143*X*Y^3*Z+33*X*Y^2*Z^2+217*X*Y^2*Z+332*X*Y*Z

Algorithm definition

The algorithm ⟨3×24×32:1696⟩ could be constructed using the following decomposition:

⟨3×24×32:1696⟩ = ⟨2×4×4:26⟩ + ⟨1×4×4:16⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨1×4×4:16⟩ + ⟨2×4×4:26⟩ + ⟨1×4×4:16⟩ + ⟨1×4×4:16⟩ + ⟨2×4×4:26⟩ + ⟨1×4×4:16⟩ + ⟨1×4×4:16⟩ + ⟨1×4×4:16⟩ + ⟨2×4×4:26⟩ + ⟨1×4×4:16⟩ + ⟨1×4×4:16⟩ + ⟨1×4×4:16⟩ + ⟨1×4×4:16⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨1×4×4:16⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨1×4×4:16⟩ + ⟨1×4×4:16⟩ + ⟨1×4×4:16⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨1×4×4:16⟩ + ⟨2×4×4:26⟩ + ⟨1×4×4:16⟩ + ⟨1×4×4:16⟩ + ⟨1×4×4:16⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨1×4×4:16⟩ + ⟨2×4×4:26⟩ + ⟨1×4×4:16⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨1×4×4:16⟩ + ⟨1×4×4:16⟩ + ⟨1×4×4:16⟩ + ⟨2×4×4:26⟩ + ⟨1×4×4:16⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨1×4×4:16⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨1×4×4:16⟩ + ⟨1×4×4:16⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩ + ⟨2×4×4:26⟩.

This decomposition is defined by the following equality:

Trace⁡Mul⁡A_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_1_24A_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_2_24A_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_3_24B_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_1_29B_1_30B_1_31B_1_32B_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_2_29B_2_30B_2_31B_2_32B_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_3_29B_3_30B_3_31B_3_32B_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_4_29B_4_30B_4_31B_4_32B_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_5_29B_5_30B_5_31B_5_32B_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_6_29B_6_30B_6_31B_6_32B_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_7_29B_7_30B_7_31B_7_32B_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_8_29B_8_30B_8_31B_8_32B_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_28B_9_29B_9_30B_9_31B_9_32B_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_10_28B_10_29B_10_30B_10_31B_10_32B_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_11_28B_11_29B_11_30B_11_31B_11_32B_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_12_28B_12_29B_12_30B_12_31B_12_32B_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_13_28B_13_29B_13_30B_13_31B_13_32B_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_14_28B_14_29B_14_30B_14_31B_14_32B_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_15_28B_15_29B_15_30B_15_31B_15_32B_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_16_28B_16_29B_16_30B_16_31B_16_32B_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_17_28B_17_29B_17_30B_17_31B_17_32B_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_18_28B_18_29B_18_30B_18_31B_18_32B_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_19_28B_19_29B_19_30B_19_31B_19_32B_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_20_28B_20_29B_20_30B_20_31B_20_32B_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_21_28B_21_29B_21_30B_21_31B_21_32B_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_22_28B_22_29B_22_30B_22_31B_22_32B_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_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