Description of fast matrix multiplication algorithm: ⟨14×19×22:3525⟩

Algorithm type

2X8Y8Z7+X8Y7Z8+X7Y8Z8+3X8Y8Z6+3X8Y7Z7+5X7Y8Z7+2X8Y7Z6+X7Y8Z6+8X7Y7Z7+X6Y7Z8+3X7Y7Z6+X11Y4Z4+2X7Y4Z8+2X8Y7Z3+X4Y10Z3+2X8Y4Z4+5X4Y8Z4+17X4Y6Z6+2X4Y4Z8+3X8Y4Z3+2X4Y8Z3+5X4Y7Z4+32X4Y6Z5+5X4Y4Z7+X3Y4Z8+2X7Y4Z3+4X4Y6Z4+X4Y4Z6+3X4Y3Z7+2X3Y8Z3+X3Y7Z4+24X2Y6Z6+X4Y5Z4+X4Y3Z6+X3Y7Z3+48X2Y6Z5+5X8Y2Z2+X6Y4Z2+2X6Y2Z4+159X4Y4Z4+3X4Y2Z6+X3Y5Z4+X3Y3Z6+4X2Y8Z2+6X2Y4Z6+7X2Y2Z8+3X7Y2Z2+X5Y2Z4+28X4Y4Z3+29X4Y3Z4+24X3Y4Z4+X3Y2Z6+2X2Y7Z2+X2YZ8+19X6Y2Z2+X5YZ4+31X4Y4Z2+73X4Y3Z3+8X4Y2Z4+15X3Y4Z3+15X3Y3Z4+32X2Y6Z2+16X2Y4Z4+13X2Y2Z6+XY3Z6+X6YZ2+3X5Y2Z2+2X4Y4Z+2X4Y3Z2+4X3Y2Z4+2X2Y6Z+5X2Y5Z2+X2Y4Z3+3X2Y3Z4+2X4Y3Z+57X4Y2Z2+103X2Y4Z2+161X2Y3Z3+55X2Y2Z4+3XY5Z2+2XY3Z4+10X4Y2Z+21X3Y2Z2+7X2Y4Z+42X2Y3Z2+22X2Y2Z3+8X2YZ4+4XY4Z2+96XY3Z3+6XY2Z4+36X4YZ+13X3Y2Z+18X3YZ2+687X2Y2Z2+26X2YZ3+33XY4Z+3XY3Z2+37XY2Z3+37XYZ4+108X3YZ+83X2Y2Z+20X2YZ2+174XY3Z+99XY2Z2+66XYZ3+132X2YZ+271XY2Z+96XYZ2+264XYZ2X8Y8Z7X8Y7Z8X7Y8Z83X8Y8Z63X8Y7Z75X7Y8Z72X8Y7Z6X7Y8Z68X7Y7Z7X6Y7Z83X7Y7Z6X11Y4Z42X7Y4Z82X8Y7Z3X4Y10Z32X8Y4Z45X4Y8Z417X4Y6Z62X4Y4Z83X8Y4Z32X4Y8Z35X4Y7Z432X4Y6Z55X4Y4Z7X3Y4Z82X7Y4Z34X4Y6Z4X4Y4Z63X4Y3Z72X3Y8Z3X3Y7Z424X2Y6Z6X4Y5Z4X4Y3Z6X3Y7Z348X2Y6Z55X8Y2Z2X6Y4Z22X6Y2Z4159X4Y4Z43X4Y2Z6X3Y5Z4X3Y3Z64X2Y8Z26X2Y4Z67X2Y2Z83X7Y2Z2X5Y2Z428X4Y4Z329X4Y3Z424X3Y4Z4X3Y2Z62X2Y7Z2X2YZ819X6Y2Z2X5YZ431X4Y4Z273X4Y3Z38X4Y2Z415X3Y4Z315X3Y3Z432X2Y6Z216X2Y4Z413X2Y2Z6XY3Z6X6YZ23X5Y2Z22X4Y4Z2X4Y3Z24X3Y2Z42X2Y6Z5X2Y5Z2X2Y4Z33X2Y3Z42X4Y3Z57X4Y2Z2103X2Y4Z2161X2Y3Z355X2Y2Z43XY5Z22XY3Z410X4Y2Z21X3Y2Z27X2Y4Z42X2Y3Z222X2Y2Z38X2YZ44XY4Z296XY3Z36XY2Z436X4YZ13X3Y2Z18X3YZ2687X2Y2Z226X2YZ333XY4Z3XY3Z237XY2Z337XYZ4108X3YZ83X2Y2Z20X2YZ2174XY3Z99XY2Z266XYZ3132X2YZ271XY2Z96XYZ2264XYZ2*X^8*Y^8*Z^7+X^8*Y^7*Z^8+X^7*Y^8*Z^8+3*X^8*Y^8*Z^6+3*X^8*Y^7*Z^7+5*X^7*Y^8*Z^7+2*X^8*Y^7*Z^6+X^7*Y^8*Z^6+8*X^7*Y^7*Z^7+X^6*Y^7*Z^8+3*X^7*Y^7*Z^6+X^11*Y^4*Z^4+2*X^7*Y^4*Z^8+2*X^8*Y^7*Z^3+X^4*Y^10*Z^3+2*X^8*Y^4*Z^4+5*X^4*Y^8*Z^4+17*X^4*Y^6*Z^6+2*X^4*Y^4*Z^8+3*X^8*Y^4*Z^3+2*X^4*Y^8*Z^3+5*X^4*Y^7*Z^4+32*X^4*Y^6*Z^5+5*X^4*Y^4*Z^7+X^3*Y^4*Z^8+2*X^7*Y^4*Z^3+4*X^4*Y^6*Z^4+X^4*Y^4*Z^6+3*X^4*Y^3*Z^7+2*X^3*Y^8*Z^3+X^3*Y^7*Z^4+24*X^2*Y^6*Z^6+X^4*Y^5*Z^4+X^4*Y^3*Z^6+X^3*Y^7*Z^3+48*X^2*Y^6*Z^5+5*X^8*Y^2*Z^2+X^6*Y^4*Z^2+2*X^6*Y^2*Z^4+159*X^4*Y^4*Z^4+3*X^4*Y^2*Z^6+X^3*Y^5*Z^4+X^3*Y^3*Z^6+4*X^2*Y^8*Z^2+6*X^2*Y^4*Z^6+7*X^2*Y^2*Z^8+3*X^7*Y^2*Z^2+X^5*Y^2*Z^4+28*X^4*Y^4*Z^3+29*X^4*Y^3*Z^4+24*X^3*Y^4*Z^4+X^3*Y^2*Z^6+2*X^2*Y^7*Z^2+X^2*Y*Z^8+19*X^6*Y^2*Z^2+X^5*Y*Z^4+31*X^4*Y^4*Z^2+73*X^4*Y^3*Z^3+8*X^4*Y^2*Z^4+15*X^3*Y^4*Z^3+15*X^3*Y^3*Z^4+32*X^2*Y^6*Z^2+16*X^2*Y^4*Z^4+13*X^2*Y^2*Z^6+X*Y^3*Z^6+X^6*Y*Z^2+3*X^5*Y^2*Z^2+2*X^4*Y^4*Z+2*X^4*Y^3*Z^2+4*X^3*Y^2*Z^4+2*X^2*Y^6*Z+5*X^2*Y^5*Z^2+X^2*Y^4*Z^3+3*X^2*Y^3*Z^4+2*X^4*Y^3*Z+57*X^4*Y^2*Z^2+103*X^2*Y^4*Z^2+161*X^2*Y^3*Z^3+55*X^2*Y^2*Z^4+3*X*Y^5*Z^2+2*X*Y^3*Z^4+10*X^4*Y^2*Z+21*X^3*Y^2*Z^2+7*X^2*Y^4*Z+42*X^2*Y^3*Z^2+22*X^2*Y^2*Z^3+8*X^2*Y*Z^4+4*X*Y^4*Z^2+96*X*Y^3*Z^3+6*X*Y^2*Z^4+36*X^4*Y*Z+13*X^3*Y^2*Z+18*X^3*Y*Z^2+687*X^2*Y^2*Z^2+26*X^2*Y*Z^3+33*X*Y^4*Z+3*X*Y^3*Z^2+37*X*Y^2*Z^3+37*X*Y*Z^4+108*X^3*Y*Z+83*X^2*Y^2*Z+20*X^2*Y*Z^2+174*X*Y^3*Z+99*X*Y^2*Z^2+66*X*Y*Z^3+132*X^2*Y*Z+271*X*Y^2*Z+96*X*Y*Z^2+264*X*Y*Z

Algorithm definition

The algorithm ⟨14×19×22:3525⟩ could be constructed using the following decomposition:

⟨14×19×22:3525⟩ = ⟨8×10×12:630⟩ + ⟨6×10×10:406⟩ + ⟨8×9×12:560⟩ + ⟨6×10×12:476⟩ + ⟨6×9×12:434⟩ + ⟨8×10×10:532⟩ + ⟨8×9×10:487⟩.

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_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_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_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_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_15A_5_16A_5_17A_5_18A_5_19A_6_1A_6_2A_6_3A_6_4A_6_5A_6_6A_6_7A_6_8A_6_9A_6_10A_6_11A_6_12A_6_13A_6_14A_6_15A_6_16A_6_17A_6_18A_6_19A_7_1A_7_2A_7_3A_7_4A_7_5A_7_6A_7_7A_7_8A_7_9A_7_10A_7_11A_7_12A_7_13A_7_14A_7_15A_7_16A_7_17A_7_18A_7_19A_8_1A_8_2A_8_3A_8_4A_8_5A_8_6A_8_7A_8_8A_8_9A_8_10A_8_11A_8_12A_8_13A_8_14A_8_15A_8_16A_8_17A_8_18A_8_19A_9_1A_9_2A_9_3A_9_4A_9_5A_9_6A_9_7A_9_8A_9_9A_9_10A_9_11A_9_12A_9_13A_9_14A_9_15A_9_16A_9_17A_9_18A_9_19A_10_1A_10_2A_10_3A_10_4A_10_5A_10_6A_10_7A_10_8A_10_9A_10_10A_10_11A_10_12A_10_13A_10_14A_10_15A_10_16A_10_17A_10_18A_10_19A_11_1A_11_2A_11_3A_11_4A_11_5A_11_6A_11_7A_11_8A_11_9A_11_10A_11_11A_11_12A_11_13A_11_14A_11_15A_11_16A_11_17A_11_18A_11_19A_12_1A_12_2A_12_3A_12_4A_12_5A_12_6A_12_7A_12_8A_12_9A_12_10A_12_11A_12_12A_12_13A_12_14A_12_15A_12_16A_12_17A_12_18A_12_19A_13_1A_13_2A_13_3A_13_4A_13_5A_13_6A_13_7A_13_8A_13_9A_13_10A_13_11A_13_12A_13_13A_13_14A_13_15A_13_16A_13_17A_13_18A_13_19A_14_1A_14_2A_14_3A_14_4A_14_5A_14_6A_14_7A_14_8A_14_9A_14_10A_14_11A_14_12A_14_13A_14_14A_14_15A_14_16A_14_17A_14_18A_14_19B_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_22C_1_1C_1_2C_1_3C_1_4C_1_5C_1_6C_1_7C_1_8C_1_9C_1_10C_1_11C_1_12C_1_13C_1_14C_2_1C_2_2C_2_3C_2_4C_2_5C_2_6C_2_7C_2_8C_2_9C_2_10C_2_11C_2_12C_2_13C_2_14C_3_1C_3_2C_3_3C_3_4C_3_5C_3_6C_3_7C_3_8C_3_9C_3_10C_3_11C_3_12C_3_13C_3_14C_4_1C_4_2C_4_3C_4_4C_4_5C_4_6C_4_7C_4_8C_4_9C_4_10C_4_11C_4_12C_4_13C_4_14C_5_1C_5_2C_5_3C_5_4C_5_5C_5_6C_5_7C_5_8C_5_9C_5_10C_5_11C_5_12C_5_13C_5_14C_6_1C_6_2C_6_3C_6_4C_6_5C_6_6C_6_7C_6_8C_6_9C_6_10C_6_11C_6_12C_6_13C_6_14C_7_1C_7_2C_7_3C_7_4C_7_5C_7_6C_7_7C_7_8C_7_9C_7_10C_7_11C_7_12C_7_13C_7_14C_8_1C_8_2C_8_3C_8_4C_8_5C_8_6C_8_7C_8_8C_8_9C_8_10C_8_11C_8_12C_8_13C_8_14C_9_1C_9_2C_9_3C_9_4C_9_5C_9_6C_9_7C_9_8C_9_9C_9_10C_9_11C_9_12C_9_13C_9_14C_10_1C_10_2C_10_3C_10_4C_10_5C_10_6C_10_7C_10_8C_10_9C_10_10C_10_11C_10_12C_10_13C_10_14C_11_1C_11_2C_11_3C_11_4C_11_5C_11_6C_11_7C_11_8C_11_9C_11_10C_11_11C_11_12C_11_13C_11_14C_12_1C_12_2C_12_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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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