Description of fast matrix multiplication algorithm: ⟨9×16×17:1524⟩

Algorithm type

16⁢X4⁢Y6⁢Z4+33⁢X4⁢Y5⁢Z4+65⁢X4⁢Y4⁢Z4+9⁢X9⁢Y⁢Z+9⁢X⁢Y⁢Z9+48⁢X6⁢Y2⁢Z2+12⁢X4⁢Y4⁢Z2+92⁢X2⁢Y6⁢Z2+71⁢X2⁢Y4⁢Z4+48⁢X2⁢Y2⁢Z6+6⁢X6⁢Y2⁢Z+38⁢X4⁢Y3⁢Z2+46⁢X2⁢Y5⁢Z2+36⁢X2⁢Y3⁢Z4+6⁢X⁢Y2⁢Z6+3⁢X6⁢Y⁢Z+22⁢X4⁢Y2⁢Z2+129⁢X2⁢Y4⁢Z2+19⁢X2⁢Y2⁢Z4+72⁢X⁢Y6⁢Z+3⁢X⁢Y⁢Z6+6⁢X3⁢Y3⁢Z+6⁢X3⁢Y2⁢Z2+18⁢X3⁢Y⁢Z3+6⁢X2⁢Y4⁢Z+3⁢X2⁢Y3⁢Z2+6⁢X2⁢Y2⁢Z3+14⁢X⁢Y5⁢Z+70⁢X⁢Y4⁢Z2+6⁢X⁢Y3⁢Z3+42⁢X3⁢Y2⁢Z+3⁢X3⁢Y⁢Z2+90⁢X2⁢Y3⁢Z+102⁢X2⁢Y2⁢Z2+3⁢X2⁢Y⁢Z3+55⁢X⁢Y4⁢Z+36⁢X⁢Y3⁢Z2+42⁢X⁢Y2⁢Z3+9⁢X3⁢Y⁢Z+9⁢X2⁢Y2⁢Z+24⁢X2⁢Y⁢Z2+6⁢X⁢Y3⁢Z+11⁢X⁢Y2⁢Z2+9⁢X⁢Y⁢Z3+12⁢X2⁢Y⁢Z+9⁢X⁢Y2⁢Z+144⁢X⁢Y⁢Z16X4Y6Z433X4Y5Z465X4Y4Z49X9YZ9XYZ948X6Y2Z212X4Y4Z292X2Y6Z271X2Y4Z448X2Y2Z66X6Y2Z38X4Y3Z246X2Y5Z236X2Y3Z46XY2Z63X6YZ22X4Y2Z2129X2Y4Z219X2Y2Z472XY6Z3XYZ66X3Y3Z6X3Y2Z218X3YZ36X2Y4Z3X2Y3Z26X2Y2Z314XY5Z70XY4Z26XY3Z342X3Y2Z3X3YZ290X2Y3Z102X2Y2Z23X2YZ355XY4Z36XY3Z242XY2Z39X3YZ9X2Y2Z24X2YZ26XY3Z11XY2Z29XYZ312X2YZ9XY2Z144XYZ16*X^4*Y^6*Z^4+33*X^4*Y^5*Z^4+65*X^4*Y^4*Z^4+9*X^9*Y*Z+9*X*Y*Z^9+48*X^6*Y^2*Z^2+12*X^4*Y^4*Z^2+92*X^2*Y^6*Z^2+71*X^2*Y^4*Z^4+48*X^2*Y^2*Z^6+6*X^6*Y^2*Z+38*X^4*Y^3*Z^2+46*X^2*Y^5*Z^2+36*X^2*Y^3*Z^4+6*X*Y^2*Z^6+3*X^6*Y*Z+22*X^4*Y^2*Z^2+129*X^2*Y^4*Z^2+19*X^2*Y^2*Z^4+72*X*Y^6*Z+3*X*Y*Z^6+6*X^3*Y^3*Z+6*X^3*Y^2*Z^2+18*X^3*Y*Z^3+6*X^2*Y^4*Z+3*X^2*Y^3*Z^2+6*X^2*Y^2*Z^3+14*X*Y^5*Z+70*X*Y^4*Z^2+6*X*Y^3*Z^3+42*X^3*Y^2*Z+3*X^3*Y*Z^2+90*X^2*Y^3*Z+102*X^2*Y^2*Z^2+3*X^2*Y*Z^3+55*X*Y^4*Z+36*X*Y^3*Z^2+42*X*Y^2*Z^3+9*X^3*Y*Z+9*X^2*Y^2*Z+24*X^2*Y*Z^2+6*X*Y^3*Z+11*X*Y^2*Z^2+9*X*Y*Z^3+12*X^2*Y*Z+9*X*Y^2*Z+144*X*Y*Z

Algorithm definition

The algorithm ⟨9×16×17:1524⟩ could be constructed using the following decomposition:

⟨9×16×17:1524⟩ = ⟨9×16×1:144⟩ + ⟨9×16×16:1380⟩.

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_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_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_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_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_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_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_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_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_16B_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_17C_1_1C_1_2C_1_3C_1_4C_1_5C_1_6C_1_7C_1_8C_1_9C_2_1C_2_2C_2_3C_2_4C_2_5C_2_6C_2_7C_2_8C_2_9C_3_1C_3_2C_3_3C_3_4C_3_5C_3_6C_3_7C_3_8C_3_9C_4_1C_4_2C_4_3C_4_4C_4_5C_4_6C_4_7C_4_8C_4_9C_5_1C_5_2C_5_3C_5_4C_5_5C_5_6C_5_7C_5_8C_5_9C_6_1C_6_2C_6_3C_6_4C_6_5C_6_6C_6_7C_6_8C_6_9C_7_1C_7_2C_7_3C_7_4C_7_5C_7_6C_7_7C_7_8C_7_9C_8_1C_8_2C_8_3C_8_4C_8_5C_8_6C_8_7C_8_8C_8_9C_9_1C_9_2C_9_3C_9_4C_9_5C_9_6C_9_7C_9_8C_9_9C_10_1C_10_2C_10_3C_10_4C_10_5C_10_6C_10_7C_10_8C_10_9C_11_1C_11_2C_11_3C_11_4C_11_5C_11_6C_11_7C_11_8C_11_9C_12_1C_12_2C_12_3C_12_4C_12_5C_12_6C_12_7C_12_8C_12_9C_13_1C_13_2C_13_3C_13_4C_13_5C_13_6C_13_7C_13_8C_13_9C_14_1C_14_2C_14_3C_14_4C_14_5C_14_6C_14_7C_14_8C_14_9C_15_1C_15_2C_15_3C_15_4C_15_5C_15_6C_15_7C_15_8C_15_9C_16_1C_16_2C_16_3C_16_4C_16_5C_16_6C_16_7C_16_8C_16_9C_17_1C_17_2C_17_3C_17_4C_17_5C_17_6C_17_7C_17_8C_17_9=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_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_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_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_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_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_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_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_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_16B_1_1B_2_1B_3_1B_4_1B_5_1B_6_1B_7_1B_8_1B_9_1B_10_1B_11_1B_12_1B_13_1B_14_1B_15_1B_16_1C_1_1C_1_2C_1_3C_1_4C_1_5C_1_6C_1_7C_1_8C_1_9+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_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_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_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_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_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_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_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_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_16B_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_17C_2_1C_2_2C_2_3C_2_4C_2_5C_2_6C_2_7C_2_8C_2_9C_3_1C_3_2C_3_3C_3_4C_3_5C_3_6C_3_7C_3_8C_3_9C_4_1C_4_2C_4_3C_4_4C_4_5C_4_6C_4_7C_4_8C_4_9C_5_1C_5_2C_5_3C_5_4C_5_5C_5_6C_5_7C_5_8C_5_9C_6_1C_6_2C_6_3C_6_4C_6_5C_6_6C_6_7C_6_8C_6_9C_7_1C_7_2C_7_3C_7_4C_7_5C_7_6C_7_7C_7_8C_7_9C_8_1C_8_2C_8_3C_8_4C_8_5C_8_6C_8_7C_8_8C_8_9C_9_1C_9_2C_9_3C_9_4C_9_5C_9_6C_9_7C_9_8C_9_9C_10_1C_10_2C_10_3C_10_4C_10_5C_10_6C_10_7C_10_8C_10_9C_11_1C_11_2C_11_3C_11_4C_11_5C_11_6C_11_7C_11_8C_11_9C_12_1C_12_2C_12_3C_12_4C_12_5C_12_6C_12_7C_12_8C_12_9C_13_1C_13_2C_13_3C_13_4C_13_5C_13_6C_13_7C_13_8C_13_9C_14_1C_14_2C_14_3C_14_4C_14_5C_14_6C_14_7C_14_8C_14_9C_15_1C_15_2C_15_3C_15_4C_15_5C_15_6C_15_7C_15_8C_15_9C_16_1C_16_2C_16_3C_16_4C_16_5C_16_6C_16_7C_16_8C_16_9C_17_1C_17_2C_17_3C_17_4C_17_5C_17_6C_17_7C_17_8C_17_9

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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