File size: 1,273 Bytes
f18e7de | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 | import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Main
import OAI.LinearAlgebra.MatrixMultiplication.ComplexBounds.CW75ReorderedRectangular
import OAI.LinearAlgebra.MatrixMultiplication.Duality.ExponentBound
import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.Conditional
import OAI.LinearAlgebra.MatrixMultiplication.Numerical.AllFieldCertificatesReplayIntegration
namespace OAI
/-! Arithmetic complexity bounds for square and rectangular matrix multiplication. -/
namespace MatrixMultiplication
/-- The complex square matrix multiplication exponent is at most `9/4`. -/
theorem complex_omega_le_nine_quarters :
Arithmetic.omega ℂ ≤ (9 : ℝ) / 4 :=
AuxiliarySeparation.omega_le_nine_quarters
/-- The complex dual matrix multiplication exponent is greater than `0.465`. -/
theorem complex_alpha_gt_93_div_200 :
(93 : ℝ) / 200 < Arithmetic.complexAlpha :=
DualExponentBound.alpha_gt
/-- Multiplication of `n × n^0.709` by `n^0.709 × n` complex matrices has
arithmetic exponent less than `2.092`. -/
theorem complex_rectangular_omega_lt_523_div_250 :
Arithmetic.rectangularOmega ℂ ((709 : ℝ) / 1000) < (523 : ℝ) / 250 :=
CW75ReorderedRectangular.rectangularOmega_lt_target
end MatrixMultiplication
end OAI
|