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cdec422 | 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 33 34 | import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Arithmetic.Exponent
import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Arithmetic.RankBound
import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Convolution.RankLowerBound
import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Determinant.Bounds
import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Growth.Floor
namespace OAI
/-!
# Matrix multiplication via auxiliary separation and polynomial multiplication
Theorem 1.1 gives a uniform arithmetic-operation bound for complex square
matrix multiplication, and consequently an upper bound of `9/4` on its exponent.
The two conclusions are stated separately in the existing arithmetic model.
-/
namespace MatrixMultiplication.AuxiliarySeparation
/-- Theorem 1.1: for every positive exponent slack, one constant bounds the cost
of correct complex matrix multiplication programs at every positive size. -/
theorem matrix_multiplication_cost_le (ε : ℝ) (hε : 0 < ε) :
∃ C : ℝ, 0 < C ∧ ∀ n : ℕ, 1 ≤ n →
∃ P : Arithmetic.MatrixAlgorithm ℂ n n n,
P.Correct ∧ (P.cost : ℝ) ≤ C * (n : ℝ) ^ ((9 : ℝ) / 4 + ε) :=
arithmeticComplexity_of_exactRankExponent_le exactRankExponent_le_nine_quarters ε hε
/-- The complex arithmetic matrix multiplication exponent is at most `9/4`. -/
theorem omega_le_nine_quarters : Arithmetic.omega ℂ ≤ (9 : ℝ) / 4 :=
omega_le_exactRankExponent.trans exactRankExponent_le_nine_quarters
end MatrixMultiplication.AuxiliarySeparation
end OAI
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