| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open scoped Matrix Matrix.Norms.L2Operator Kronecker | |
| namespace DirectCrouzeix | |
| def numericalRange {n : ℕ} (A : Matrix (Fin n) (Fin n) ℂ) : Set ℂ := | |
| {z | ∃ u : EuclideanSpace ℂ (Fin n), ‖u‖ = 1 ∧ | |
| inner ℂ u (Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℂ) A u) = z} | |
| def polynomialValue {m d : ℕ} (B : Fin (d + 1) → Matrix (Fin m) (Fin m) ℂ) | |
| (z : ℂ) : Matrix (Fin m) (Fin m) ℂ := | |
| ∑ k : Fin (d + 1), z ^ (k : ℕ) • B k | |
| def tensorEvaluation {n m d : ℕ} (A : Matrix (Fin n) (Fin n) ℂ) | |
| (B : Fin (d + 1) → Matrix (Fin m) (Fin m) ℂ) : | |
| Matrix (Fin n × Fin m) (Fin n × Fin m) ℂ := | |
| ∑ k : Fin (d + 1), (A ^ (k : ℕ)) ⊗ₖ (B k) | |
| def rangeMaximum {n m d : ℕ} (A : Matrix (Fin n) (Fin n) ℂ) | |
| (B : Fin (d + 1) → Matrix (Fin m) (Fin m) ℂ) : ℝ := | |
| sSup ((fun z => ‖polynomialValue B z‖) '' numericalRange A) | |
| def UniversalBound (c : ℝ) : Prop := | |
| ∀ (n m d : ℕ), 0 < n → 0 < m → | |
| ∀ (A : Matrix (Fin n) (Fin n) ℂ) | |
| (B : Fin (d + 1) → Matrix (Fin m) (Fin m) ℂ), | |
| ‖tensorEvaluation A B‖ ≤ c * rangeMaximum A B | |
| end DirectCrouzeix | |
| end | |
| end OAI | |