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ddc84c5 | 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 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import OAI.Analysis.DirectCrouzeix.DomainBound
namespace OAI
noncomputable section
open scoped Matrix Matrix.Norms.L2Operator Kronecker
noncomputable section
open MeasureTheory Set Filter Metric
open scoped Topology Interval ENNReal NNReal ComplexConjugate
noncomputable section
open Filter Metric Set
open scoped Topology ComplexConjugate
noncomputable section
open Set Filter Metric
open scoped Topology ComplexConjugate
noncomputable section
open Set Filter Metric
open scoped Topology ComplexConjugate
noncomputable section
open Set Filter Metric
open scoped Topology ComplexConjugate
noncomputable section
open Set Filter Metric
open scoped Topology ComplexConjugate
noncomputable section
open Set Filter Metric
open scoped Topology ComplexConjugate
noncomputable section
open Set
open scoped ComplexConjugate Matrix
namespace DirectCrouzeix
open Set Metric
theorem convex_numericalRange {n : ℕ} (A : Matrix (Fin n) (Fin n) ℂ) :
Convex ℝ (numericalRange A) :=
Geometry.convex_quadratic_range (Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℂ) A).toLinearMap
theorem complete_crouzeix : UniversalBound 2 := by
intro n m d hn hm A B
let M := rangeMaximum A B
have hM : 0 ≤ M := by
obtain ⟨z,hz,he⟩ := rangeMaximum_attained hn A B
rw [show M = ‖polynomialValue B z‖ from he.symm]
exact norm_nonneg _
apply le_of_forall_pos_le_add
intro ε hε
let ρ := M+ε/2
have hρ : 0 < ρ := by dsimp [ρ]; positivity
have hMρ : M < ρ := by dsimp [ρ]; linarith
let V := {z : ℂ | ‖polynomialValue B z‖ < ρ}
have hV : IsOpen V := isOpen_lt (continuous_polynomialValue B).norm continuous_const
have hWV : numericalRange A ⊆ V := by
intro z hz
exact lt_of_le_of_lt (norm_polynomialValue_le_rangeMaximum A B hz) hMρ
obtain ⟨δ,hδ,hδV⟩ := (isCompact_numericalRange A).exists_thickening_subset_open hV hWV
obtain ⟨D,v,b,hW,hc,hnear⟩ := Geometry.exists_exponential_outer (isCompact_numericalRange A)
(numericalRange_nonempty hn A) (convex_numericalRange A) hδ
have hbound : ∀ z, Geometry.expLevel v b z ≤ 1 →
‖matrixPolynomialValue z (coefficientPolynomial B)‖ ≤ ρ := by
intro z hz
rw [matrixPolynomialValue_coefficientPolynomial]
apply le_of_lt
apply hδV
obtain ⟨w,hw,hzw⟩ := hnear z hz
exact mem_thickening_iff.mpr ⟨w,hw,by simpa only [dist_eq_norm] using hzw⟩
have hh := expLevel_domain_bound_scaled hn hm A v b hc hW (coefficientPolynomial B) hρ hbound
rw [tensorPolynomial_coefficientPolynomial] at hh
calc
‖tensorEvaluation A B‖ ≤ 2*ρ := hh
_ = 2*rangeMaximum A B+ε := by dsimp [ρ,M]; ring
end DirectCrouzeix
end
end
end
end
end
end
end
end
end
end OAI
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