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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 | import OAI.Analysis.DirectCrouzeix.OuterApproximation
namespace OAI
universe u_144 u_145
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 Filter Metric
theorem expLevel_domain_bound {n m : ℕ} (hn : 0 < n) (hm : 0 < m)
(A : Matrix (Fin n) (Fin n) ℂ) {ι : Type u_144} [Fintype ι]
(v : ι → ℂ) (b : ι → ℝ) (hc : IsCompact {z | Geometry.expLevel v b z ≤ 1})
(hW : ∀ z ∈ numericalRange A, Geometry.expLevel v b z < 1)
(F : MatrixPolynomial m)
(hF : ∀ z, Geometry.expLevel v b z ≤ 1 → ‖matrixPolynomialValue z F‖ ≤ 1) :
‖tensorPolynomial A F‖ ≤ 2 := by
obtain ⟨a,ha⟩ := numericalRange_nonempty hn A
obtain ⟨R,hR,c,g,hc0,hg,hinj,hder,hcircle,hout⟩ :=
Geometry.exists_expLevel_exterior v b hc (hW a ha)
have hge (ζ : ℂ) (hζ : 1 ≤ ‖ζ‖) : 1 ≤ Geometry.expLevel v b (Faber.exterior c g ζ) := by
rcases eq_or_lt_of_le hζ with he|hl
· exact le_of_eq (hcircle ζ he.symm).symm
· exact le_of_lt (hout ζ hl)
have hs (t : Faber.Angle) (w : ℂ) (hw : Geometry.expLevel v b w ≤ 1) :
0 ≤ (conj (Faber.boundaryNormal c g t)*(Faber.boundary c g t-w)).re := by
have hdt := Faber.hasDerivAt_exterior hg c (Faber.unitPoint_ne_zero t) (Faber.unitPoint_inv_mem hR t)
have hh := Geometry.expLevel_holomorphic_support v b (hW a ha) (Faber.unitPoint_norm t)
hdt.differentiableAt.hasDerivAt (hder _ (Faber.unitPoint_ne_zero t) (Faber.unitPoint_inv_mem hR t)) hcircle hge hw
have he : Faber.unitPoint t*deriv (Faber.exterior c g) (Faber.unitPoint t) = Faber.boundaryNormal c g t := by
rw [hdt.deriv]; rfl
rw [he,real_inner_eq_re_inner ℂ,RCLike.inner_apply] at hh
change 0 ≤ ((Faber.exterior c g (Faber.unitPoint t)-w)*conj (Faber.boundaryNormal c g t)).re at hh
simpa only [Faber.boundary,mul_comm] using hh
apply exterior_collar_bound hn hm A hR hg hc0 hinj hder
· intro t s
exact hs t _ (le_of_eq (hcircle _ (Faber.unitPoint_norm s)))
· intro ζ hζ hz
have h1 := hge ζ hζ
have h2 := hW _ hz
linarith
· intro t w hw
exact hs t w (le_of_lt (hW w hw))
· intro t
exact hF _ (le_of_eq (hcircle _ (Faber.unitPoint_norm t)))
theorem expLevel_domain_bound_scaled {n m : ℕ} (hn : 0 < n) (hm : 0 < m)
(A : Matrix (Fin n) (Fin n) ℂ) {ι : Type u_145} [Fintype ι]
(v : ι → ℂ) (b : ι → ℝ) (hc : IsCompact {z | Geometry.expLevel v b z ≤ 1})
(hW : ∀ z ∈ numericalRange A, Geometry.expLevel v b z < 1)
(F : MatrixPolynomial m) {ρ : ℝ} (hρ : 0 < ρ)
(hF : ∀ z, Geometry.expLevel v b z ≤ 1 → ‖matrixPolynomialValue z F‖ ≤ ρ) :
‖tensorPolynomial A F‖ ≤ 2*ρ := by
have hb := expLevel_domain_bound hn hm A v b hc hW (((ρ⁻¹:ℝ):ℂ) • F) (by
intro z hz
rw [matrixPolynomialValue_smul,norm_smul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hρ)]
exact (mul_le_mul_of_nonneg_left (hF z hz) (le_of_lt (inv_pos.mpr hρ))).trans_eq (inv_mul_cancel₀ (ne_of_gt hρ)))
rw [tensorPolynomial_smul,norm_smul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hρ)] at hb
have hh := mul_le_mul_of_nonneg_left hb (le_of_lt hρ)
rw [← mul_assoc,mul_inv_cancel₀ (ne_of_gt hρ),one_mul] at hh
simpa only [mul_comm] using hh
end DirectCrouzeix
namespace DirectCrouzeix
open Set Metric
end DirectCrouzeix
end
end
end
end
end
end
end
end
end
end OAI
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