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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.Geometry
theorem exists_exponential_outer {K : Set ℂ} (hK : IsCompact K) (hne : K.Nonempty)
(hv : Convex ℝ K) {ε : ℝ} (hε : 0 < ε) :
∃ (D : Finset ℂ) (v : D → ℂ) (b : D → ℝ),
(∀ z ∈ K, expLevel v b z < 1) ∧
IsCompact {z | expLevel v b z ≤ 1} ∧
(∀ z, expLevel v b z ≤ 1 → ∃ w ∈ K, ‖z-w‖ < ε) := by
classical
obtain ⟨M,hM0,hM⟩ := hK.isBounded.exists_pos_norm_le
let L := 2*(M+1)
let C := 1+L+M
have hC : 0 < C := by dsimp [C,L]; linarith
let δ := min (1/2) (ε/(2*C))
have hδ : 0 < δ := lt_min (by norm_num) (div_pos hε (by positivity))
have hδ1 : δ < 1 := lt_of_le_of_lt (min_le_left _ _) (by norm_num)
have hδε : δ*C < ε := by
have hh : δ ≤ ε/(2*C) := min_le_right _ _
have hc := (le_div_iff₀ (by positivity : 0 < 2*C)).mp hh
nlinarith
obtain ⟨D,hD,h1,hm1,hI,hmI,hnet⟩ := exists_axis_net hδ
let N : ℝ := Fintype.card D
have hN : 0 ≤ N := by dsimp [N]; positivity
let t := (N+1)/δ
have ht : 0 < t := div_pos (by linarith) hδ
have htδ : t*δ = N+1 := div_mul_cancel₀ _ (ne_of_gt hδ)
let v : D → ℂ := fun ν => t • (ν:ℂ)
let b : D → ℝ := fun ν => -t*(convexSupport K ν+δ)
have he (z : ℂ) (ν : D) : inner ℝ (v ν) z+b ν = t*(inner ℝ (ν:ℂ) z-convexSupport K ν-δ) := by
dsimp only [v,b]
rw [real_inner_smul_left]
ring
have hsumlt : N*Real.exp (-t*δ) < 1 := by
rw [neg_mul,htδ,Real.exp_neg,mul_inv_lt_iff₀ (Real.exp_pos _),one_mul]
linarith [Real.add_one_le_exp (N+1)]
have hinside (z : ℂ) (hz : z ∈ K) : expLevel v b z < 1 := by
calc
expLevel v b z ≤ ∑ _ν : D, Real.exp (-t*δ) := by
apply Finset.sum_le_sum
intro ν hν
rw [he]
apply Real.exp_le_exp.mpr
have hh := mul_le_mul_of_nonneg_left (le_convexSupport hK hz (ν:ℂ)) (le_of_lt ht)
nlinarith
_ = N*Real.exp (-t*δ) := by simp [N]
_ < 1 := hsumlt
have hsupport (z : ℂ) (hz : expLevel v b z ≤ 1) (ν : ℂ) (hν : ν ∈ D) :
inner ℝ ν z ≤ convexSupport K ν+δ := by
have hh : Real.exp (inner ℝ (v ⟨ν,hν⟩) z+b ⟨ν,hν⟩) ≤ expLevel v b z :=
Finset.single_le_sum (f := fun i : D => Real.exp (inner ℝ (v i) z+b i))
(fun i _ => Real.exp_nonneg _) (Finset.mem_univ (⟨ν,hν⟩ : D))
have hex := (Real.exp_le_one_iff).mp (hh.trans hz)
rw [he] at hex
nlinarith
have hbound (z : ℂ) (hz : expLevel v b z ≤ 1) : ‖z‖ ≤ L := by
have hh := axis_support_bound hne hM h1 hm1 hI hmI (hsupport z hz)
dsimp [L]
linarith
refine ⟨D,v,b,hinside,?_,?_⟩
· apply (isCompact_closedBall (0:ℂ) L).of_isClosed_subset
· exact isClosed_le (expLevel_contDiff v b).continuous continuous_const
· intro z hz
exact mem_closedBall_zero_iff.mpr (hbound z hz)
· intro z hz
obtain ⟨w,hw,hzw⟩ := direction_net_distance hK hne hv hδ hM hnet (hbound z hz) (hsupport z hz)
exact ⟨w,hw,lt_of_le_of_lt hzw hδε⟩
end DirectCrouzeix.Geometry
namespace DirectCrouzeix
@[simp] theorem matrixPolynomialValue_coefficientPolynomial {m d : ℕ}
(B : Fin (d+1) → Matrix (Fin m) (Fin m) ℂ) (z : ℂ) :
matrixPolynomialValue z (coefficientPolynomial B) = polynomialValue B z := by
simp only [coefficientPolynomial,map_sum,map_mul,map_pow,matrixPolynomialValue_C,
matrixPolynomialValue_X,smul_pow,one_pow,Matrix.mul_smul,Matrix.mul_one,polynomialValue]
@[simp] theorem matrixPolynomialValue_smul {m : ℕ} (F : MatrixPolynomial m) (s z : ℂ) :
matrixPolynomialValue z (s • F) = s • matrixPolynomialValue z F := by
rw [← smul_one_smul (Matrix (Fin m) (Fin m) ℂ) s F,Polynomial.smul_eq_C_mul,
map_mul,matrixPolynomialValue_C,Matrix.smul_mul,Matrix.one_mul]
@[simp] theorem tensorPolynomial_smul {n m : ℕ} (A : Matrix (Fin n) (Fin n) ℂ)
(F : MatrixPolynomial m) (s : ℂ) :
tensorPolynomial A (s • F) = s • tensorPolynomial A F := by
rw [← smul_one_smul (Matrix (Fin m) (Fin m) ℂ) s F,Polynomial.smul_eq_C_mul,
map_mul,tensorPolynomial_C,Matrix.kronecker_smul,Matrix.one_kronecker_one,
Matrix.smul_mul,Matrix.one_mul]
end DirectCrouzeix
namespace DirectCrouzeix
open Set Filter Metric
end DirectCrouzeix
end
end
end
end
end
end
end
end
end
end OAI
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