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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 | import OAI.Analysis.DirectCrouzeix.Resolvent
namespace OAI
noncomputable section
open scoped Matrix Matrix.Norms.L2Operator Kronecker
namespace DirectCrouzeix
open scoped MatrixOrder ComplexOrder
def exteriorCorrection (h q : ℂ → ℂ) (ζ μ : ℂ) : ℂ :=
q ζ / (h ζ - h μ) - ζ / (ζ - μ)
def exteriorKernel (h q : ℂ → ℂ) (ζ μ : ℂ) : ℝ :=
1 + (exteriorCorrection h q ζ μ).re + (star (exteriorCorrection h q ζ μ)).re
theorem re_circle_fraction {ζ μ : ℂ} (hζ : ‖ζ‖ = 1) (hμ : ‖μ‖ = 1)
(hne : ζ ≠ μ) : 2 * (ζ / (ζ - μ)).re = 1 := by
have hnζ : ζ.re ^ 2 + ζ.im ^ 2 = 1 := by
simpa [Complex.normSq_apply, pow_two] using
(show Complex.normSq ζ = 1 by rw [Complex.normSq_eq_norm_sq, hζ]; norm_num)
have hnμ : μ.re ^ 2 + μ.im ^ 2 = 1 := by
simpa [Complex.normSq_apply, pow_two] using
(show Complex.normSq μ = 1 by rw [Complex.normSq_eq_norm_sq, hμ]; norm_num)
have hd : Complex.normSq (ζ - μ) ≠ 0 := by
exact mt Complex.normSq_eq_zero.mp (sub_ne_zero.mpr hne)
rw [Complex.div_re]
field_simp
simp [Complex.normSq_apply, Complex.sub_re, Complex.sub_im] at *
nlinarith
theorem exteriorKernel_nonneg_off_diagonal (h q : ℂ → ℂ)
{ζ μ : ℂ} (hζ : ‖ζ‖ = 1) (hμ : ‖μ‖ = 1) (hne : ζ ≠ μ)
(hs : 0 ≤ (star (q ζ) * (h ζ - h μ)).re) :
0 ≤ exteriorKernel h q ζ μ := by
have hfrac := re_circle_fraction hζ hμ hne
have hp : 0 ≤ (q ζ / (h ζ - h μ)).re := by
rw [Complex.div_re, ← add_div]
apply div_nonneg _ (Complex.normSq_nonneg _)
simpa [Complex.mul_re] using hs
simp only [exteriorKernel, exteriorCorrection, Complex.sub_re, Complex.star_def,
Complex.conj_re]
linarith
def coefficientEmbedding (n m : ℕ) :
Matrix (Fin m) (Fin m) ℂ →+*
Matrix (Fin n × Fin m) (Fin n × Fin m) ℂ where
toFun B := (1 : Matrix (Fin n) (Fin n) ℂ) ⊗ₖ B
map_zero' := Matrix.kronecker_zero _
map_one' := Matrix.one_kronecker_one
map_add' B C := Matrix.kronecker_add _ B C
map_mul' B C := by
simpa using Matrix.mul_kronecker_mul
(1 : Matrix (Fin n) (Fin n) ℂ) 1 B C
def tensorPolynomial {n m : ℕ} (A : Matrix (Fin n) (Fin n) ℂ) :
Polynomial (Matrix (Fin m) (Fin m) ℂ) →+*
Matrix (Fin n × Fin m) (Fin n × Fin m) ℂ :=
Polynomial.eval₂RingHom' (coefficientEmbedding n m)
(A ⊗ₖ (1 : Matrix (Fin m) (Fin m) ℂ)) (by
intro B
change (1 ⊗ₖ B) * (A ⊗ₖ 1) = (A ⊗ₖ 1) * (1 ⊗ₖ B)
rw [← Matrix.mul_kronecker_mul, ← Matrix.mul_kronecker_mul]
simp)
def coefficientPolynomial {m d : ℕ}
(B : Fin (d + 1) → Matrix (Fin m) (Fin m) ℂ) :
Polynomial (Matrix (Fin m) (Fin m) ℂ) :=
∑ k : Fin (d + 1), Polynomial.C (B k) * Polynomial.X ^ (k : ℕ)
theorem kronecker_pow_one {n m : ℕ} (A : Matrix (Fin n) (Fin n) ℂ) (k : ℕ) :
(A ⊗ₖ (1 : Matrix (Fin m) (Fin m) ℂ)) ^ k = (A ^ k) ⊗ₖ 1 := by
induction k with
| zero => simp
| succ k ih =>
rw [pow_succ, ih, ← Matrix.mul_kronecker_mul]
simp [pow_succ]
@[simp] theorem tensorPolynomial_C {n m : ℕ} (A : Matrix (Fin n) (Fin n) ℂ)
(B : Matrix (Fin m) (Fin m) ℂ) :
tensorPolynomial A (Polynomial.C B) = (1 : Matrix (Fin n) (Fin n) ℂ) ⊗ₖ B := by
exact Polynomial.eval₂_C _ _
@[simp] theorem tensorPolynomial_X {n m : ℕ} (A : Matrix (Fin n) (Fin n) ℂ) :
tensorPolynomial (m := m) A Polynomial.X = A ⊗ₖ (1 : Matrix (Fin m) (Fin m) ℂ) := by
exact Polynomial.eval₂_X _ _
theorem tensorPolynomial_coefficientPolynomial {n m d : ℕ}
(A : Matrix (Fin n) (Fin n) ℂ)
(B : Fin (d + 1) → Matrix (Fin m) (Fin m) ℂ) :
tensorPolynomial A (coefficientPolynomial B) = tensorEvaluation A B := by
simp only [coefficientPolynomial, map_sum, map_mul, map_pow,
tensorPolynomial_C, tensorPolynomial_X, kronecker_pow_one,
← Matrix.mul_kronecker_mul, Matrix.one_mul, Matrix.mul_one, tensorEvaluation]
theorem tensorPolynomial_mul {n m : ℕ} (A : Matrix (Fin n) (Fin n) ℂ)
(F G : Polynomial (Matrix (Fin m) (Fin m) ℂ)) :
tensorPolynomial A (F * G) = tensorPolynomial A F * tensorPolynomial A G :=
map_mul _ _ _
end DirectCrouzeix
end
end OAI
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