File size: 3,072 Bytes
8e4d533 | 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 | import Mathlib
namespace OAI
namespace BorsukNine
open scoped RealInnerProductSpace
section ReflectionTransport
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
noncomputable def householder (c x : E) : E ≃ₗᵢ[ℝ] E :=
(Submodule.span ℝ {c+x}).reflection
lemma householder_apply (c x v : E) :
householder c x v = (2 * (⟪c+x,v⟫ / ‖c+x‖ ^ 2)) • (c+x) - v := by
rw [householder,Submodule.reflection_singleton_apply]
simp only [RCLike.ofReal_real_eq_id, id_eq, two_smul, two_mul, add_smul]
lemma householder_send {c x : E} (h : ‖c‖ = ‖x‖) : householder c x c = x := by
have hh := Submodule.reflection_sub (v := c) (w := -x) (by simpa using h)
rw [sub_neg_eq_add,Submodule.reflection_orthogonal_apply] at hh
exact neg_injective hh
lemma householder_orthogonal {c x v : E} (h : ‖c‖ = ‖x‖) (hv : ⟪c,v⟫ = 0) :
⟪x,householder c x v⟫ = 0 := by
calc
⟪x,householder c x v⟫ = ⟪householder c x c,householder c x v⟫ := by rw [householder_send h]
_ = ⟪c,v⟫ := (householder c x).inner_map_map c v
_ = 0 := hv
lemma householder_at_self {c v : E} (hv : ⟪c,v⟫ = 0) :
householder c c v = -v := by
rw [householder_apply]
simp [inner_add_left,hv]
lemma householder_joint_continuousOn (c : E) :
ContinuousOn (fun p : E × E => householder c p.1 p.2) {p | c+p.1 ≠ 0} := by
have hc : Continuous (fun p : E × E => c+p.1) := continuous_const.add continuous_fst
have hz : Continuous (fun p : E × E => p.2) := continuous_snd
have hn : Continuous (fun p : E × E => ⟪c+p.1,p.2⟫) := hc.inner hz
have hd : Continuous (fun p : E × E => ‖c+p.1‖ ^ 2) := hc.norm.pow 2
have hq : ContinuousOn (fun p : E × E => ⟪c+p.1,p.2⟫ / ‖c+p.1‖ ^ 2) {p | c+p.1 ≠ 0} :=
hn.continuousOn.div hd.continuousOn (fun _ hp => pow_ne_zero 2 (norm_ne_zero_iff.mpr hp))
have hmul : ContinuousOn (fun p : E × E => (2 : ℝ) * (⟪c+p.1,p.2⟫ / ‖c+p.1‖ ^ 2))
{p | c+p.1 ≠ 0} := continuous_const.continuousOn.mul hq
have hs : ContinuousOn (fun p : E × E => ((2 : ℝ) * (⟪c+p.1,p.2⟫ / ‖c+p.1‖ ^ 2)) • (c+p.1) - p.2)
{p | c+p.1 ≠ 0} := (hmul.smul hc.continuousOn).sub hz.continuousOn
simpa only [householder_apply] using hs
lemma householder_joint_contDiffOn (c : E) :
ContDiffOn ℝ (⊤ : ℕ∞) (fun p : E × E => householder c p.1 p.2) {p | c+p.1 ≠ 0} := by
have he : (fun p : E × E => ‖c+p.1‖ ^ 2) = fun p => ⟪c+p.1,c+p.1⟫ := by
ext p
exact (real_inner_self_eq_norm_sq _).symm
simp_rw [householder_apply]
apply ContDiffOn.sub _ contDiff_snd.contDiffOn
apply ContDiffOn.smul _ (contDiff_const.add contDiff_fst).contDiffOn
apply contDiff_const.contDiffOn.mul
apply ContDiffOn.div
(((contDiff_const.add contDiff_fst).inner ℝ contDiff_snd).contDiffOn)
(by rw [he]; exact ((contDiff_const.add contDiff_fst).inner ℝ
(contDiff_const.add contDiff_fst)).contDiffOn)
intro p hp
exact pow_ne_zero 2 (norm_ne_zero_iff.mpr hp)
end ReflectionTransport
end BorsukNine
end OAI
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