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Update circular.jl
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@alanedelman
alanedelman committed Sep 30, 2024
1 parent 794fb77 commit f255bf4
Showing 1 changed file with 42 additions and 63 deletions.
105 changes: 42 additions & 63 deletions notebooks/circular.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -15,7 +15,7 @@ macro bind(def, element)
end

# ╔═╡ 6486e004-c95b-4fcd-936e-6508a0e0283c
using LinearAlgebra, StatsBase, PlutoUI, GenericLinearAlgebra
using LinearAlgebra, StatsBase, PlutoUI, SpecialFunctions

# ╔═╡ fab8681e-0be3-4d97-a0cb-2ad57472282a
using Plots
Expand Down Expand Up @@ -114,6 +114,12 @@ let
end
╠═╡ =#

# ╔═╡ d426a597-d198-4d16-9ca1-4600c84d968f
Z = circular4q(3)

# ╔═╡ b17edbda-3a23-48a1-8413-2a5350a29e86
eigvals(ℍ2ℂ(Z))

# ╔═╡ 686fae20-6ed7-444c-9085-0f97d8432303
function circularm(n,β)
e = eigvals(circular(n,β))
Expand All @@ -129,18 +135,20 @@ end
let
n = 3
t = 10000
β = 5 #-4/5
# β = 1 # -1
# β = 2 # -1
β = 1

v = [ circularm(n,β) for i=1:t]
# m2 = mean(first.(v))
# m11 = mean(last.(v))
# mean( first.(v) .* conj.(last.(v)))
# 3/mean(abs.(last.(v)).^2) - 1 # 3/|m11|²-1 = β

mean(first.(v).*conj(last.(v))) , -6β/( (β+1)*(β+2))
m2 = first.(v)
m11 = last.(v)

# 3/mean(abs.(m11).^2) - 1 , β # 3/|m11|²-1 = β
# mean(m2.*conj(m11)) , -2β/(β+2), -6β/( (β+1)*(β+2)) # n = 2, 3
mean(abs.(m2).^2), (2π)^(-n)* (β/4+.5)^2 * gamma(1+n*β/2)/gamma(1+β/2)^n
end

# ╔═╡ 6666dce5-8772-4436-a116-fc9b6173bc28
10/7

# ╔═╡ dd7e4afa-323e-4287-a8a0-f1448fb669ac
5/7

Expand Down Expand Up @@ -169,31 +177,10 @@ end
# ╔═╡ 2443d72e-42f4-4f66-8170-04633f9b9031
M = circular4(2)

# ╔═╡ 662e1079-13d9-4c26-b359-96556bbf75ee
ℍ2ℂ(q::Quaternion) = [q.s+im*q.v1 q.v2+im*q.v3
-q.v2+im*q.v3 q.s-im*q.v1 ]

# ╔═╡ b11d81bd-b6d7-4325-9012-c65efb859d02
ℍ2ℂ(M::Matrix) = hvcat(size(M,1),ℍ2ℂ.(M)...)

# ╔═╡ a173bcd2-c02a-4897-9fbe-b7bde52f354b
negatej(q::Quaternion) = Quaternion(q.s,q.v1,-q.v2,q.v3)

# ╔═╡ 61e9b1b5-95c5-4974-83b6-53387e93ded6
function jdual(M)
negatej.(transpose(M))
end

# ╔═╡ b58a06f2-ad63-4742-abf8-bf491842b1f2
function circular4q(n) #β=4
A = quat.( randn(n,n), randn(n,n), randn(n,n), randn(n,n))
U = quat(0,0,1,0) * Matrix(qr(A).Q) * quat(0,0,1,0) * jdual(Matrix(qr(A).Q)) #<-- this might be right
end

# ╔═╡ 5e4d9a97-db38-445e-8982-7813b9656546
let
n = 5
t = 5000
t = 15000
m = [ smallabs(circular4(n)) for i=1:t]

stephist(m,normalize=true,label="dense")
Expand All @@ -204,33 +191,21 @@ let
stephist!(mq,normalize=true,label="Quaternionic Way")
end

# ╔═╡ d426a597-d198-4d16-9ca1-4600c84d968f
Z = circular4q(3)

# ╔═╡ b17edbda-3a23-48a1-8413-2a5350a29e86
eigvals(ℍ2ℂ(Z))

# ╔═╡ 228f2a0c-2a52-432d-9dda-45f22dda4a4a
A = [quat( [rand(1:5) for i=1:4]...) for i=1:2,j=1:2]

# ╔═╡ bc6bb958-7b18-4324-bc1d-644d543f9842
jdual(A)

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
GenericLinearAlgebra = "14197337-ba66-59df-a3e3-ca00e7dcff7a"
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
Quaternions = "94ee1d12-ae83-5a48-8b1c-48b8ff168ae0"
SpecialFunctions = "276daf66-3868-5448-9aa4-cd146d93841b"
StatsBase = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
[compat]
GenericLinearAlgebra = "~0.3.13"
Plots = "~1.40.8"
PlutoUI = "~0.7.60"
Quaternions = "~0.7.5"
SpecialFunctions = "~2.4.0"
StatsBase = "~0.34.3"
"""

Expand All @@ -240,7 +215,7 @@ PLUTO_MANIFEST_TOML_CONTENTS = """
julia_version = "1.10.4"
manifest_format = "2.0"
project_hash = "d5477210c2247950b3cde5a6471f10197fb68f21"
project_hash = "4aadc0bfa083dbcc0fec18591f4522480020ff68"
[[deps.AbstractPlutoDingetjes]]
deps = ["Pkg"]
Expand Down Expand Up @@ -298,13 +273,11 @@ deps = ["ColorTypes", "FixedPointNumbers", "LinearAlgebra", "Requires", "Statist
git-tree-sha1 = "a1f44953f2382ebb937d60dafbe2deea4bd23249"
uuid = "c3611d14-8923-5661-9e6a-0046d554d3a4"
version = "0.10.0"
weakdeps = ["SpecialFunctions"]
[deps.ColorVectorSpace.extensions]
SpecialFunctionsExt = "SpecialFunctions"
[deps.ColorVectorSpace.weakdeps]
SpecialFunctions = "276daf66-3868-5448-9aa4-cd146d93841b"
[[deps.Colors]]
deps = ["ColorTypes", "FixedPointNumbers", "Reexport"]
git-tree-sha1 = "362a287c3aa50601b0bc359053d5c2468f0e7ce0"
Expand Down Expand Up @@ -455,12 +428,6 @@ git-tree-sha1 = "a8863b69c2a0859f2c2c87ebdc4c6712e88bdf0d"
uuid = "d2c73de3-f751-5644-a686-071e5b155ba9"
version = "0.73.7+0"
[[deps.GenericLinearAlgebra]]
deps = ["LinearAlgebra", "Printf", "Random", "libblastrampoline_jll"]
git-tree-sha1 = "f47136cac29a9b7a8a88dbce1195394978091edb"
uuid = "14197337-ba66-59df-a3e3-ca00e7dcff7a"
version = "0.3.13"
[[deps.Gettext_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "9b02998aba7bf074d14de89f9d37ca24a1a0b046"
Expand Down Expand Up @@ -778,6 +745,12 @@ git-tree-sha1 = "7493f61f55a6cce7325f197443aa80d32554ba10"
uuid = "458c3c95-2e84-50aa-8efc-19380b2a3a95"
version = "3.0.15+1"
[[deps.OpenSpecFun_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "13652491f6856acfd2db29360e1bbcd4565d04f1"
uuid = "efe28fd5-8261-553b-a9e1-b2916fc3738e"
version = "0.5.5+0"
[[deps.Opus_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "6703a85cb3781bd5909d48730a67205f3f31a575"
Expand Down Expand Up @@ -987,6 +960,18 @@ deps = ["Libdl", "LinearAlgebra", "Random", "Serialization", "SuiteSparse_jll"]
uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
version = "1.10.0"
[[deps.SpecialFunctions]]
deps = ["IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"]
git-tree-sha1 = "2f5d4697f21388cbe1ff299430dd169ef97d7e14"
uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
version = "2.4.0"
[deps.SpecialFunctions.extensions]
SpecialFunctionsChainRulesCoreExt = "ChainRulesCore"
[deps.SpecialFunctions.weakdeps]
ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
[[deps.Statistics]]
deps = ["LinearAlgebra", "SparseArrays"]
uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
Expand Down Expand Up @@ -1400,16 +1385,10 @@ version = "1.4.1+1"
# ╠═686fae20-6ed7-444c-9085-0f97d8432303
# ╠═5ede4dd3-c432-4108-b22d-ab72200f91f1
# ╠═aaa6835c-7d8f-4208-ab5d-0b06de61dbb8
# ╠═6666dce5-8772-4436-a116-fc9b6173bc28
# ╠═dd7e4afa-323e-4287-a8a0-f1448fb669ac
# ╠═5fcb9846-0a0d-4517-8eff-90f580172b82
# ╠═fb3ec55c-db9f-4897-b4d0-3a07a3ba859c
# ╠═37236855-5f7d-4d4a-8046-116a51dbae7f
# ╠═662e1079-13d9-4c26-b359-96556bbf75ee
# ╠═b11d81bd-b6d7-4325-9012-c65efb859d02
# ╠═a173bcd2-c02a-4897-9fbe-b7bde52f354b
# ╠═61e9b1b5-95c5-4974-83b6-53387e93ded6
# ╠═b58a06f2-ad63-4742-abf8-bf491842b1f2
# ╠═228f2a0c-2a52-432d-9dda-45f22dda4a4a
# ╠═bc6bb958-7b18-4324-bc1d-644d543f9842
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002

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