diff --git a/notebooks/circular.jl b/notebooks/circular.jl index 737121e..380fc06 100644 --- a/notebooks/circular.jl +++ b/notebooks/circular.jl @@ -47,12 +47,27 @@ end # ╔═╡ db8c16a0-79df-4cc0-9c69-a6bcfcd23a1c circular(3,2) -# ╔═╡ f9f00c9d-633a-4734-838f-3ffdcef6efe0 -@bind β Slider([.5,1,4,10,100,1000,1e10]; default=1, show_value=true) +# ╔═╡ 02db4122-8e3c-477c-b1ef-d9c236787910 +smallabs(U) = minimum(abs.(angle.(eigvals(Matrix(U))))) + +# ╔═╡ 14b4a088-ff74-43f3-b301-4057593c0447 +function circular2(n) + A = randn(n,n) .+ im * randn(n,n) + U = qr(A).Q .* cis.(2π*rand(n)) +end + +# ╔═╡ 148317c1-e48f-4d39-833c-b51d2c43e5fd +function circular1(n) #β=1 + A = randn(n,n) .+ im * randn(n,n) + U = qr(A).Q * transpose( Matrix(qr(A).Q )) +end # ╔═╡ e4418523-234c-4faa-a29b-43484013b322 @bind n Slider(2:17; default=3, show_value=true) +# ╔═╡ f9f00c9d-633a-4734-838f-3ffdcef6efe0 +@bind β Slider([.5,1,4,10,50,100,1000,1e10]; default=1, show_value=true) + # ╔═╡ 63cd9672-d3d2-4b2a-8076-f98b60dc11e6 begin e = eigvals(circular(n,β)) @@ -64,36 +79,23 @@ begin plot!(cis.(θ),label=false) end -# ╔═╡ 02db4122-8e3c-477c-b1ef-d9c236787910 -smallabs(U) = minimum(abs.(angle.(eigvals(Matrix(U))))) +# ╔═╡ 5cd13d1b-9bb4-4a19-9e6a-07e214572639 +begin -# ╔═╡ 14b4a088-ff74-43f3-b301-4057593c0447 -function circular2(n) - A = randn(n,n) .+ im * randn(n,n) - U = qr(A).Q .* cis.(2π*rand(n)) -end + t = 20 -# ╔═╡ 148317c1-e48f-4d39-833c-b51d2c43e5fd -function circular1(n) #β=1 - A = randn(n,n) .+ im * randn(n,n) - U = qr(A).Q * transpose( Matrix(qr(A).Q )) + # m = [ smallabs(circular2(n)) for i=1:t] + # display((mean(m),var(m))) + # stephist(m,normalize=true,label="dense") + mh = [ smallabs(circular(n,β)) for i=1:t] + #display((mean(mh),var(mh))) + + stephist(mh,normalize=true,label="Hessenberg") + title!("β=$β n=$n t=$t") end -# ╔═╡ 5cd13d1b-9bb4-4a19-9e6a-07e214572639 -# ╠═╡ disabled = true -#=╠═╡ -let - n = 5 - t = 50 - m = [ smallabs(circular2(n)) for i=1:t] - display((mean(m),var(m))) - stephist(m,normalize=true,label="dense") - mh = [ smallabs(circular(n,2)) for i=1:t] - display((mean(mh),var(mh))) - title!("β=2 n=$n t=$t") - stephist!(mh,normalize=true,label="Hessenberg") -end - ╠═╡ =# +# ╔═╡ a6be0162-84fd-408c-b3d6-d3cee3b58b61 + # ╔═╡ 63bf73bf-b312-49cc-a909-c043bfc80fd1 circular1(5) @@ -1369,14 +1371,15 @@ version = "1.4.1+1" # ╠═fab8681e-0be3-4d97-a0cb-2ad57472282a # ╠═5455f492-7c41-11ef-3315-19809ebaffc3 # ╠═db8c16a0-79df-4cc0-9c69-a6bcfcd23a1c -# ╠═f9f00c9d-633a-4734-838f-3ffdcef6efe0 -# ╠═e4418523-234c-4faa-a29b-43484013b322 # ╠═63cd9672-d3d2-4b2a-8076-f98b60dc11e6 # ╠═02db4122-8e3c-477c-b1ef-d9c236787910 # ╠═14b4a088-ff74-43f3-b301-4057593c0447 # ╠═148317c1-e48f-4d39-833c-b51d2c43e5fd # ╠═2443d72e-42f4-4f66-8170-04633f9b9031 +# ╠═e4418523-234c-4faa-a29b-43484013b322 +# ╠═f9f00c9d-633a-4734-838f-3ffdcef6efe0 # ╠═5cd13d1b-9bb4-4a19-9e6a-07e214572639 +# ╠═a6be0162-84fd-408c-b3d6-d3cee3b58b61 # ╠═63bf73bf-b312-49cc-a909-c043bfc80fd1 # ╠═82ce98e9-37a9-4023-ac62-5c092a8fa932 # ╠═5e4d9a97-db38-445e-8982-7813b9656546 diff --git a/notebooks/geometric_sequence b/notebooks/geometric_sequence new file mode 100644 index 0000000..ebe707d --- /dev/null +++ b/notebooks/geometric_sequence @@ -0,0 +1 @@ +a,r,n refers implicitly to \rightarrow a,ar,ar^2,\ldots,ar^{n-1}$$ \ No newline at end of file diff --git a/notebooks/geometric_sequence.jl b/notebooks/geometric_sequence.jl new file mode 100644 index 0000000..5e39f5a --- /dev/null +++ b/notebooks/geometric_sequence.jl @@ -0,0 +1,25 @@ +# g(a,r,n) = geometric_sequence +# refers implicitly to → a,ar,ar²,…, arⁿ⁻¹ + +# state(i,t) = iteration number and current term (i.e. t = arⁱ) +# start = state(0,i) +# next = (i,t) → (i+1, r*t) +# done when i=n + +struct geometric_sequence + a :: Float64 # start + r :: Float64 # ratio + n :: Int # number of terms +end + +struct geometric_sequence_state + i :: Int # iteration number + t :: Float64 # current term +end + +iterate(g::geometric_sequence) = geometric_sequence_state(0,g.a) # start + +# next should return term and state +function iterate(g::geometric_sequence, s::geometric_sequence_state) + s.t, geometric_sequence_state(s.i+1, g.r * s.t ) +end \ No newline at end of file diff --git a/notebooks/jacobi.jl b/notebooks/jacobi.jl new file mode 100644 index 0000000..7fecc7e --- /dev/null +++ b/notebooks/jacobi.jl @@ -0,0 +1,1473 @@ +### A Pluto.jl notebook ### +# v0.19.46 + +using Markdown +using InteractiveUtils + +# ╔═╡ df9a4768-81ae-11ef-3f3e-e1408e0fd133 +using LinearAlgebra,PlutoUI, Distributions, Plots, Quaternions, GenericLinearAlgebra + +# ╔═╡ 59adca81-b8a5-4bc4-9b77-bc01be31e834 +TableOfContents(title="📚 Table of Contents", indent=true, depth=4, aside=true) + +# ╔═╡ f31b98a5-69d9-440a-b0e3-23cb64e5984c +md""" +# Jacobi Ensembles +""" + +# ╔═╡ 4520eabe-5fba-42bb-a917-25f39f408cc6 +function J1E(a,b,n) + A = randn(n+a,n) + B = randn(n+b,n) + sqrt.( 1 ./ (1 .+1 ./ (svdvals(A,B)).^2)) +end + +# ╔═╡ ca028a51-2f9b-475b-9db9-470b87cd0f32 +function J2E(a,b,n) + A = randn(Complex{Float64},n+a,n) + B = randn(Complex{Float64},n+b,n) + sqrt.( 1 ./ (1 .+1 ./ (svdvals(A,B)).^2)) +end + +# ╔═╡ 3fba3ecf-4779-4c4b-b367-b12c00d13dff +function orthogonal(n) + +Q = qr(randn(n,n)).Q .* rand([-1,1],n) +while( det(Q)<0) + Q = qr(randn(n,n)).Q .* rand([-1,1],n) +end +e = real.(eigvals(Q))[1:2:n] +e = sqrt.((e .+ 1)/2) + +end + +# ╔═╡ 709c796d-80da-4eea-8c58-0c51e1aff570 +ℍ2ℂ(q::Quaternion) = [q.s+im*q.v1 q.v2+im*q.v3 + -q.v2+im*q.v3 q.s-im*q.v1 ] + +# ╔═╡ c89113f5-55a3-4fdf-a0fe-c4550487fe00 +function symp_unitary(n) + + + +Q = qr(randn(QuaternionF64,n,n)).Q + Q = hvcat(n,ℍ2ℂ.(Matrix(Q))...) + +# e = eigvals(hvcat(n,Q...)) + +# e = real.(e)[1:2:2n] +# e = sqrt.((e .+ 1)/2) + +end + +# ╔═╡ 2c23ab90-6623-48fc-bac4-296bb5cbc917 +let + Q = symp_unitary(2) + Q'Q +end + +# ╔═╡ 5e58d900-6a42-4899-a6d3-ff79a99f9990 +let + n=8 +symp_unitary(n) +end + +# ╔═╡ c7229a4d-2b1c-4bd5-aac3-68ec2ee5212f +let + t = 500 + n = 2 + m_orth = [ minimum(abs.(symp_unitary(n))) for i = 1:t] + a = b = .5 + + + + + # m1 = [ minimum(abs.(JβE(a,b,n÷2,2))) for i = 1:t] + # stephist(m_orth,label="orth") + + # stephist!(m1,label="Jacobi") + # title!("a=$a, b=$b, β=2") + + + +end + +# ╔═╡ 0dba0b42-4df5-4142-ad92-62ca603c3941 +symp_unitary(2) + +# ╔═╡ d07354ab-8834-4194-97a2-263c71b4ff4a +randn(QuaternionF64,1) + +# ╔═╡ 84e64bbc-12a7-4ce6-9721-dc3098939f2a +orthogonal(4) + +# ╔═╡ 96ee753d-4843-4418-a1c5-477707a0fa14 +7÷2 + +# ╔═╡ 0808d006-811f-4084-8cbe-de4d6d9b5c55 +function J4E(a,b,n) + A = randn(Quaternion{Float64},n+a,n) + B = randn(Quaternion{Float64},n+b,n) + sqrt.( 1 ./ (1 .+1 ./ (svdvals(A,B)).^2)) +end + +# ╔═╡ e48ef58a-6a1a-4862-885f-890e877cb0f7 +J1E(2,2,2) + +# ╔═╡ c4c0876c-8389-4262-80ae-1cf0b358d2b6 +function JβE(a,b,n,β) + # exponents (β/2)(a+1)-1, (β/2)(b+1)-1 on [0,1] + i = 1:n + c² = rand.( Beta.( (β/2)*(a.+i),(β/2)*(b.+i)) ) + s² = 1 .- c² + i = 1:n-1 + c′² = rand.( Beta.( (β/2)*i,(β/2)*(a+b+1 .+i)) ) + s′² = 1 .- c′² + c = sqrt.(c²); s = sqrt.(s²); c′=sqrt.(c′²); s′=sqrt.(s′²) + σ = svdvals(Bidiagonal(reverse(c .* [s′;1]), reverse(-s[2:n].*c′),:U)) + +end + +# ╔═╡ d933b5a4-6838-4cdb-92ab-9d244b470787 +let + t = 50000 + n = 4 + m_orth = [ minimum(abs.(orthogonal(n))) for i = 1:t] + a = -.5 + + + n%2 == 0 ? b=-.5 : b = .5 + + + m1 = [ minimum(abs.(JβE(a,b,n÷2,2))) for i = 1:t] + stephist(m_orth,label="orth") + + stephist!(m1,label="Jacobi") + title!("a=$a, b=$b, β=2") + + + +end + +# ╔═╡ 5d955227-162c-4faf-a05b-3583437924df +begin + t = 50 + a = 5 + b = 2 + n = 2 + β = .5 + + m2 = [ minimum(JβE(a,b,n,β)) for i = 1:t] + stephist(m2) + + if β==1 + m1 = [ minimum(J1E(a,b,n)) for i = 1:t]; stephist!(m1) + end + if β==2 + m1 = [ minimum(J2E(a,b,n)) for i = 1:t]; stephist!(m1) + end + title!("a=$a, b=$b, β=$β") + plot!(legend=false) + + +end + +# ╔═╡ 7b6486a9-4487-4d81-9c78-3c86c641431c +minimum(abs.(orthogonal(4))) + +# ╔═╡ 00000000-0000-0000-0000-000000000001 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+# ╠═e48ef58a-6a1a-4862-885f-890e877cb0f7 +# ╠═c4c0876c-8389-4262-80ae-1cf0b358d2b6 +# ╠═5d955227-162c-4faf-a05b-3583437924df +# ╠═7b6486a9-4487-4d81-9c78-3c86c641431c +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002