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A Julia Notebook on a diffferential
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| ### A Pluto.jl notebook ### | |
| # v0.17.7 | |
| using Markdown | |
| using InteractiveUtils | |
| # ╔═╡ b8662ee7-3af9-4703-978f-6ce42408b581 | |
| using Manifolds, Manopt, Plots, Random | |
| # ╔═╡ b0dbce8a-c0c8-4bd9-8f60-e9218fc3058a | |
| begin | |
| using LinearAlgebra | |
| import Manifolds.get_basis | |
| function get_tridiagonal_elements(trian) | |
| N = size(trian, 1) | |
| res = zeros(N) | |
| down = true | |
| for i in 1:N | |
| if i == N && down | |
| elem = 0 | |
| else | |
| elem = trian[i + (down ? +1 : -1), i] | |
| end | |
| if elem ≈ 0 | |
| res[i] = 0 | |
| else | |
| res[i] = elem | |
| down = !down | |
| end | |
| end | |
| return res | |
| end | |
| function ev_diagonal(tridiagonal_elements, unitary; i) | |
| a,b = [unitary[:,i] for i in [i, i+1]] | |
| evec = [[-b a] * [a b]' ./ sqrt(2)] | |
| evals = [0.0] | |
| return (values = evals, vectors = evec) | |
| end | |
| function ev_offdiagonal(tridiagonal_elements, unitary; i, j) | |
| a,b,c,d = [unitary[:,i] for i in [i, i+1, j, j+1]] | |
| ref = transpose(view(unitary, :, [i,i+1, j,j+1])) | |
| evec = [ | |
| [-c -d a b] * ref ./ sqrt(4), | |
| [-c d a -b] * ref ./ sqrt(4), | |
| [-d -c b a] * ref ./ sqrt(4), | |
| [ d -c b -a] * ref ./ sqrt(4), | |
| ] | |
| evals = [(tridiagonal_elements[i] - tridiagonal_elements[j])^2 / 4, | |
| (tridiagonal_elements[i] + tridiagonal_elements[j])^2 / 4, | |
| (tridiagonal_elements[i] + tridiagonal_elements[j])^2 / 4, | |
| (tridiagonal_elements[i] - tridiagonal_elements[j])^2 / 4] | |
| return (values = evals, vectors = evec) | |
| end | |
| function ev_zero(tridiagonal_elements, unitary; i) | |
| evec = Vector{Array{Float64,2}}() | |
| evals = Vector{Float64}() | |
| ref = unitary[:, i] | |
| for idx in 1:size(unitary,1) | |
| if idx != i | |
| push!(evec, (ref * unitary[:, idx]' - unitary[:, idx] * ref') ./ sqrt(2)) | |
| push!(evals, tridiagonal_elements[idx]^2 / 4) | |
| end | |
| end | |
| return (values = evals, vectors = evec) | |
| end | |
| function get_basis(M::Rotations{N}, p, B::DiagonalizingOrthonormalBasis{ℝ}) where N | |
| trian, unitary, ev = schur(B.frame_direction) | |
| trian_elem = get_tridiagonal_elements(trian) | |
| evec = Vector{Array{Float64,2}}() | |
| evals = Vector{Float64}() | |
| i = 1 | |
| while i <= N | |
| if trian_elem[i] == 0 | |
| evs = ev_zero(trian_elem, unitary, i=i) | |
| push!(evec, evs[:vectors]...) | |
| push!(evals, evs[:values]...) | |
| i += 1 | |
| else | |
| evs = ev_diagonal(trian_elem, unitary, i=i) | |
| push!(evec, evs[:vectors]...) | |
| push!(evals, evs[:values]...) | |
| j = 1 | |
| while j < i | |
| if trian_elem[j] == 0 | |
| j += 1 | |
| else | |
| evs = ev_offdiagonal(trian_elem, unitary, i=i, j=j) | |
| push!(evec, evs[:vectors]...) | |
| push!(evals, evs[:values]...) | |
| j += 2 | |
| end | |
| end | |
| i += 2 | |
| end | |
| end | |
| return CachedBasis(B, evals, evec) | |
| end | |
| end | |
| # ╔═╡ a474d350-9689-11ec-2178-65e3d33c9f05 | |
| md""" | |
| # A small note on a certain differential | |
| """ | |
| # ╔═╡ d3f21250-521d-4438-a774-3fcd09fd315f | |
| Random.seed!(42) | |
| # ╔═╡ 610a9702-524e-4294-957c-fc601f7061d1 | |
| md""" | |
| Let ``\mathcal M`` be a manifold, ``p,q \in \mathcal M`` and ``t \in \mathbb R`` be given. | |
| We consider the function $f \colon T_p\mathcal M \to \mathbb R$ | |
| ```math | |
| f(X) = \frac{1}{2}d_{\mathcal M}^2(q, \exp_p(tX)),\qquad X \in T_p\mathcal M | |
| ``` | |
| Hence the differential is a map from ``T_p\mathcal M`` to ``\mathbb R`` since both domain and codomain are their own tangent spaces. | |
| """ | |
| # ╔═╡ 35cdd1a3-ba0f-47e3-83b3-2258a480a957 | |
| md""" | |
| If we use this within minimisation, we can even specify the minmiser ``X^* = \frac{1}{t}\log_pq`` aince then the distance is zero. | |
| """ | |
| # ╔═╡ 996217a9-83db-47f9-b6d5-f28eedbf6e67 | |
| md""" | |
| We decompose ``f`` into the functions | |
| ```math | |
| \begin{align} | |
| g(s) &= \frac{1}{2}d_{\mathcal M}(q, s), \qquad &D_s g(s)[X] &= \langle -\log_sq, X \rangle_s, \quad X \in T_s\mathcal M | |
| \\ | |
| h(Y) &= \exp_p(Y), \qquad &D_Y h(Y)[Z] &= D_Y \exp_p(Y)[Z] | |
| \\ | |
| i(X) &= tX, \qquad &D_X i(X)[Y] &= tY. | |
| \end{align} | |
| ``` | |
| And we use the Chain rule on manifolds, i.e. for ``f, g`` the concatenation ``f(g(p)) = (f\circ g)(p)`` then the chain rule reads ``D (f\circ g)(p)[X] = D f(g(p))\bigl[ Dg(p)[X] \bigr]`` (cf. AMS08 p. 195). | |
| """ | |
| # ╔═╡ 9c17c5ee-7a4d-4bc4-a86e-d606f76fcd40 | |
| md""" | |
| For our case this reads ``f(X) = g(h(i(X)))`` so that the differential reads | |
| ```math | |
| Df(X)[Y] = D\, g\bigl( h(i(X)) \bigr)\Bigl[ D h(i(X)) \bigl[ Di(X)[Y] \bigr] \Bigr] | |
| ``` | |
| where | |
| * ``Di(X)[Y] = tY`` | |
| * ``h(i(X)) = \exp_p(tX)`` | |
| * hence ``D h(i(X)) \bigl[ Di(X)[Y] \bigr] = D_exp_p(tX)[tY]`` | |
| so that we finally have with ``s=\exp_ptX`` that | |
| ```math | |
| \begin{split} | |
| Df(X)[Y] &= D \frac{1}{2}d^2_{\mathcal M}\bigl(q, \exp_p (tX)\bigr) [Y] = D \frac{1}{2}d^2_{\mathcal M}(q,\exp_p tX)\bigl[D\exp_p(tX)[tY]\bigr] | |
| \\ | |
| &= \langle -\log_{\exp_p (tX)} q, D \exp_p(tX)[tY] \rangle_{\exp_p(tX)} | |
| \end{split} | |
| ``` | |
| """ | |
| # ╔═╡ dc9bbe53-abcf-4e02-922b-632234ecff74 | |
| md"Let's check this in code" | |
| # ╔═╡ c9cc1f8c-f9b0-4417-8b46-ddaf70681c1f | |
| M = Sphere(2) | |
| # ╔═╡ 040f9163-cdb8-4b67-925f-6d4938bc4007 | |
| begin | |
| sp_p = random_point(M) | |
| sp_q = random_point(M) | |
| end | |
| # ╔═╡ ad5e006b-7173-49c5-b3f7-e3d95045403e | |
| t = 0.3 | |
| # ╔═╡ 86423703-b26c-4f25-89ae-139f5572e7c4 | |
| sp_Xstar = 1/t * log(M, sp_p, sp_q) | |
| # ╔═╡ 5f7192e9-c56a-4d55-ab31-550b7bb358e6 | |
| sp_f(X) = 0.5 * distance(M, sp_q, exp(M, sp_p, t*X)).^2 | |
| # ╔═╡ d0cfa119-3e93-468d-b039-d5928cadf2b4 | |
| sp_f(sp_Xstar) | |
| # ╔═╡ ffeb8c41-0fd8-47f2-8f76-a6b85a2a98b3 | |
| sp_Df(X, Y) = inner(M, | |
| exp(M, sp_p, t*X), | |
| -log(M, exp(M, sp_p, t*X), sp_q), | |
| differential_exp_argument(M, sp_p, t*X, t*Y), | |
| ) | |
| # ╔═╡ 7e88a79e-3d94-4ab4-a841-f3cc055b245f | |
| md""" the following ready checks, that in ``X^*`` the differential is zero """ | |
| # ╔═╡ 43b84248-8a24-4a54-810c-ca88344224ba | |
| sp_Df(sp_Xstar, random_tangent(M, sp_p)) | |
| # ╔═╡ bd26abad-62e7-49c5-9694-3e80b4a8992f | |
| function check_diff(M::AbstractManifold, cost_fnc, diff_fnc; | |
| p = random_point(M), v = random_tangent(M, p), | |
| log_space_range = range(-8,0, length = 51)) | |
| f0 = cost_fnc(p) | |
| df0 = diff_fnc(p, v) | |
| log_space = 10 .^log_space_range | |
| # f(R_x(t * v)) = f(x) + t * <grad f(x), v)> + O(t^2) | |
| lhs = [cost_fnc(geodesic(M, p, v, t)) for t in log_space] | |
| rhs = [f0 + t * df0 for t in log_space] | |
| err = abs.(rhs - lhs) | |
| plt = scatter(log10.(log_space), log10.(err), legend = false); | |
| Plots.abline!(plt, 1, 0); Plots.abline!(plt, 2, 0) | |
| return plt | |
| end | |
| # ╔═╡ 7ab20b27-3d4f-450b-b6c3-ef1fb3daf3a7 | |
| check_diff(TangentSpace(M, sp_p), sp_f, sp_Df, p = sp_Xstar, v = random_tangent(M, sp_p)) | |
| # ╔═╡ 82f1553b-a7ae-4374-b210-e0be52344fd2 | |
| md""" | |
| # Rotation Manifolds | |
| After demonstrating that the hand-derived differential works well for Spheres, let's check if it also works for Rotation manifolds | |
| """ | |
| # ╔═╡ 3daab236-66cf-4ab1-9bf6-16069c2ea62f | |
| md""" | |
| We start of by defining some helper function for the ortho-normal basis decomposition of the Jacobi operator for Rotation manifolds. | |
| """ | |
| # ╔═╡ f058edf3-b3ef-4fe4-8a5b-85218b82fecc | |
| md""" | |
| Next, we create the set of points and tangent vectors that we will use for checking our differential. | |
| """ | |
| # ╔═╡ 2981c0df-e480-488c-a99b-71e2e58bd09e | |
| begin | |
| R = Rotations(3) | |
| rot_p = random_point(R) | |
| rot_q = random_point(R) | |
| rot_Xstar = 1/t * log(R, rot_p, rot_q) | |
| end | |
| # ╔═╡ 8f23143a-434f-4171-9810-1f124e8ca243 | |
| md""" | |
| And we copy the definitions of the cost functions and the differential (because Pluto doesn't allow the redefinition of exisiting variables). | |
| """ | |
| # ╔═╡ 256eac60-1392-4955-8457-34a748552d0b | |
| rot_f(X) = 0.5 * distance(R, rot_q, exp(R, rot_p, t*X)).^2 | |
| # ╔═╡ 26492b35-86bd-4de8-a4b1-ce5fcab7572f | |
| rot_Df(X, Y) = inner(R, | |
| exp(R, rot_p, t*X), | |
| -log(R, exp(R, rot_p, t*X), rot_q), | |
| differential_exp_argument(R, rot_p, t*X, t*Y), | |
| ) | |
| # ╔═╡ 7e6ff61a-2a67-4e35-b133-05b9fce7fa23 | |
| rot_f(rot_Xstar) | |
| # ╔═╡ ef82cdc3-71f9-43b3-b966-5e8251e5a261 | |
| rot_Df(rot_Xstar, random_tangent(R, sp_p)) | |
| # ╔═╡ 6590aeaa-d63d-4296-aab5-9aab2bc9397f | |
| check_diff(TangentSpace(R, rot_p), rot_f, rot_Df, | |
| p = rot_Xstar, v = random_tangent(R, rot_p)) | |
| # ╔═╡ 24634ffe-b07e-4abd-9618-e0d1b731884b | |
| md""" | |
| So far the checks look excellent. However, we are currently testing at the ``X^*``, the optimum of ``f``. | |
| """ | |
| # ╔═╡ f91f7d84-8a87-48aa-8215-043c632b9bef | |
| md""" | |
| # Calling `check_diff()` at random points | |
| """ | |
| # ╔═╡ a57498a8-8628-4ca0-90e6-c13bea5c3b3d | |
| md""" | |
| While the differential `sp_Df` works for Spheres... | |
| """ | |
| # ╔═╡ 89c16b4d-1f9b-4de8-8a93-a057a64d0188 | |
| check_diff(TangentSpace(M, sp_p), sp_f, sp_Df, | |
| p = random_tangent(M, sp_p), v = random_tangent(M, sp_p)) | |
| # ╔═╡ c309b6a9-90ce-4bf5-a4bf-a24e73e8aefc | |
| md""" | |
| ... the same code fails for Rotation manifolds. | |
| """ | |
| # ╔═╡ 28822962-d73d-4a8a-a507-ac1aa48ae9a0 | |
| check_diff(TangentSpace(R, rot_p), rot_f, rot_Df, | |
| p = random_tangent(R, rot_p), v = random_tangent(R, rot_p)) | |
| # ╔═╡ 5ca90811-725b-4248-affb-65bafaf04a60 | |
| md""" | |
| # Modifying `βdifferential_exp_arg` | |
| Through more or less random experiments, I found last week that modifying `βdifferential_exp_arg` to take the `t` argument into account, seems to improve the results. | |
| """ | |
| # ╔═╡ 1e0ddc01-1e1d-4dbd-9e7d-366e5244e363 | |
| function βdifferential_exp_arg_with_time(κ, t, d) | |
| (d == 0 || t == 0) && return 1 | |
| (κ < 0) && return sinh(sqrt(-κ) * t * d) / (t * d * sqrt(-κ)) | |
| (κ > 0) && return sin(sqrt(κ) * t * d) / (t * d * sqrt(κ)) | |
| return 1 # curvature zero | |
| end | |
| # ╔═╡ 9f1e733c-ea30-41b4-a5d8-c78968f57f30 | |
| rot_Df2(X, Y) = inner(R, | |
| exp(R, rot_p, t*X), | |
| -log(R, exp(R, rot_p, t*X), rot_q), | |
| jacobi_field(R, rot_p, exp(R, rot_p, X), t, t * Y, βdifferential_exp_arg_with_time) | |
| ) | |
| # ╔═╡ a7bb5819-ae74-4071-8024-aff36ef2acb0 | |
| check_diff(TangentSpace(R, rot_p), rot_f, rot_Df2, | |
| p = random_tangent(R, rot_p), v = random_tangent(R, rot_p)) | |
| # ╔═╡ 00000000-0000-0000-0000-000000000001 | |
| PLUTO_PROJECT_TOML_CONTENTS = """ | |
| [deps] | |
| LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" | |
| Manifolds = "1cead3c2-87b3-11e9-0ccd-23c62b72b94e" | |
| Manopt = "0fc0a36d-df90-57f3-8f93-d78a9fc72bb5" | |
| Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" | |
| Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" | |
| [compat] | |
| Manifolds = "~0.7.5" | |
| Manopt = "~0.3.17" | |
| Plots = "~1.24.3" | |
| """ | |
| # ╔═╡ 00000000-0000-0000-0000-000000000002 | |
| PLUTO_MANIFEST_TOML_CONTENTS = """ | |
| # This file is machine-generated - editing it directly is not advised | |
| [[AbstractFFTs]] | |
| deps = ["ChainRulesCore", "LinearAlgebra"] | |
| git-tree-sha1 = "6f1d9bc1c08f9f4a8fa92e3ea3cb50153a1b40d4" | |
| uuid = "621f4979-c628-5d54-868e-fcf4e3e8185c" | |
| version = "1.1.0" | |
| [[Adapt]] | |
| deps = ["LinearAlgebra"] | |
| git-tree-sha1 = "af92965fb30777147966f58acb05da51c5616b5f" | |
| uuid = "79e6a3ab-5dfb-504d-930d-738a2a938a0e" | |
| version = "3.3.3" | |
| [[ArgTools]] | |
| git-tree-sha1 = "bdf73eec6a88885256f282d48eafcad25d7de494" | |
| uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f" | |
| version = "1.1.1" | |
| [[ArnoldiMethod]] | |
| deps = ["LinearAlgebra", "Random", "StaticArrays"] | |
| git-tree-sha1 = "62e51b39331de8911e4a7ff6f5aaf38a5f4cc0ae" | |
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| version = "0.2.0" | |
| [[ArrayInterface]] | |
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| git-tree-sha1 = "1ee88c4c76caa995a885dc2f22a5d548dfbbc0ba" | |
| uuid = "4fba245c-0d91-5ea0-9b3e-6abc04ee57a9" | |
| version = "3.2.2" | |
| [[Artifacts]] | |
| deps = ["Pkg"] | |
| git-tree-sha1 = "c30985d8821e0cd73870b17b0ed0ce6dc44cb744" | |
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| [[Base64]] | |
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| [[Bzip2_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "c3598e525718abcc440f69cc6d5f60dda0a1b61e" | |
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| version = "1.0.6+5" | |
| [[Cairo_jll]] | |
| deps = ["Artifacts", "Bzip2_jll", "Fontconfig_jll", "FreeType2_jll", "Glib_jll", "JLLWrappers", "LZO_jll", "Libdl", "Pixman_jll", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll", "Zlib_jll", "libpng_jll"] | |
| git-tree-sha1 = "e2f47f6d8337369411569fd45ae5753ca10394c6" | |
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| [[Calculus]] | |
| deps = ["LinearAlgebra"] | |
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| uuid = "49dc2e85-a5d0-5ad3-a950-438e2897f1b9" | |
| version = "0.5.1" | |
| [[ChainRulesCore]] | |
| deps = ["Compat", "LinearAlgebra", "SparseArrays"] | |
| git-tree-sha1 = "32ad4ece064a61855a35bdc34e3da0b495e01169" | |
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| version = "1.12.2" | |
| [[ChangesOfVariables]] | |
| deps = ["ChainRulesCore", "LinearAlgebra", "Test"] | |
| git-tree-sha1 = "bf98fa45a0a4cee295de98d4c1462be26345b9a1" | |
| uuid = "9e997f8a-9a97-42d5-a9f1-ce6bfc15e2c0" | |
| version = "0.1.2" | |
| [[ColorSchemes]] | |
| deps = ["ColorTypes", "Colors", "FixedPointNumbers", "Random"] | |
| git-tree-sha1 = "12fc73e5e0af68ad3137b886e3f7c1eacfca2640" | |
| uuid = "35d6a980-a343-548e-a6ea-1d62b119f2f4" | |
| version = "3.17.1" | |
| [[ColorTypes]] | |
| deps = ["FixedPointNumbers", "Random"] | |
| git-tree-sha1 = "024fe24d83e4a5bf5fc80501a314ce0d1aa35597" | |
| uuid = "3da002f7-5984-5a60-b8a6-cbb66c0b333f" | |
| version = "0.11.0" | |
| [[Colors]] | |
| deps = ["ColorTypes", "FixedPointNumbers", "Reexport"] | |
| git-tree-sha1 = "417b0ed7b8b838aa6ca0a87aadf1bb9eb111ce40" | |
| uuid = "5ae59095-9a9b-59fe-a467-6f913c188581" | |
| version = "0.12.8" | |
| [[Compat]] | |
| deps = ["Base64", "Dates", "DelimitedFiles", "Distributed", "InteractiveUtils", "LibGit2", "Libdl", "LinearAlgebra", "Markdown", "Mmap", "Pkg", "Printf", "REPL", "Random", "SHA", "Serialization", "SharedArrays", "Sockets", "SparseArrays", "Statistics", "Test", "UUIDs", "Unicode"] | |
| git-tree-sha1 = "44c37b4636bc54afac5c574d2d02b625349d6582" | |
| uuid = "34da2185-b29b-5c13-b0c7-acf172513d20" | |
| version = "3.41.0" | |
| [[CompilerSupportLibraries_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "8e695f735fca77e9708e795eda62afdb869cbb70" | |
| uuid = "e66e0078-7015-5450-92f7-15fbd957f2ae" | |
| version = "0.3.4+0" | |
| [[Contour]] | |
| deps = ["StaticArrays"] | |
| git-tree-sha1 = "9f02045d934dc030edad45944ea80dbd1f0ebea7" | |
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| [[CovarianceEstimation]] | |
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| [[Distributions]] | |
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| deps = ["FFMPEG_jll"] | |
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| [[GLFW_jll]] | |
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| [[GR]] | |
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| [[LaTeXStrings]] | |
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| [[LibSSH2_jll]] | |
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| [[Libffi_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
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| [[Libgcrypt_jll]] | |
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| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
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| git-tree-sha1 = "291dd857901f94d683973cdf679984cdf73b56d0" | |
| uuid = "89763e89-9b03-5906-acba-b20f662cd828" | |
| version = "4.1.0+2" | |
| [[Libuuid_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
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| uuid = "38a345b3-de98-5d2b-a5d3-14cd9215e700" | |
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| [[LogExpFunctions]] | |
| deps = ["ChainRulesCore", "ChangesOfVariables", "DocStringExtensions", "InverseFunctions", "IrrationalConstants", "LinearAlgebra"] | |
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| uuid = "2ab3a3ac-af41-5b50-aa03-7779005ae688" | |
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| [[Logging]] | |
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| version = "0.5.9" | |
| [[Manifolds]] | |
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| [[Markdown]] | |
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| [[MbedTLS]] | |
| deps = ["Dates", "MbedTLS_jll", "Random", "Sockets"] | |
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| uuid = "739be429-bea8-5141-9913-cc70e7f3736d" | |
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| [[MbedTLS_jll]] | |
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| [[Measures]] | |
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| [[Missings]] | |
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| [[Mmap]] | |
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| [[MozillaCACerts_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
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| [[NetworkOptions]] | |
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| [[Ogg_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
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| [[OpenLibm_jll]] | |
| deps = ["Libdl", "Pkg"] | |
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| uuid = "05823500-19ac-5b8b-9628-191a04bc5112" | |
| version = "0.7.1+0" | |
| [[OpenSSL_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
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| uuid = "458c3c95-2e84-50aa-8efc-19380b2a3a95" | |
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| [[OpenSpecFun_jll]] | |
| deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Pkg"] | |
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| uuid = "efe28fd5-8261-553b-a9e1-b2916fc3738e" | |
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| [[Opus_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
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| uuid = "91d4177d-7536-5919-b921-800302f37372" | |
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| [[OrderedCollections]] | |
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| [[PCRE_jll]] | |
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| uuid = "2f80f16e-611a-54ab-bc61-aa92de5b98fc" | |
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| [[PDMats]] | |
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| [[Parsers]] | |
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| [[Pixman_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
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| uuid = "30392449-352a-5448-841d-b1acce4e97dc" | |
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| [[Pkg]] | |
| deps = ["Dates", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "UUIDs"] | |
| uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f" | |
| [[PlotThemes]] | |
| deps = ["PlotUtils", "Requires", "Statistics"] | |
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| [[PlotUtils]] | |
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| [[Plots]] | |
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| [[Preferences]] | |
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| [[Printf]] | |
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| [[Qt5Base_jll]] | |
| deps = ["Artifacts", "CompilerSupportLibraries_jll", "Fontconfig_jll", "Glib_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "OpenSSL_jll", "Pkg", "Xorg_libXext_jll", "Xorg_libxcb_jll", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_keysyms_jll", "Xorg_xcb_util_renderutil_jll", "Xorg_xcb_util_wm_jll", "Zlib_jll", "xkbcommon_jll"] | |
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| [[Random]] | |
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| [[RecipesBase]] | |
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| [[Reexport]] | |
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| [[SHA]] | |
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| [[Scratch]] | |
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| [[Showoff]] | |
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| [[SpecialFunctions]] | |
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| [[Static]] | |
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| [[StatsAPI]] | |
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| [[StatsBase]] | |
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| [[StatsFuns]] | |
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| [[TOML]] | |
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| [[URIs]] | |
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| [[UUIDs]] | |
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| [[Unicode]] | |
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| [[UnicodeFun]] | |
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| [[Wayland_jll]] | |
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| git-tree-sha1 = "dc643a9b774da1c2781413fd7b6dcd2c56bb8056" | |
| uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89" | |
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| [[Wayland_protocols_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
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| uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91" | |
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| [[XML2_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"] | |
| git-tree-sha1 = "be0db24f70aae7e2b89f2f3092e93b8606d659a6" | |
| uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a" | |
| version = "2.9.10+3" | |
| [[XSLT_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Pkg", "XML2_jll"] | |
| git-tree-sha1 = "2b3eac39df218762d2d005702d601cd44c997497" | |
| uuid = "aed1982a-8fda-507f-9586-7b0439959a61" | |
| version = "1.1.33+4" | |
| [[Xorg_libX11_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"] | |
| git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527" | |
| uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc" | |
| version = "1.6.9+4" | |
| [[Xorg_libXau_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
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| uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec" | |
| version = "1.0.9+4" | |
| [[Xorg_libXcursor_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"] | |
| git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd" | |
| uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724" | |
| version = "1.2.0+4" | |
| [[Xorg_libXdmcp_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4" | |
| uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05" | |
| version = "1.1.3+4" | |
| [[Xorg_libXext_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"] | |
| git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3" | |
| uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3" | |
| version = "1.3.4+4" | |
| [[Xorg_libXfixes_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"] | |
| git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4" | |
| uuid = "d091e8ba-531a-589c-9de9-94069b037ed8" | |
| version = "5.0.3+4" | |
| [[Xorg_libXi_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"] | |
| git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246" | |
| uuid = "a51aa0fd-4e3c-5386-b890-e753decda492" | |
| version = "1.7.10+4" | |
| [[Xorg_libXinerama_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"] | |
| git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123" | |
| uuid = "d1454406-59df-5ea1-beac-c340f2130bc3" | |
| version = "1.1.4+4" | |
| [[Xorg_libXrandr_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"] | |
| git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631" | |
| uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484" | |
| version = "1.5.2+4" | |
| [[Xorg_libXrender_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"] | |
| git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96" | |
| uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa" | |
| version = "0.9.10+4" | |
| [[Xorg_libpthread_stubs_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb" | |
| uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74" | |
| version = "0.1.0+3" | |
| [[Xorg_libxcb_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"] | |
| git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6" | |
| uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b" | |
| version = "1.13.0+3" | |
| [[Xorg_libxkbfile_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"] | |
| git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2" | |
| uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a" | |
| version = "1.1.0+4" | |
| [[Xorg_xcb_util_image_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] | |
| git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97" | |
| uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b" | |
| version = "0.4.0+1" | |
| [[Xorg_xcb_util_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"] | |
| git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1" | |
| uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5" | |
| version = "0.4.0+1" | |
| [[Xorg_xcb_util_keysyms_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] | |
| git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00" | |
| uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7" | |
| version = "0.4.0+1" | |
| [[Xorg_xcb_util_renderutil_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] | |
| git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e" | |
| uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e" | |
| version = "0.3.9+1" | |
| [[Xorg_xcb_util_wm_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] | |
| git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67" | |
| uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361" | |
| version = "0.4.1+1" | |
| [[Xorg_xkbcomp_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"] | |
| git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b" | |
| uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4" | |
| version = "1.4.2+4" | |
| [[Xorg_xkeyboard_config_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"] | |
| git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d" | |
| uuid = "33bec58e-1273-512f-9401-5d533626f822" | |
| version = "2.27.0+4" | |
| [[Xorg_xtrans_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845" | |
| uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10" | |
| version = "1.4.0+3" | |
| [[Zlib_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "320228915c8debb12cb434c59057290f0834dbf6" | |
| uuid = "83775a58-1f1d-513f-b197-d71354ab007a" | |
| version = "1.2.11+18" | |
| [[Zstd_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "2c1332c54931e83f8f94d310fa447fd743e8d600" | |
| uuid = "3161d3a3-bdf6-5164-811a-617609db77b4" | |
| version = "1.4.8+0" | |
| [[ZygoteRules]] | |
| deps = ["MacroTools"] | |
| git-tree-sha1 = "8c1a8e4dfacb1fd631745552c8db35d0deb09ea0" | |
| uuid = "700de1a5-db45-46bc-99cf-38207098b444" | |
| version = "0.2.2" | |
| [[libass_jll]] | |
| deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"] | |
| git-tree-sha1 = "acc685bcf777b2202a904cdcb49ad34c2fa1880c" | |
| uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0" | |
| version = "0.14.0+4" | |
| [[libfdk_aac_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "7a5780a0d9c6864184b3a2eeeb833a0c871f00ab" | |
| uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280" | |
| version = "0.1.6+4" | |
| [[libpng_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"] | |
| git-tree-sha1 = "6abbc424248097d69c0c87ba50fcb0753f93e0ee" | |
| uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f" | |
| version = "1.6.37+6" | |
| [[libvorbis_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"] | |
| git-tree-sha1 = "fa14ac25af7a4b8a7f61b287a124df7aab601bcd" | |
| uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a" | |
| version = "1.3.6+6" | |
| [[nghttp2_jll]] | |
| deps = ["Libdl", "Pkg"] | |
| git-tree-sha1 = "8e2c44ab4d49ad9518f359ed8b62f83ba8beede4" | |
| uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d" | |
| version = "1.40.0+2" | |
| [[x264_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "d713c1ce4deac133e3334ee12f4adff07f81778f" | |
| uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a" | |
| version = "2020.7.14+2" | |
| [[x265_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "487da2f8f2f0c8ee0e83f39d13037d6bbf0a45ab" | |
| uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76" | |
| version = "3.0.0+3" | |
| [[xkbcommon_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"] | |
| git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6" | |
| uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd" | |
| version = "0.9.1+5" | |
| """ | |
| # ╔═╡ Cell order: | |
| # ╟─a474d350-9689-11ec-2178-65e3d33c9f05 | |
| # ╠═b8662ee7-3af9-4703-978f-6ce42408b581 | |
| # ╠═d3f21250-521d-4438-a774-3fcd09fd315f | |
| # ╟─610a9702-524e-4294-957c-fc601f7061d1 | |
| # ╟─35cdd1a3-ba0f-47e3-83b3-2258a480a957 | |
| # ╟─996217a9-83db-47f9-b6d5-f28eedbf6e67 | |
| # ╟─9c17c5ee-7a4d-4bc4-a86e-d606f76fcd40 | |
| # ╟─dc9bbe53-abcf-4e02-922b-632234ecff74 | |
| # ╠═c9cc1f8c-f9b0-4417-8b46-ddaf70681c1f | |
| # ╠═040f9163-cdb8-4b67-925f-6d4938bc4007 | |
| # ╠═ad5e006b-7173-49c5-b3f7-e3d95045403e | |
| # ╠═86423703-b26c-4f25-89ae-139f5572e7c4 | |
| # ╠═5f7192e9-c56a-4d55-ab31-550b7bb358e6 | |
| # ╠═d0cfa119-3e93-468d-b039-d5928cadf2b4 | |
| # ╠═ffeb8c41-0fd8-47f2-8f76-a6b85a2a98b3 | |
| # ╟─7e88a79e-3d94-4ab4-a841-f3cc055b245f | |
| # ╠═43b84248-8a24-4a54-810c-ca88344224ba | |
| # ╟─bd26abad-62e7-49c5-9694-3e80b4a8992f | |
| # ╠═7ab20b27-3d4f-450b-b6c3-ef1fb3daf3a7 | |
| # ╟─82f1553b-a7ae-4374-b210-e0be52344fd2 | |
| # ╟─3daab236-66cf-4ab1-9bf6-16069c2ea62f | |
| # ╟─b0dbce8a-c0c8-4bd9-8f60-e9218fc3058a | |
| # ╟─f058edf3-b3ef-4fe4-8a5b-85218b82fecc | |
| # ╠═2981c0df-e480-488c-a99b-71e2e58bd09e | |
| # ╟─8f23143a-434f-4171-9810-1f124e8ca243 | |
| # ╠═256eac60-1392-4955-8457-34a748552d0b | |
| # ╠═26492b35-86bd-4de8-a4b1-ce5fcab7572f | |
| # ╠═7e6ff61a-2a67-4e35-b133-05b9fce7fa23 | |
| # ╠═ef82cdc3-71f9-43b3-b966-5e8251e5a261 | |
| # ╠═6590aeaa-d63d-4296-aab5-9aab2bc9397f | |
| # ╟─24634ffe-b07e-4abd-9618-e0d1b731884b | |
| # ╟─f91f7d84-8a87-48aa-8215-043c632b9bef | |
| # ╟─a57498a8-8628-4ca0-90e6-c13bea5c3b3d | |
| # ╠═89c16b4d-1f9b-4de8-8a93-a057a64d0188 | |
| # ╟─c309b6a9-90ce-4bf5-a4bf-a24e73e8aefc | |
| # ╠═28822962-d73d-4a8a-a507-ac1aa48ae9a0 | |
| # ╟─5ca90811-725b-4248-affb-65bafaf04a60 | |
| # ╠═1e0ddc01-1e1d-4dbd-9e7d-366e5244e363 | |
| # ╠═9f1e733c-ea30-41b4-a5d8-c78968f57f30 | |
| # ╠═a7bb5819-ae74-4071-8024-aff36ef2acb0 | |
| # ╟─00000000-0000-0000-0000-000000000001 | |
| # ╟─00000000-0000-0000-0000-000000000002 |
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