### A Pluto.jl notebook ###
# v0.19.19

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
end

# ╔═╡ a22d543c-6a50-11ed-002d-7f2fe5bd7276
using Plots, PlutoUI, LaTeXStrings, ShortCodes

# ╔═╡ 4de22c75-0d00-4838-8ba4-52bab4dab89e
md"""
# Polynomial interpolation

In this notebook, we will look at different approaches to approximate a given function with polynomial interpolation. We know the interpolation error ``E_n`` of a polynomial of degree ``n`` and for a function ``f \in C^{n+1}[a, b]`` to be
```math
E_n[f](x) = \frac{\omega_{n+1}}{(n + 1)!} f^{(n+1)}(\xi),
```
where ``\xi \in [a, b]`` is an unknown location and
```math
\omega_{n+1}(x) = \prod\limits_{j = 0}^{n} (x - x_j).
```
Thus, the total error is based on both the function to interpolate ``f^{(n+1)}(\xi)`` and the function-independent node locations ``\frac{\omega_{n+1}}{(n + 1)!}``.

Below, the left plot shows the function and its interpolation, while the right plot shows the nodal error (and its optimal bounds given by the Chebyshev nodes). We also give the computed maximum error and the integral error of the interpolant on ``[a, b]``.
"""

# ╔═╡ 43ef7734-1c6b-41f8-b581-44c77252a595
md"""
##### Node positions for `free` node distribution
Note: extra nodes not currently used are ignored.
"""

# ╔═╡ 6e4a15eb-8c68-4837-81dd-caac7c81ba7e
md"""
##### How to add a new function
To add your own function (and show the corresponding polynomial interpolation), you need to do the following things:
* Implement a new function of a single variable `x` (let's call it `myfun`).
* Add `myfun` to the dropdown menu above (under "Interpolation settings").
One example, for which this has already been done in this notebook, is `bell`. To get started, simply copy `bell`, rename it to something unique, and implement away 🙂
"""

# ╔═╡ 6b49c667-b16e-44f3-9f0d-2b8a62f5efda
md"""
## Auxiliary functions and internals
If you are only interested in *using* the interactive plotting features in this notebook, there is **no need to read any further**. The remainder of this document contains auxiliary functions for computing the Taylor polynomial. Furthermore, implementation details that are required for this workbook to execute correctly are included here.
"""

# ╔═╡ b8f81a3f-d773-4e67-b8d7-e1734b04abaa
md"""
### Load required packages
This noteboook requires some additional packages to work:
* Plots.jl is used for plotting the results
* PlutoUI.jl is used to create the interactive GUI elements
* LaTeXStrings.jl is used to make some mathematical notations look nicer
* ShortCodes.jl allows to embed references to papers etc. more easily
"""

# ╔═╡ 94c7aca0-1f23-4b2a-94ae-1aef90fed8d9
md"""
### Definitions for additional functions to interpolate
"""

# ╔═╡ 9e81ff03-1e93-4664-a030-32c593b2be0a
function bell(x)
	return 1/(1 + x^2)
end

# ╔═╡ f87f000f-e0c9-46eb-a242-89a4290bd84f
md"""
##### Interpolation settings
Function to interpolate: $(@bind selected_function Select([cos, sin, bell]))

Number of nodes ``N``: $(@bind N Slider(3:7, default=5, show_value=true))

Node distribution: $(@bind node_distribution Select(["equidistant", "Chebyshev", "free"]))

In case `free` is selected, use the sliders below to chose the node positions.

Plot range: $(@bind x_min Scrubbable(-5.0)) to $(@bind x_max Scrubbable(5.0))
"""

# ╔═╡ 018a4c32-c7d4-4b47-9a87-91b972f2eb27
md"""
Node 1: $(@bind node1 Slider(x_min:0.05:x_max, default=x_min, show_value=true))

Node 2: $(@bind node2 Slider(x_min:0.05:x_max, default=1/6*(x_max-x_min)+x_min, show_value=true))

Node 3: $(@bind node3 Slider(x_min:0.05:x_max, default=2/6*(x_max-x_min)+x_min, show_value=true))

Node 4: $(@bind node4 Slider(x_min:0.05:x_max, default=3/6*(x_max-x_min)+x_min, show_value=true))

Node 5: $(@bind node5 Slider(x_min:0.05:x_max, default=4/6*(x_max-x_min)+x_min, show_value=true))

Node 6: $(@bind node6 Slider(x_min:0.05:x_max, default=5/6*(x_max-x_min)+x_min, show_value=true))

Node 7: $(@bind node7 Slider(x_min:0.05:x_max, default=x_max, show_value=true))
"""

# ╔═╡ cfc107a8-adda-418a-a468-20efeeecaed5
md"""
### Definitions for Lagrange interpolation functions
The following includes definitions for computing the Lagrange interpolating polynomial for a given set of nodes, using the second form (also called true form) of the barycentric interpolation formula. For more details, see, e.g., 

* $(DOI("10.1137/S0036144502417715"))
* [https://en.wikipedia.org/wiki/Lagrange_polynomial](https://en.wikipedia.org/wiki/Lagrange_polynomial), under _Barycentric Form_

The actual implementations are based on the file [`basis_lobatto_legendre.jl`](https://github.com/trixi-framework/Trixi.jl/blob/5e28148c349dc6f14632a3fac801b89e23b85096/src/solvers/dgsem/basis_lobatto_legendre.jl) of the Julia-based simulation framework [Trixi.jl](https://github.com/trixi-framework/Trixi.jl).
"""

# ╔═╡ 242b9f3f-392b-4f2e-af66-b58db0004224
"""
    barycentric_weights(nodes)

Compute the barycentric weights ``\\omega_j`` for a given list of ``N`` node locations `nodes` as
```math
 \\omega_j = \\frac{1}{\\prod\\limits_{j \\neq k} (x_j - x_k)}, \\qquad j = 1, \\ldots, N
```
and return them as an array.
"""
function barycentric_weights(nodes)
	# Initialize all weights with `1.0`
    N = length(nodes)
    weights = ones(N)

	# Compute the product in the denominator for all weights simultaneously,
	# making use of the symmetry to speed things up
    for j = 2:N, k = 1:(j-1)
        weights[k] *= nodes[k] - nodes[j]
        weights[j] *= nodes[j] - nodes[k]
    end

	# Compute the inverse for each weight
    for j in 1:N
        weights[j] = 1 / weights[j]
    end

    return weights
end

# ╔═╡ 46cf9285-3a01-4ad5-b0f1-c077363fbb5f
"""
    lagrange_interpolating_polynomials(x, nodes, wbary)

Evaluate the Lagrange interpolating polynomials at position `x`. The ``N``
polynomials are defined through the node locations `nodes` and the corresponding
barycentric weights `wbary`.
"""
function lagrange_interpolating_polynomials(x, nodes, wbary)
	# Initialize all polynomials with `0.0`
    N = length(nodes)
    polynomials = zeros(N)

	# Early exit if `x` matches one of the node locations to avoid catastrophic
	# "∞/∞" division
    for i in 1:N
        if isapprox(x, nodes[i], rtol=eps(x))
            polynomials[i] = 1
            return polynomials
        end
    end

	# Otherwise compute each polynomial at position `x`
    for i in 1:N
       polynomials[i] = wbary[i] / (x - nodes[i])
    end
    total = sum(polynomials)

    for i in 1:N
        polynomials[i] /= total
    end

    return polynomials
end

# ╔═╡ 8156b2cc-bd11-4714-82c5-00f97eceec8c
md"""
### Definitions for functions to compute node locations
The following functions compute the node locations with an equidistant distribution or as the roots of the Chebyshev polynomial.
"""

# ╔═╡ e560e608-ac7a-427d-9e2f-715ad01283da
"""
    equidistant_nodes(a, b, N)

Compute ``N`` equidistant node locations (including end points) on the interval
``[a, b]``.
"""
function equidistant_nodes(a, b, N)
	# Sanity check on the input
	@assert N > 1 "No equidistant nodes can be found for less than two locations."
	@assert a < b "Interval boundary `a` must be less than `b`"

	# Initialize nodes to zero and compute spacing `h`
	nodes = zeros(N)
	h = (b - a)/(N - 1)

	# Compute node locations
	for i in 1:N
		nodes[i] = a + (i - 1) * h
	end

	return nodes
end

# ╔═╡ 4327ee5b-2491-44b1-8330-6c1121f8bdb1
"""
    chebyshev_nodes(a, b, N)

Compute ``N`` node locations as the roots of the corresponding Chebyshev polynomial
and transform them to the interval ``[a, b]``.
"""
function chebyshev_nodes(a, b, N)
	# Sanity check on the input
	@assert N > 0 "No Chebyshev nodes can be found for less than one location."
	@assert a < b "Interval boundary `a` must be less than `b`"

	# Initialize nodes to zero
	nodes = zeros(N)

	# Compute node locations on interval [-1, 1] and transform to interval [a, b]
	for i in 1:N
		normalized = cos((2*i - 1) / (2*N) * pi)
		nodes[i] = (normalized + 1) / 2 * (b - a) + a
	end

	# Sort nodes from smaller to larger values
	sort!(nodes)

	return nodes
end

# ╔═╡ 4d297775-84a7-42e3-ac72-792338351354
md"""
### Definitions of some helper functions
"""

# ╔═╡ 0ff277d6-1cb8-4bc7-9a37-4d8f6f6d2e27
"""
    omega(x, nodes)

Compute the function
```math
\\omega_{N}(x) = \\prod\\limits_{j = 0}^{N-1} (x - x_j)
```
for ``N`` node positions `nodes` and at location `x`.
"""
function omega(x, nodes)
	result = 1.0
	for i in eachindex(nodes)
	    result *= (x - nodes[i])
	end
	return result
end

# ╔═╡ ac216c60-8288-465b-ab2b-2ba3e7701d02
begin
	# Auxiliary variables
	n = N - 1
	y_min = -2.0
	y_max = 2.0
	xvalues = range(x_min, x_max, 1001)
	
	# Compute exact function values
	exact_values = selected_function.(xvalues)

	# Compute nodes and barycentric weights
	nodes_cheby = chebyshev_nodes(x_min, x_max, N)
	if node_distribution == "equidistant"
		nodes = equidistant_nodes(x_min, x_max, N)
	elseif node_distribution == "Chebyshev"
		nodes = nodes_cheby
	else
		nodes = sort([node1, node2, node3, node4, node5, node6, node7][1:N])
	end
	wbary = barycentric_weights(nodes)

	# Compute polynomial interpolation values
	yvalues = selected_function.(nodes)
	interpolated_values = zeros(length(xvalues))
	for i in eachindex(xvalues)
		polynomials = lagrange_interpolating_polynomials(xvalues[i], nodes, wbary)
		interpolated_values[i] = sum(yvalues .* polynomials)
	end

	# Compute ω/(n+1)! as the nodal contribution to the total approximation error
	nodal_errors_real = omega.(xvalues, Ref(nodes))/factorial(N)
	nodal_errors_cheby = omega.(xvalues, Ref(nodes_cheby))/factorial(N)
	max_nodal_error_cheby = abs(maximum(nodal_errors_cheby))

	# Compute difference between exact and interpolated values
	difference = exact_values .- interpolated_values
	difference_abs = abs.(difference)
	linf_error = maximum(difference_abs)
	l1_error = sum(difference_abs) * (x_max - x_min)/length(xvalues)
	
	# Plot canvas 1 and set some default options
	p1 = plot(
		xlims=(x_min, x_max),
		ylims=(y_min, y_max),
		legend=:bottomright,
		legendfontsize=10,
		tickfontsize=10,
		gridlinewidth=1.5,
		title=L"\textrm{Interpolant}",
	)
	
	# Plot exact function
	label_exact = L"%$(nameof(selected_function))$\,(x)$"
    plot!(p1, xvalues, exact_values, label=label_exact, linewidth=3)

	# Plot polynomial
	label_polynomial = L"I_{%$n}[%$selected_function](x)"
    plot!(p1, xvalues, interpolated_values, label=label_polynomial, linewidth=3, linecolor=:green)
	
	# Plot fillrange
    plot!(p1, xvalues, interpolated_values, fillrange=exact_values, fillalpha=0.3, fillcolor=:green, label="")

	# Plot markers at node locations
	scatter!(p1, nodes, yvalues, label="nodes", markersize=5, markercolor=:red)

	# Plot canvas 2 and set some default options
	y_min2 = min(y_min, -(ceil(max_nodal_error_cheby)+1))
	y_max2 = max(y_max, +(ceil(max_nodal_error_cheby)+1))
	p2 = plot(
		xlims=(x_min, x_max),
		ylims=(y_min2, y_max2),
		legend=:bottomright,
		legendfontsize=10,
		tickfontsize=10,
		gridlinewidth=1.5,
		title=L"\textrm{Error}\, \frac{\omega_{n+1}}{(n+1)!}",
	)

	# Plot nodal error
	plot!(p2, xvalues, nodal_errors_real, label="nodal error", linewidth=3, linecolor=:purple)

	# Plot optimal error bounds
	plot!([x_min, x_max], [max_nodal_error_cheby, max_nodal_error_cheby],
		  linestyle=:dash, linecolor=:gray, label="")
	plot!([x_min, x_max], [-max_nodal_error_cheby, -max_nodal_error_cheby],
		  linestyle=:dash, linecolor=:gray, label="optimal bounds")

	# Plot markers at node locations on the x-axis
	scatter!(p2, nodes, zeros(size(nodes)), label="nodes", markersize=5, markercolor=:red)

	# Create joint plot
	plot(p1, p2, size=(1200,400))
end

# ╔═╡ 146e4de4-8a68-4d63-bc56-234ef0ebad3f
md"""
Maximum error ``L^\infty``: $(linf_error)

Integral error ``L^1``: $(l1_error)
"""

# ╔═╡ bf205421-a919-4090-acb1-9b2f2555b056
md"""
### Notebook style adjustments
Make notebook wider to show plots side by side:
"""

# ╔═╡ 282da2f2-c0da-4310-9bc4-54b5addc6b11
html"""
<style>
	main {
		margin: 0 auto;
		max-width: 2000px;
    	padding-left: max(160px, 10%);
    	padding-right: max(160px, 10%);
	}
</style>
"""

# ╔═╡ 00000000-0000-0000-0000-000000000001
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[compat]
LaTeXStrings = "~1.3.0"
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"""

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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"]
git-tree-sha1 = "0c03844e2231e12fda4d0086fd7cbe4098ee8dc5"
uuid = "ea2cea3b-5b76-57ae-a6ef-0a8af62496e1"
version = "5.15.3+2"

[[deps.REPL]]
deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"]
uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb"

[[deps.Random]]
deps = ["SHA", "Serialization"]
uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"

[[deps.RecipesBase]]
deps = ["SnoopPrecompile"]
git-tree-sha1 = "d12e612bba40d189cead6ff857ddb67bd2e6a387"
uuid = "3cdcf5f2-1ef4-517c-9805-6587b60abb01"
version = "1.3.1"

[[deps.RecipesPipeline]]
deps = ["Dates", "NaNMath", "PlotUtils", "RecipesBase", "SnoopPrecompile"]
git-tree-sha1 = "e974477be88cb5e3040009f3767611bc6357846f"
uuid = "01d81517-befc-4cb6-b9ec-a95719d0359c"
version = "0.6.11"

[[deps.Reexport]]
git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b"
uuid = "189a3867-3050-52da-a836-e630ba90ab69"
version = "1.2.2"

[[deps.RelocatableFolders]]
deps = ["SHA", "Scratch"]
git-tree-sha1 = "90bc7a7c96410424509e4263e277e43250c05691"
uuid = "05181044-ff0b-4ac5-8273-598c1e38db00"
version = "1.0.0"

[[deps.Requires]]
deps = ["UUIDs"]
git-tree-sha1 = "838a3a4188e2ded87a4f9f184b4b0d78a1e91cb7"
uuid = "ae029012-a4dd-5104-9daa-d747884805df"
version = "1.3.0"

[[deps.SHA]]
uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce"
version = "0.7.0"

[[deps.Scratch]]
deps = ["Dates"]
git-tree-sha1 = "f94f779c94e58bf9ea243e77a37e16d9de9126bd"
uuid = "6c6a2e73-6563-6170-7368-637461726353"
version = "1.1.1"

[[deps.Serialization]]
uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b"

[[deps.ShortCodes]]
deps = ["Base64", "CodecZlib", "HTTP", "JSON3", "Memoize", "UUIDs"]
git-tree-sha1 = "95479a28f0bb4ad37ec7c7ece7fdbfc400c643e0"
uuid = "f62ebe17-55c5-4640-972f-b59c0dd11ccf"
version = "0.3.5"

[[deps.Showoff]]
deps = ["Dates", "Grisu"]
git-tree-sha1 = "91eddf657aca81df9ae6ceb20b959ae5653ad1de"
uuid = "992d4aef-0814-514b-bc4d-f2e9a6c4116f"
version = "1.0.3"

[[deps.SimpleBufferStream]]
git-tree-sha1 = "874e8867b33a00e784c8a7e4b60afe9e037b74e1"
uuid = "777ac1f9-54b0-4bf8-805c-2214025038e7"
version = "1.1.0"

[[deps.SnoopPrecompile]]
git-tree-sha1 = "f604441450a3c0569830946e5b33b78c928e1a85"
uuid = "66db9d55-30c0-4569-8b51-7e840670fc0c"
version = "1.0.1"

[[deps.Sockets]]
uuid = "6462fe0b-24de-5631-8697-dd941f90decc"

[[deps.SortingAlgorithms]]
deps = ["DataStructures"]
git-tree-sha1 = "a4ada03f999bd01b3a25dcaa30b2d929fe537e00"
uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c"
version = "1.1.0"

[[deps.SparseArrays]]
deps = ["LinearAlgebra", "Random"]
uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"

[[deps.SpecialFunctions]]
deps = ["ChainRulesCore", "IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"]
git-tree-sha1 = "d75bda01f8c31ebb72df80a46c88b25d1c79c56d"
uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
version = "2.1.7"

[[deps.Statistics]]
deps = ["LinearAlgebra", "SparseArrays"]
uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"

[[deps.StatsAPI]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "f9af7f195fb13589dd2e2d57fdb401717d2eb1f6"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.5.0"

[[deps.StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "d1bf48bfcc554a3761a133fe3a9bb01488e06916"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.21"

[[deps.StructTypes]]
deps = ["Dates", "UUIDs"]
git-tree-sha1 = "ca4bccb03acf9faaf4137a9abc1881ed1841aa70"
uuid = "856f2bd8-1eba-4b0a-8007-ebc267875bd4"
version = "1.10.0"

[[deps.TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"
version = "1.0.0"

[[deps.Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"
version = "1.10.1"

[[deps.TensorCore]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "1feb45f88d133a655e001435632f019a9a1bcdb6"
uuid = "62fd8b95-f654-4bbd-a8a5-9c27f68ccd50"
version = "0.1.1"

[[deps.Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[deps.TranscodingStreams]]
deps = ["Random", "Test"]
git-tree-sha1 = "8a75929dcd3c38611db2f8d08546decb514fcadf"
uuid = "3bb67fe8-82b1-5028-8e26-92a6c54297fa"
version = "0.9.9"

[[deps.Tricks]]
git-tree-sha1 = "6bac775f2d42a611cdfcd1fb217ee719630c4175"
uuid = "410a4b4d-49e4-4fbc-ab6d-cb71b17b3775"
version = "0.1.6"

[[deps.URIs]]
git-tree-sha1 = "e59ecc5a41b000fa94423a578d29290c7266fc10"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.4.0"

[[deps.UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[deps.Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[deps.UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
version = "0.4.1"

[[deps.Unzip]]
git-tree-sha1 = "ca0969166a028236229f63514992fc073799bb78"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.2.0"

[[deps.Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.19.0+0"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4528479aa01ee1b3b4cd0e6faef0e04cf16466da"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.25.0+0"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "58443b63fb7e465a8a7210828c91c08b92132dff"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.14+0"

[[deps.XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[deps.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"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[deps.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"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.12+3"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.2+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "868e669ccb12ba16eaf50cb2957ee2ff61261c56"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.29.0+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.4.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.1.1+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.48.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+0"
"""

# ╔═╡ Cell order:
# ╟─4de22c75-0d00-4838-8ba4-52bab4dab89e
# ╟─ac216c60-8288-465b-ab2b-2ba3e7701d02
# ╟─146e4de4-8a68-4d63-bc56-234ef0ebad3f
# ╟─f87f000f-e0c9-46eb-a242-89a4290bd84f
# ╟─43ef7734-1c6b-41f8-b581-44c77252a595
# ╟─018a4c32-c7d4-4b47-9a87-91b972f2eb27
# ╟─6e4a15eb-8c68-4837-81dd-caac7c81ba7e
# ╟─6b49c667-b16e-44f3-9f0d-2b8a62f5efda
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