### A Pluto.jl notebook ###
# v0.19.40

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, LinearAlgebra

# ╔═╡ f87f000f-e0c9-46eb-a242-89a4290bd84f
md"""
### Numerical solution
In the discretized equation $(*)$ above, each temperature value $u_{i,j}$ depends on its neighboring values. Thus, we end up with a linear system,
```math
A u = b
```
which we need to solve to obtain the temperature values at each inner node. Here, $A$ is the coefficient matrix, $u$ contains our unknown temperature values $u_{i,j}$ that we want to solve for, and $b$ holds the right-hand side of the system, i.e., $-h^2 f_{i,j}$.

We will solve the system using one of Julia's built-in linear solvers from the `LinearAlgebra` module. For simplicity, we will use $f(x,y) = 2$ everywhere:
"""

# ╔═╡ a7e57f8e-2032-419f-b957-12e90c3b9773
md"""
In the plot below, you can see a surface plot of the numerical solution, where the height indicates the value of the temperature at a given point. Increasing the number of intervals $N$, and thus decreasing the spacing between the nodes $h$, will lead to longer computing times but also a more accurate solution.
"""

# ╔═╡ 25a2120b-e9cd-48bc-901e-5611a2330f99
md"""
Number of intervals ``N``: $(@bind N Slider(4:30, default=5, show_value=true)) | 
Camera angle: $(@bind angle Slider(1:90, default=45, show_value=false))
"""

# ╔═╡ 4de22c75-0d00-4838-8ba4-52bab4dab89e
md"""
# Finite difference method

In this notebook, we will approximate the solution to the steady state heat equation in two dimensions,
```math
\begin{equation}
\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = -\frac{1}{\kappa} f,
\end{equation}
```
using a **finite difference method**. Here, ``u(x, y)`` is the temperature, ``x`` and ``y`` are the spatial directions, ``\kappa`` is the thermal conductivity, and ``f(x, y)`` is a known heat source. For simplicity, we will assume $\kappa = 1$ in the following sections and drop it from the equations. The steady state heat equation is a prototypical example for a so-called *Poisson problem*.

Our computational domain is $\Omega = [0, 1]^2$, and we assume $u$ to be zero on all boundaries, i.e., $u(x,y) = 0$ if $x \in \{0,1\}$ or $y \in \{0, 1\}$. We discretize our domain with $N + 1$ nodes in each spatial direction on a rectangular grid. The nodes $(x_i, y_j)$ with $i, j = 0, 1, \ldots, N$ are equidistantly placed and their positions are computed as
```math
\begin{align}
x_i &= ih,\\
y_j &= jh,
\end{align}
```
where the $h = 1/N$ is the distance between the nodes in each axis direction.

To discretize the heat equation, we use second-order central finite differences to approximate the partial derivative in the $x$-direction,
```math
\frac{\partial^2 u(x, y)}{\partial x^2} \approx \frac{u(x-h, y) - 2 u(x, y) + u(x+h, y)}{h^2}
```
and likewise for the derivative in the $y$-direction. Inserted in the heat equation above, we obtain
```math
\begin{equation}
\frac{u_{i-1,j} - 2 u_{i,j} + u_{i+1,j}}{h^2} + \frac{u_{i,j-1} - 2 u_{i,j} + u_{i,j+1}}{h^2} = -f_{i,j},\quad i,j = 1, \ldots, N-1,
\end{equation}
```
where $u_{i,j} = u(x_i, y_j)$ is the value of the temperature at node $(i,j$), and $f_{i,j} = f(x_i, y_j)$. We can further simplify this equation to
```math
\begin{equation}
u_{i-1,j} + u_{i,j-1} - 4 u_{i,j} + u_{i+1,j} + u_{i,j+1} = -h^2 f_{i,j},\quad i,j = 1, \ldots, N-1, \quad (*)
\end{equation}
```
In the plot below, you can see the domain $\Omega$ and the node locations for the chosen $N$ (violet circles for inner nodes, red diamonds for boundary nodes). The green shape denotes the numerical stencil for a user-selected $(i, j)$ pair, i.e., it includes all neighbor nodes that are used to compute approximate the solution at the central node.

Index ``i``: $(@bind idx_i Slider(1:(N-1), default=3, show_value=true)) | Index ``j``: $(@bind idx_j Slider(1:(N-1), default=1, show_value=true))
"""

# ╔═╡ c5689d78-a0ae-432d-b7cd-a3de894d4443
begin
	h = 1 / N
	plot([0.0, 1.0, 1.0, 0.0, 0.0], [0.0, 0.0, 1.0, 1.0, 0.0],
		aspect_ratio=:equal, xlims=(-0.1, 1.1), ylims=(-0.1, 1.1), label=L"\Omega", color=:black, lw=3)
	plot!([0.0, 1.0], [1.0, 1.0], fillrange=0, fillcolor=:black, fillalpha=0.1, label="")
	scatter!([(x, y) for x in (0.0+h):h:(1.0-h) for y in (0.0+h):h:(1.0-h)],
		color=:violet, label="inner nodes")
	scatter!([(x, y) for x in 0.0:h:1.0 for y in 0:1],
		color=:red, label="boundary nodes", markershape=:diamond)
	scatter!([(x, y) for x in 0:1 for y in (0.0+h):h:(1.0-h)],
		color=:red, label="", markershape=:diamond)

	cx, cy = (idx_i*h, idx_j*h)
    plot!([
		(cx-0.2h, cy-1.2h),
		(cx+0.2h, cy-1.2h),
		(cx+0.2h, cy-0.2h),
		(cx+0.2h, cy-0.2h),
		(cx+1.2h, cy-0.2h),
		(cx+1.2h, cy+0.2h),
		(cx+0.2h, cy+0.2h),
		(cx+0.2h, cy+1.2h),
		(cx-0.2h, cy+1.2h),
		(cx-0.2h, cy+0.2h),
		(cx-1.2h, cy+0.2h),
		(cx-1.2h, cy-0.2h),
		(cx-0.2h, cy-0.2h),
		(cx-0.2h, cy-1.2h),
	],
	lw=2, label="stencil", color=:green)
end

# ╔═╡ 138b0481-7d42-4197-8ca2-b3fb073366a0
md"""
### Assembling the linear system
From equation $(*)$ above, we can assemble our coefficient matrix $A$ that represents our linear system of equations (see class notes):
"""

# ╔═╡ 8e2710f6-9e2c-47a9-8a1e-33c6df4386cf
begin
	m = (N - 1)^2 # Matrix size
	
	# Banded matrix with full bands
	A = diagm(    0  => fill(-4, m),
	              1  => fill( 1, m - 1),
		        N-1  => fill( 1, m - (N - 1)),
		         -1  => fill( 1, m - 1),
		      -(N-1) => fill( 1, m - (N - 1)),
	)
	
	# Reset values to obtain block structure
	for i in (N - 1):(N - 1):(m - 1)
		A[i, i+1] = 0
		A[i+1, i] = 0
	end
	
	nothing # hide output
end

# ╔═╡ f185aacd-0334-4217-8fa9-9fd0776c12c6
begin
	b = -h^2 * 2 * ones(m) # build right-hand side
	u = A \ b # solve linear system
	nothing # hide output
end

# ╔═╡ eb184f17-09c4-426e-80bb-f7d5867a01a4
begin
	surface(reshape(u, (N - 1, N - 1)), camera=(angle, 45), zlims=(0.0, 0.15)) # Plot
end

# ╔═╡ 52e5b8cc-d18a-4e20-bc66-54c010e2cc1f
md"""
The following plot shows the sparsity pattern of the matrix $A$ that represents the discretization with a finite difference method as a linear system.
"""

# ╔═╡ ac216c60-8288-465b-ab2b-2ba3e7701d02
begin
	markersize = if N == 4
	    15
	elseif N == 5
		11
	elseif N == 6
		7
	elseif N == 7
		4
	elseif N == 8
		3
	else
		2
	end
	markersizes_small = Dict(5=>11, 6=>7, 7=>4, 8=>3)
	
	spy(A + UniformScaling(5), markersize=markersize, markershape=:square)

	#Highlight the row corresponding to the selected DOF
	idx = idx_i + (idx_j - 1) * (N-1)
	row = findall(!iszero,Int.(abs.(A[idx, :]) .> 1e-3))
	scatter!(row, row.^0 * idx, markershape=:square, markersize=markersize, label = nothing) 
end

# ╔═╡ 4376970c-ea81-415e-bdb8-c35d36751536
md"""
Below, you can also see the numerical values of the matrix A:
"""

# ╔═╡ 625a431f-9187-4c17-9411-3e9078499c38
A

# ╔═╡ 95fdd1b4-3e76-4dbd-976b-2243aea1e8b5
md"""
Note that for simplicity we use a classical or *dense* matrix representation here, where each entry of the matrix is stored explicitly. In a real application, one would use a *sparse* matrix representation that only stores the non-zero elements explicitly.
"""

# ╔═╡ 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. 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
* `LinearAlgebra` is a Julia standard library module for linear algebra operations
"""

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[[deps.HarfBuzz_jll]]
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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 = "3bac05bc7e74a75fd9cba4295cde4045d9fe2386"
uuid = "6c6a2e73-6563-6170-7368-637461726353"
version = "1.2.1"

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

[[deps.ShortCodes]]
deps = ["Base64", "CodecZlib", "Downloads", "JSON3", "Memoize", "URIs", "UUIDs"]
git-tree-sha1 = "5844ee60d9fd30a891d48bab77ac9e16791a0a57"
uuid = "f62ebe17-55c5-4640-972f-b59c0dd11ccf"
version = "0.3.6"

[[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]]
deps = ["Preferences"]
git-tree-sha1 = "e760a70afdcd461cf01a575947738d359234665c"
uuid = "66db9d55-30c0-4569-8b51-7e840670fc0c"
version = "1.0.3"

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

[[deps.SortingAlgorithms]]
deps = ["DataStructures"]
git-tree-sha1 = "66e0a8e672a0bdfca2c3f5937efb8538b9ddc085"
uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c"
version = "1.2.1"

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

[[deps.Statistics]]
deps = ["LinearAlgebra", "SparseArrays"]
uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
version = "1.9.0"

[[deps.StatsAPI]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "1ff449ad350c9c4cbc756624d6f8a8c3ef56d3ed"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.7.0"

[[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.SuiteSparse_jll]]
deps = ["Artifacts", "Libdl", "Pkg", "libblastrampoline_jll"]
uuid = "bea87d4a-7f5b-5778-9afe-8cc45184846c"
version = "5.10.1+6"

[[deps.TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"
version = "1.0.3"

[[deps.Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"
version = "1.10.0"

[[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]]
git-tree-sha1 = "a09c933bebed12501890d8e92946bbab6a1690f1"
uuid = "3bb67fe8-82b1-5028-8e26-92a6c54297fa"
version = "0.10.5"
weakdeps = ["Random", "Test"]

    [deps.TranscodingStreams.extensions]
    TestExt = ["Test", "Random"]

[[deps.Tricks]]
git-tree-sha1 = "eae1bb484cd63b36999ee58be2de6c178105112f"
uuid = "410a4b4d-49e4-4fbc-ab6d-cb71b17b3775"
version = "0.1.8"

[[deps.URIs]]
git-tree-sha1 = "67db6cc7b3821e19ebe75791a9dd19c9b1188f2b"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.5.1"

[[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", "EpollShim_jll", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "7558e29847e99bc3f04d6569e82d0f5c54460703"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.21.0+1"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "93f43ab61b16ddfb2fd3bb13b3ce241cafb0e6c9"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.31.0+0"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Zlib_jll"]
git-tree-sha1 = "07e470dabc5a6a4254ffebc29a1b3fc01464e105"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.12.5+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", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "afead5aba5aa507ad5a3bf01f58f82c8d1403495"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.8.6+0"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "6035850dcc70518ca32f012e46015b9beeda49d8"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.11+0"

[[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"]
git-tree-sha1 = "34d526d318358a859d7de23da945578e8e8727b7"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.4+0"

[[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"]
git-tree-sha1 = "8fdda4c692503d44d04a0603d9ac0982054635f9"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.1+0"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "b4bfde5d5b652e22b9c790ad00af08b6d042b97d"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.15.0+0"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "730eeca102434283c50ccf7d1ecdadf521a765a4"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.2+0"

[[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", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "330f955bc41bb8f5270a369c473fc4a5a4e4d3cb"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.6+0"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "691634e5453ad362044e2ad653e79f3ee3bb98c3"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.39.0+0"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "e92a1a012a10506618f10b7047e478403a046c77"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.5.0+0"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+0"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "49ce682769cd5de6c72dcf1b94ed7790cd08974c"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.5+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "a68c9655fbe6dfcab3d972808f1aafec151ce3f8"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.43.0+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"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.8.0+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", "Zlib_jll"]
git-tree-sha1 = "d7015d2e18a5fd9a4f47de711837e980519781a4"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.43+1"

[[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 = "1.52.0+1"

[[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 = "9c304562909ab2bab0262639bd4f444d7bc2be37"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+1"
"""

# ╔═╡ Cell order:
# ╟─4de22c75-0d00-4838-8ba4-52bab4dab89e
# ╟─c5689d78-a0ae-432d-b7cd-a3de894d4443
# ╟─f87f000f-e0c9-46eb-a242-89a4290bd84f
# ╠═f185aacd-0334-4217-8fa9-9fd0776c12c6
# ╟─a7e57f8e-2032-419f-b957-12e90c3b9773
# ╟─eb184f17-09c4-426e-80bb-f7d5867a01a4
# ╟─25a2120b-e9cd-48bc-901e-5611a2330f99
# ╟─138b0481-7d42-4197-8ca2-b3fb073366a0
# ╠═8e2710f6-9e2c-47a9-8a1e-33c6df4386cf
# ╟─52e5b8cc-d18a-4e20-bc66-54c010e2cc1f
# ╟─ac216c60-8288-465b-ab2b-2ba3e7701d02
# ╟─4376970c-ea81-415e-bdb8-c35d36751536
# ╟─625a431f-9187-4c17-9411-3e9078499c38
# ╟─95fdd1b4-3e76-4dbd-976b-2243aea1e8b5
# ╟─6b49c667-b16e-44f3-9f0d-2b8a62f5efda
# ╟─b8f81a3f-d773-4e67-b8d7-e1734b04abaa
# ╠═a22d543c-6a50-11ed-002d-7f2fe5bd7276
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002