### A Pluto.jl notebook ###
# v0.19.14

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

# ╔═╡ d3c9d60e-19d1-11eb-064d-61814bb84582
using Plots, PlutoUI, BenchmarkTools, FFTW, Test; gr(size = (640, 360)); PlutoUI.TableOfContents(depth=4)

# ╔═╡ 147ff08c-19cf-11eb-341d-ed1c6568c0c7
md"""
# Numerical Methods of Medical Imaging - Exercise 2

[Institute for Biomedical Imaging](https://www.tuhh.de/ibi/home.html), Hamburg University of Technology

* 👨‍🏫 Prof. Dr. Tobias Knopp
* 👨‍🏫 Dr. Martin Möddel
"""

# ╔═╡ 58b50b7a-19cf-11eb-146e-a5405521ff5d
md"""
# Discrete and fast Fourier transform

* 📅 Due date: 22.11.2022, 12 a.m. 
* You can earn 1 point for each task (6 points in total).

![](https://imgs.xkcd.com/comics/fourier.jpg)

"Fourier" taken from [xkcd](https://xkcd.com/26/) is licensed under [CC](https://en.wikipedia.org/wiki/en:Creative_Commons) [BY-NC 2.5](https://creativecommons.org/licenses/by-nc/2.5/) license.
"""

# ╔═╡ 4bec237c-19d1-11eb-27bd-13b1f2086f69
md"""
## Fourier Expansion

It is possible to expand piecewise continuous functions $f:\mathbb{R}\rightarrow\mathbb{R}$ into a so-called Fourier series

$f(x) = \sum_{k=-\infty}^{\infty} \hat{f}_k e^{-2\pi i k x}.$

For 1-perioic functions, the Fourier coefficients $\hat f_k$ are given by the scalar product of $f$ with elements from the Fourier basis $\{e^{-2\pi i k x}\}_k$, i.e.

$\hat f_k = \int_{0}^{1} f(x)e^{-2\pi i k x}dx$

### 1. The Discrete Fourier Transform

Whenever these coefficients can not be obtained analytically, discrete Fourier transformation provides an approximation up to a constant, which depends on the number of sampling points $N$

$\hat f_k \propto \sum_{n=0}^{N-1} f_n e^{-\frac{2 \pi i k n}{N}},$

with $f_n = f(\frac n N)$ and $n=0,1,\dots,N-1$.
"""

# ╔═╡ 800f28ce-19f3-11eb-0765-fd1c43b4bc19
md"""
Let

$r(x) = 
	  \begin{cases} 
	    1 &\text{if }  \vert x \vert < \frac 1 4\\
            \frac 1 2 & \text{if }  \vert x \vert = \frac 1 4\\
	    0 & \text{else}
          \end{cases}$

be the 1-periodic rectangular function $r:\mathbb R \rightarrow \left[ 0,1 \right]$ given on the domain $x \in [-\frac 1 2,\frac 1 2 )$.
"""

# ╔═╡ 91726846-19d1-11eb-36e3-b1dc8f478323
function r(x)
    y = rem(abs(x)+0.25,1.0)
	if y==0.5 || y==0.0
        return 0.5
	elseif y<0.5
        return 1.0
    else
		return 0.0
    end
end

# ╔═╡ e8936f28-19d1-11eb-32aa-efda10cd8210
plot(range(-2,stop=2,length=500),r,
	xlabel = "x",
	ylabel = "y",
	title = "rectangular function", 
	label = "r",
	lw = 2
)

# ╔═╡ 4e339c26-19f6-11eb-2296-e1e0f2fda2fe
md"""
#### 🎓 Task
Implement a function `sample(f::Function, N::Integer=128)` to discretize $f$ into a vector $f_n = f(\frac n N)$ and $n=0,1,\dots,N-1$.

Implement a function `dft(fns::Vector)` returning the discrete Fourier coefficients $\hat{f}_k$, $k=0,1,\dots,N-1$ as a vector.
"""

# ╔═╡ c3d9ef84-19fb-11eb-24ad-6f6671b3dec0
@bind N Slider(1:128, default=32)

# ╔═╡ d14ff6d4-19f3-11eb-17e6-773096878044
begin
	function sample(f::Function, N::Integer=128)
		# this does not seem to be correct
		return ones(Float64,N)
	end
	
	function dft(fns::Vector)
		# this does not seem to be correct
	    return ones(ComplexF64,length(fns))
	end
end

# ╔═╡ 018e0d2a-1b63-11eb-0841-6d88fc93a49f
fns = sample(r,N)

# ╔═╡ 6215a130-1b63-11eb-3b4b-99fd03bf750a
scatter((0:N-1)/N,abs.(fns),
	xlabel = "k",
	ylabel = "abs.(fₖ)",
	title = "rectangular function", 
	label = "sampled function r (N = $N)",
	lw = 2)

# ╔═╡ 730e092a-19f4-11eb-0ac8-e946f31b3363
fks = dft(fns)

# ╔═╡ dbacee5a-19f5-11eb-1c1b-0d1ae9b92f0e
scatter(0:N-1,abs.(fks),
	xlabel = "k",
	ylabel = "abs.(fₖ)",
	title = "rectangular function", 
	label = "discrete Fourier coefficients (N = $N)",
	lw = 2,
	yscale = :log10)

# ╔═╡ f918a2f0-1a85-11eb-2807-51f70d8984d7
md"""
### 2. The Inverse Discrete Fourier Transform

The inverse discrete Fourier Transform for $n=0,\dots,N-1$ is given by

$f_n = \frac{1}{N} \sum_{k=0}^{N-1} \hat f_k e^{\frac{2\pi i k n}{N}}.$

#### 🎓 Task
Implement the inverse of the discrete Fourier transform `idft(fks::Vector)` and test if `idft(dft(fns)) ≈ fns`.
"""

# ╔═╡ db0c98aa-1a8c-11eb-01b1-5dbfff26d33b
function idft(fks::AbstractVector)
	# this does not seem to be correct
    return ones(ComplexF64,length(fks))
end

# ╔═╡ fb8ab86e-1a8c-11eb-0779-d9215f2c3f14
fnsi = idft(fks)

# ╔═╡ 975c3e5c-1b64-11eb-3136-4b9f0c1922d5
@test fns ≈ fnsi

# ╔═╡ 3103d282-1a8d-11eb-2b36-e508a89b3171
scatter((0:N-1)/N,abs.(fnsi),
	xlabel = "k",
	ylabel = "abs.(fₖ)",
	title = "rectangular function", 
	label = "idft(dft(fns)) (N = $N)",
	lw = 2)

# ╔═╡ 32a544fe-19fc-11eb-2309-8509537cdb4d
md"""
### 3. The Complex Trigonometric Polynomial of Degree N

The complex trigonometric polynomial of degree $N$ given by

$p_N(x) = \begin{cases}
  \frac{1}{N} \left[ \hat{f}_0 + \hat{f}_1 e^{i 2\pi t} + \cdots + \hat{f}_{N/2-1} e^{i(N/2-1) 2\pi t} + \hat{f}_{N/2} \cos(N\pi t) + \hat{f}_{N/2+1} e^{i(-N/2+1) 2\pi t} + \cdots + \hat{f}_{N-1} e^{-i 2\pi t} \right],
  & N\text{ even} \\
  \frac{1}{N} \left[ \hat{f}_0 + \hat{f}_1 e^{i 2\pi t} + \cdots + \hat{f}_{\lfloor N/2 \rfloor} e^{i \lfloor N/2 \rfloor 2\pi t} + \hat{f}_{\lfloor N/2 \rfloor+1} e^{-i \lfloor N/2 \rfloor 2\pi t} + \cdots + \hat{f}_{N-1} e^{-i 2\pi t} \right],
  & N\text{ odd}
\end{cases}$

satisfies the interpolation property $p(n/N) = f_n$ for $n=0,\ldots,N-1$ and provides an approximation to the function from which the original DFT coefficients stem from.

#### 🎓 Task
Implement the `complexTrigonometricPolynomial(fks::AbstractVector,x)` and see if it approximates the original function using the Fourier coefficients of the rectangle function.
"""

# ╔═╡ ca622d7a-1a8d-11eb-3960-7b2cef5b540d
function complexTrigonometricPolynomial(fks::AbstractVector,x)
	# this does not seem to be correct
	return one(ComplexF64)
end

# ╔═╡ e55a43a2-1a8f-11eb-32cc-e73ad807a202
pr(x) = real(complexTrigonometricPolynomial(fks,x))

# ╔═╡ 1d074a4a-1a90-11eb-2b14-ad6654f2a9af
begin
	plot(range(-2,stop=2,length=500),r,
		xlabel = "x",
		ylabel = "y",
		title = "rectangular function",
		label = "r",
		lw = 2
	)
	plot!(range(-2,stop=2,length=500),pr,
		xlabel = "x",
		ylabel = "y",
		label = "p_N (N = $N)",
		lw = 2
	)
end

# ╔═╡ 867639d4-1ac3-11eb-0236-d50896ecbd11
md"""
### 4. Fourier Filter

A Fourier filter is a type of signal filter which is based on the manipulation of specific frequency components of a signal, where all Fourier coefficients $\hat f_k$ are multiplied by a frequency specific factor $w_k$ for all $k=0,1,\dots,N-1$

$\hat f_k \mapsto w_k \hat f_k.$

Often so-called window functions are used to define the coefficients. One well-known window function is the the Hann window

$w_k={\frac {1}{2}}\left[1+\cos \left({\frac {2\pi k}{N-1}}\right)\right].$

#### 🎓 Task
The overshot of the complex trigonometric polynomial at the edges of the recangle function is known as Gibbs phenomena. Check if filtering can help to reduce this phenomena.

Implement the `hannFilter(N)` and filter the Fourier coefficients of the rectangle function. Use these and the original coefficients to compare their respective complex trigonometric polynomials. 
"""

# ╔═╡ e37cdb34-1ac4-11eb-323e-f1d316774b5d
function hannFilter(N)
	# this does not seem to be correct
	return ones(ComplexF64,N)
end

# ╔═╡ ea15a9c0-1ac5-11eb-31df-d16368be8d92
fksFiltered = fks.*hannFilter(N)

# ╔═╡ 3c74354a-1b7c-11eb-0936-671d831ae74d
begin
	scatter(0:N-1,map(x->iszero(x) ? missing : x, abs.(fks)),
		xlabel = "k",
		ylabel = "abs.(fₖ)",
		title = "rectangular function", 
		label = "discrete Fourier coefficients (N = $N)",
		lw = 2,
		yscale = :log10)
	scatter!(0:N-1,map(x->iszero(x) ? missing : x, abs.(fksFiltered)),
		xlabel = "k",
		ylabel = "abs.(fₖ)",
		label = "filtered discrete Fourier coefficients",
		lw = 2,
		yscale = :log10)
end

# ╔═╡ 00137b6c-1ac6-11eb-1874-c54f9797b035
pFiltered(x) = real(complexTrigonometricPolynomial(fksFiltered,x))

# ╔═╡ 1e728862-1ac6-11eb-204c-8feaf3874724
begin
	plot(range(-2,stop=2,length=500),r,
		xlabel = "x",
		ylabel = "y",
		title = "rectangular function",
		label = "r",
		lw = 2
	)
	plot!(range(-2,stop=2,length=500),pr,
		xlabel = "x",
		ylabel = "y", 
		label = "p_N (N = $N)",
		lw = 2
	)
	plot!(range(-2,stop=2,length=500),pFiltered,
		xlabel = "x",
		ylabel = "y", 
		label = "p_N filtered",
		lw = 2
	)
end

# ╔═╡ 0e635470-1ab5-11eb-3ead-c96330845afb
md"""
### 5. Fast Fourier Transformation
For a fast and efficient calculation of the Fourier coefficients the function `fft` of the `FFTW` package can be used.

The following Benchmark shows the speedup.
"""

# ╔═╡ dc756532-1ab4-11eb-116d-0502731ad456
@test fft(fns) ≈ dft(fns)

# ╔═╡ ed60fce0-1ab3-11eb-09e4-7dfdcd494259
b_dft = @benchmark dft($fns)

# ╔═╡ 3e9ff912-1ab4-11eb-0977-1d5a23618723
b_fft = @benchmark fft($fns)

# ╔═╡ f3da4924-1b7a-11eb-277d-dbf45b22f5af
md"""
#### 🎓 Task

Write a function `dftfaster(fns::Vector)` which speeds up your original implementation if this is possible. Try to avoid computations of `exp(im*ϕ)` and unneccessary allocations of memory. Further tips on performance can be found in the [julia documentation](https://docs.julialang.org/en/v1/manual/performance-tips/).
"""

# ╔═╡ 6c059a14-1ab5-11eb-2ca2-7d8bf4299714
function dftfaster(fns::Vector)
	# this does not seem to be correct
    return ones(ComplexF64,length(fks))
end

# ╔═╡ e48966e0-1ab8-11eb-19ed-c7e8752b9838
@test dftfaster(fns) ≈ fks

# ╔═╡ ef01a9a4-1ab9-11eb-1796-cfa7726948d0
b_dft2 = @benchmark dftfaster($fns)

# ╔═╡ fcff99e0-1abf-11eb-2dbf-850b32db7108
md"""
### 6. Spectral Leakage
Discrete Fourier transformation of a periodic signal assumes that a whole period of the signal is taken. But what happens if this assumption is violated?

#### 🎓 Task
Take the functions $\sin (2\pi x)$ and $\sin (2.1\pi x)$ and calculate their Fourier coefficients assuming that both functions are periodic with period length $1$. Plot and compare both spectra and the respective complex trigonometric polynomial of degree $N$.
"""

# ╔═╡ 1930182c-1ac0-11eb-1bda-3ff057b25970
fkssin1 = dft(sample(x->sin(2*π*x),N))

# ╔═╡ 8cc698e2-1ac0-11eb-0844-837ad17db528
fkssin2 = dft(sample(x->sin(2.1*π*x),N))

# ╔═╡ 9d1231ca-1ac0-11eb-04c5-4571e9851823
begin
	scatter(0:N-1,abs.(fkssin2),
		xlabel = "k",
		ylabel = "abs.(fₖ)",
		title = "Fourier coefficients", 
		label = "DFT(sin(2.1πx))",
		lw = 2,
		yscale = :log10)
	
	scatter!(0:N-1,abs.(fkssin1),
		xlabel = "k",
		ylabel = "abs.(fₖ)",
		title = "Fourier coefficients", 
		label = "DFT(sin(2πx))",
		lw = 2,
		yscale = :log10)
end

# ╔═╡ 0ad8cbd0-1ac1-11eb-14c3-7b20fd63c023
psin1(x) = real(complexTrigonometricPolynomial(fkssin1,x))

# ╔═╡ 31fb3e26-1ac1-11eb-3529-21c456cba268
psin2(x) = real(complexTrigonometricPolynomial(fkssin2,x))

# ╔═╡ 44fced44-1ac1-11eb-163e-c9fd874bffd9
begin
	plot(range(-2,stop=2,length=500),psin1,
		xlabel = "x",
		ylabel = "y",
		title = "complex trigonometric polynomial", 
		label = "p_sin(2πx)",
		lw = 2
	)
	
	plot!(range(-2,stop=2,length=500),psin2,
		xlabel = "x",
		ylabel = "y",
		title = "complex trigonometric polynomial", 
		label = "p_sin(2.1πx)",
		lw = 2
	)
end

# ╔═╡ d03112a7-0073-4290-90c3-b2e1f4089d9f
begin
	hint(text) = Markdown.MD(Markdown.Admonition("hint", "Hint", [text]))
	not_defined(variable_name) = Markdown.MD(Markdown.Admonition("danger", "Oh, oh! 😱", [md"Variable **$(Markdown.Code(string(variable_name)))** is not defined. You should probably do something about this."]))
	failed_test(text=md"Some tests failed. Keep working on it!") = Markdown.MD(Markdown.Admonition("danger", "Oh, oh! 😱", [text]))
	still_missing(text=md"Replace `missing` with your solution.") = Markdown.MD(Markdown.Admonition("warning", "Let's go!", [text]))
	keep_working(text=md"The answer is not quite right.") = Markdown.MD(Markdown.Admonition("danger", "Keep working on it!", [text]));
	yays = [md"Great! 🥳", md"Correct! 👏", md"Tada! 🎉"]
	correct(text=md"$(rand(yays)) Let's move on to the next task.") = Markdown.MD(Markdown.Admonition("correct", "Got it!", [text]))
end;

# ╔═╡ 22c63176-05d1-4430-a92d-308a02b5e502
# invisible test
let 
	fn32 = sample(r,32)
	fks32 = dft(fn32)
	if fn32[rand(union(1:8,26:32))]!=1.0 || fn32[9]!=0.5 || fn32[rand(10:24)]!=0.0
		keep_working(md"Your version of `sample` is not quite right.")
	elseif !isapprox(fks32[1], 16.0+0.0im, atol=1e-6) || !isapprox(fks32[7], -6.05072e-15+7.77156e-16im, atol=1e-6) || !isapprox(fks32[16], -0.0984914-1.10467e-14im, atol=1e-6) || !isapprox(fks32[23], 9.60343e-15-9.65894e-15im, atol=1e-6)
		keep_working(md"Your version of `dft` is not quite right.")
	else
		correct()
	end
end

# ╔═╡ e7416b55-7513-4601-8fea-85688663a0ac
# invisible test
let
	x = ones(ComplexF64,32)/32
	y = zeros(ComplexF64,32); y[1]=1
	if !isapprox(idft(y), x, atol=1e-6)
		keep_working(md"Your version of `idft` is not quite right.")
	else
		correct()
	end
end

# ╔═╡ 01f1afc4-a70f-40f2-a55e-eb64f6946539
# invisible test
let
	fn32 = sample(r,32)
	fks32 = dft(fn32)
	pr32(x) = real(complexTrigonometricPolynomial(fks32,x))
	if !isapprox(pr32(0.0),1.0,atol=1e-6) || !isapprox(pr32(0.25),0.5,atol=1e-6) || !isapprox(pr32(0.3),-0.0253537,atol=1e-6) || !isapprox(pr32(0.77),0.886791,atol=1e-6)
		keep_working(md"Your version of `complexTrigonometricPolynomial` is not quite right.")
	else
		correct()
	end
end

# ╔═╡ 0abdaefe-2c16-4315-a8c9-37287ce965b5
# invisible test
let
	hf32 = hannFilter(32)
	if !isapprox(hf32[1],1.0,atol=1e-6) || !isapprox(hf32[7],0.673653,atol=1e-6) || !isapprox(hf32[16],0.00256534,atol=1e-6) || !isapprox(hf32[23],0.374674,atol=1e-6)
		keep_working(md"Your version of `hannFilter` is not quite right.")
	else
		correct()
	end
end

# ╔═╡ b2e5f6d2-3095-4a2d-bdb2-6cbd53ade757
# invisible test
let
	if !isapprox(dftfaster(fns), fks, atol=1e-6)
		keep_working(md"`dftfaster` and `dft` do not yield the same results.")
	elseif median(b_dft2).time >= median(b_dft).time 
		keep_working(md"`dftfaster` does not appear to be faster then `dft`.")
	else
		correct()
	end
end

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
BenchmarkTools = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf"
FFTW = "7a1cc6ca-52ef-59f5-83cd-3a7055c09341"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[compat]
BenchmarkTools = "~1.3.1"
FFTW = "~1.5.0"
Plots = "~1.35.2"
PlutoUI = "~0.7.43"
"""

# ╔═╡ 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 = "69f7020bd72f069c219b5e8c236c1fa90d2cb409"
uuid = "621f4979-c628-5d54-868e-fcf4e3e8185c"
version = "1.2.1"

[[AbstractPlutoDingetjes]]
deps = ["Pkg"]
git-tree-sha1 = "8eaf9f1b4921132a4cff3f36a1d9ba923b14a481"
uuid = "6e696c72-6542-2067-7265-42206c756150"
version = "1.1.4"

[[ArgTools]]
uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f"
version = "1.1.1"

[[Artifacts]]
uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33"

[[Base64]]
uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f"

[[BenchmarkTools]]
deps = ["JSON", "Logging", "Printf", "Profile", "Statistics", "UUIDs"]
git-tree-sha1 = "4c10eee4af024676200bc7752e536f858c6b8f93"
uuid = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf"
version = "1.3.1"

[[BitFlags]]
git-tree-sha1 = "84259bb6172806304b9101094a7cc4bc6f56dbc6"
uuid = "d1d4a3ce-64b1-5f1a-9ba4-7e7e69966f35"
version = "0.1.5"

[[Bzip2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "19a35467a82e236ff51bc17a3a44b69ef35185a2"
uuid = "6e34b625-4abd-537c-b88f-471c36dfa7a0"
version = "1.0.8+0"

[[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 = "4b859a208b2397a7a623a03449e4636bdb17bcf2"
uuid = "83423d85-b0ee-5818-9007-b63ccbeb887a"
version = "1.16.1+1"

[[ChainRulesCore]]
deps = ["Compat", "LinearAlgebra", "SparseArrays"]
git-tree-sha1 = "e7ff6cadf743c098e08fca25c91103ee4303c9bb"
uuid = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
version = "1.15.6"

[[ChangesOfVariables]]
deps = ["ChainRulesCore", "LinearAlgebra", "Test"]
git-tree-sha1 = "38f7a08f19d8810338d4f5085211c7dfa5d5bdd8"
uuid = "9e997f8a-9a97-42d5-a9f1-ce6bfc15e2c0"
version = "0.1.4"

[[CodecZlib]]
deps = ["TranscodingStreams", "Zlib_jll"]
git-tree-sha1 = "ded953804d019afa9a3f98981d99b33e3db7b6da"
uuid = "944b1d66-785c-5afd-91f1-9de20f533193"
version = "0.7.0"

[[ColorSchemes]]
deps = ["ColorTypes", "ColorVectorSpace", "Colors", "FixedPointNumbers", "Random"]
git-tree-sha1 = "1fd869cc3875b57347f7027521f561cf46d1fcd8"
uuid = "35d6a980-a343-548e-a6ea-1d62b119f2f4"
version = "3.19.0"

[[ColorTypes]]
deps = ["FixedPointNumbers", "Random"]
git-tree-sha1 = "eb7f0f8307f71fac7c606984ea5fb2817275d6e4"
uuid = "3da002f7-5984-5a60-b8a6-cbb66c0b333f"
version = "0.11.4"

[[ColorVectorSpace]]
deps = ["ColorTypes", "FixedPointNumbers", "LinearAlgebra", "SpecialFunctions", "Statistics", "TensorCore"]
git-tree-sha1 = "d08c20eef1f2cbc6e60fd3612ac4340b89fea322"
uuid = "c3611d14-8923-5661-9e6a-0046d554d3a4"
version = "0.9.9"

[[Colors]]
deps = ["ColorTypes", "FixedPointNumbers", "Reexport"]
git-tree-sha1 = "417b0ed7b8b838aa6ca0a87aadf1bb9eb111ce40"
uuid = "5ae59095-9a9b-59fe-a467-6f913c188581"
version = "0.12.8"

[[Compat]]
deps = ["Dates", "LinearAlgebra", "UUIDs"]
git-tree-sha1 = "3ca828fe1b75fa84b021a7860bd039eaea84d2f2"
uuid = "34da2185-b29b-5c13-b0c7-acf172513d20"
version = "4.3.0"

[[CompilerSupportLibraries_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "e66e0078-7015-5450-92f7-15fbd957f2ae"
version = "0.5.2+0"

[[Contour]]
git-tree-sha1 = "d05d9e7b7aedff4e5b51a029dced05cfb6125781"
uuid = "d38c429a-6771-53c6-b99e-75d170b6e991"
version = "0.6.2"

[[DataAPI]]
git-tree-sha1 = "46d2680e618f8abd007bce0c3026cb0c4a8f2032"
uuid = "9a962f9c-6df0-11e9-0e5d-c546b8b5ee8a"
version = "1.12.0"

[[DataStructures]]
deps = ["Compat", "InteractiveUtils", "OrderedCollections"]
git-tree-sha1 = "d1fff3a548102f48987a52a2e0d114fa97d730f0"
uuid = "864edb3b-99cc-5e75-8d2d-829cb0a9cfe8"
version = "0.18.13"

[[Dates]]
deps = ["Printf"]
uuid = "ade2ca70-3891-5945-98fb-dc099432e06a"

[[DelimitedFiles]]
deps = ["Mmap"]
uuid = "8bb1440f-4735-579b-a4ab-409b98df4dab"

[[DocStringExtensions]]
deps = ["LibGit2"]
git-tree-sha1 = "5158c2b41018c5f7eb1470d558127ac274eca0c9"
uuid = "ffbed154-4ef7-542d-bbb7-c09d3a79fcae"
version = "0.9.1"

[[Downloads]]
deps = ["ArgTools", "FileWatching", "LibCURL", "NetworkOptions"]
uuid = "f43a241f-c20a-4ad4-852c-f6b1247861c6"
version = "1.6.0"

[[Expat_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "bad72f730e9e91c08d9427d5e8db95478a3c323d"
uuid = "2e619515-83b5-522b-bb60-26c02a35a201"
version = "2.4.8+0"

[[FFMPEG]]
deps = ["FFMPEG_jll"]
git-tree-sha1 = "b57e3acbe22f8484b4b5ff66a7499717fe1a9cc8"
uuid = "c87230d0-a227-11e9-1b43-d7ebe4e7570a"
version = "0.4.1"

[[FFMPEG_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "JLLWrappers", "LAME_jll", "Libdl", "Ogg_jll", "OpenSSL_jll", "Opus_jll", "PCRE2_jll", "Pkg", "Zlib_jll", "libaom_jll", "libass_jll", "libfdk_aac_jll", "libvorbis_jll", "x264_jll", "x265_jll"]
git-tree-sha1 = "74faea50c1d007c85837327f6775bea60b5492dd"
uuid = "b22a6f82-2f65-5046-a5b2-351ab43fb4e5"
version = "4.4.2+2"

[[FFTW]]
deps = ["AbstractFFTs", "FFTW_jll", "LinearAlgebra", "MKL_jll", "Preferences", "Reexport"]
git-tree-sha1 = "90630efff0894f8142308e334473eba54c433549"
uuid = "7a1cc6ca-52ef-59f5-83cd-3a7055c09341"
version = "1.5.0"

[[FFTW_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "c6033cc3892d0ef5bb9cd29b7f2f0331ea5184ea"
uuid = "f5851436-0d7a-5f13-b9de-f02708fd171a"
version = "3.3.10+0"

[[FileWatching]]
uuid = "7b1f6079-737a-58dc-b8bc-7a2ca5c1b5ee"

[[FixedPointNumbers]]
deps = ["Statistics"]
git-tree-sha1 = "335bfdceacc84c5cdf16aadc768aa5ddfc5383cc"
uuid = "53c48c17-4a7d-5ca2-90c5-79b7896eea93"
version = "0.8.4"

[[Fontconfig_jll]]
deps = ["Artifacts", "Bzip2_jll", "Expat_jll", "FreeType2_jll", "JLLWrappers", "Libdl", "Libuuid_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "21efd19106a55620a188615da6d3d06cd7f6ee03"
uuid = "a3f928ae-7b40-5064-980b-68af3947d34b"
version = "2.13.93+0"

[[Formatting]]
deps = ["Printf"]
git-tree-sha1 = "8339d61043228fdd3eb658d86c926cb282ae72a8"
uuid = "59287772-0a20-5a39-b81b-1366585eb4c0"
version = "0.4.2"

[[FreeType2_jll]]
deps = ["Artifacts", "Bzip2_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "87eb71354d8ec1a96d4a7636bd57a7347dde3ef9"
uuid = "d7e528f0-a631-5988-bf34-fe36492bcfd7"
version = "2.10.4+0"

[[FriBidi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "aa31987c2ba8704e23c6c8ba8a4f769d5d7e4f91"
uuid = "559328eb-81f9-559d-9380-de523a88c83c"
version = "1.0.10+0"

[[GLFW_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pkg", "Xorg_libXcursor_jll", "Xorg_libXi_jll", "Xorg_libXinerama_jll", "Xorg_libXrandr_jll"]
git-tree-sha1 = "d972031d28c8c8d9d7b41a536ad7bb0c2579caca"
uuid = "0656b61e-2033-5cc2-a64a-77c0f6c09b89"
version = "3.3.8+0"

[[GR]]
deps = ["Base64", "DelimitedFiles", "GR_jll", "HTTP", "JSON", "Libdl", "LinearAlgebra", "Pkg", "Preferences", "Printf", "Random", "Serialization", "Sockets", "Test", "UUIDs"]
git-tree-sha1 = "cf7bf90e483228f6c988e474b420064e5351b892"
uuid = "28b8d3ca-fb5f-59d9-8090-bfdbd6d07a71"
version = "0.69.4"

[[GR_jll]]
deps = ["Artifacts", "Bzip2_jll", "Cairo_jll", "FFMPEG_jll", "Fontconfig_jll", "GLFW_jll", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Libtiff_jll", "Pixman_jll", "Pkg", "Qt5Base_jll", "Zlib_jll", "libpng_jll"]
git-tree-sha1 = "bc9f7725571ddb4ab2c4bc74fa397c1c5ad08943"
uuid = "d2c73de3-f751-5644-a686-071e5b155ba9"
version = "0.69.1+0"

[[Gettext_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "9b02998aba7bf074d14de89f9d37ca24a1a0b046"
uuid = "78b55507-aeef-58d4-861c-77aaff3498b1"
version = "0.21.0+0"

[[Glib_jll]]
deps = ["Artifacts", "Gettext_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Libiconv_jll", "Libmount_jll", "PCRE2_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "fb83fbe02fe57f2c068013aa94bcdf6760d3a7a7"
uuid = "7746bdde-850d-59dc-9ae8-88ece973131d"
version = "2.74.0+1"

[[Graphite2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "344bf40dcab1073aca04aa0df4fb092f920e4011"
uuid = "3b182d85-2403-5c21-9c21-1e1f0cc25472"
version = "1.3.14+0"

[[Grisu]]
git-tree-sha1 = "53bb909d1151e57e2484c3d1b53e19552b887fb2"
uuid = "42e2da0e-8278-4e71-bc24-59509adca0fe"
version = "1.0.2"

[[HTTP]]
deps = ["Base64", "CodecZlib", "Dates", "IniFile", "Logging", "LoggingExtras", "MbedTLS", "NetworkOptions", "OpenSSL", "Random", "SimpleBufferStream", "Sockets", "URIs", "UUIDs"]
git-tree-sha1 = "4abede886fcba15cd5fd041fef776b230d004cee"
uuid = "cd3eb016-35fb-5094-929b-558a96fad6f3"
version = "1.4.0"

[[HarfBuzz_jll]]
deps = ["Artifacts", "Cairo_jll", "Fontconfig_jll", "FreeType2_jll", "Glib_jll", "Graphite2_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg"]
git-tree-sha1 = "129acf094d168394e80ee1dc4bc06ec835e510a3"
uuid = "2e76f6c2-a576-52d4-95c1-20adfe4de566"
version = "2.8.1+1"

[[Hyperscript]]
deps = ["Test"]
git-tree-sha1 = "8d511d5b81240fc8e6802386302675bdf47737b9"
uuid = "47d2ed2b-36de-50cf-bf87-49c2cf4b8b91"
version = "0.0.4"

[[HypertextLiteral]]
deps = ["Tricks"]
git-tree-sha1 = "c47c5fa4c5308f27ccaac35504858d8914e102f9"
uuid = "ac1192a8-f4b3-4bfe-ba22-af5b92cd3ab2"
version = "0.9.4"

[[IOCapture]]
deps = ["Logging", "Random"]
git-tree-sha1 = "f7be53659ab06ddc986428d3a9dcc95f6fa6705a"
uuid = "b5f81e59-6552-4d32-b1f0-c071b021bf89"
version = "0.2.2"

[[IniFile]]
git-tree-sha1 = "f550e6e32074c939295eb5ea6de31849ac2c9625"
uuid = "83e8ac13-25f8-5344-8a64-a9f2b223428f"
version = "0.5.1"

[[IntelOpenMP_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "d979e54b71da82f3a65b62553da4fc3d18c9004c"
uuid = "1d5cc7b8-4909-519e-a0f8-d0f5ad9712d0"
version = "2018.0.3+2"

[[InteractiveUtils]]
deps = ["Markdown"]
uuid = "b77e0a4c-d291-57a0-90e8-8db25a27a240"

[[InverseFunctions]]
deps = ["Test"]
git-tree-sha1 = "49510dfcb407e572524ba94aeae2fced1f3feb0f"
uuid = "3587e190-3f89-42d0-90ee-14403ec27112"
version = "0.1.8"

[[IrrationalConstants]]
git-tree-sha1 = "7fd44fd4ff43fc60815f8e764c0f352b83c49151"
uuid = "92d709cd-6900-40b7-9082-c6be49f344b6"
version = "0.1.1"

[[JLFzf]]
deps = ["Pipe", "REPL", "Random", "fzf_jll"]
git-tree-sha1 = "f377670cda23b6b7c1c0b3893e37451c5c1a2185"
uuid = "1019f520-868f-41f5-a6de-eb00f4b6a39c"
version = "0.1.5"

[[JLLWrappers]]
deps = ["Preferences"]
git-tree-sha1 = "abc9885a7ca2052a736a600f7fa66209f96506e1"
uuid = "692b3bcd-3c85-4b1f-b108-f13ce0eb3210"
version = "1.4.1"

[[JSON]]
deps = ["Dates", "Mmap", "Parsers", "Unicode"]
git-tree-sha1 = "3c837543ddb02250ef42f4738347454f95079d4e"
uuid = "682c06a0-de6a-54ab-a142-c8b1cf79cde6"
version = "0.21.3"

[[JpegTurbo_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b53380851c6e6664204efb2e62cd24fa5c47e4ba"
uuid = "aacddb02-875f-59d6-b918-886e6ef4fbf8"
version = "2.1.2+0"

[[LAME_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "f6250b16881adf048549549fba48b1161acdac8c"
uuid = "c1c5ebd0-6772-5130-a774-d5fcae4a789d"
version = "3.100.1+0"

[[LERC_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "bf36f528eec6634efc60d7ec062008f171071434"
uuid = "88015f11-f218-50d7-93a8-a6af411a945d"
version = "3.0.0+1"

[[LZO_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e5b909bcf985c5e2605737d2ce278ed791b89be6"
uuid = "dd4b983a-f0e5-5f8d-a1b7-129d4a5fb1ac"
version = "2.10.1+0"

[[LaTeXStrings]]
git-tree-sha1 = "f2355693d6778a178ade15952b7ac47a4ff97996"
uuid = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f"
version = "1.3.0"

[[Latexify]]
deps = ["Formatting", "InteractiveUtils", "LaTeXStrings", "MacroTools", "Markdown", "OrderedCollections", "Printf", "Requires"]
git-tree-sha1 = "ab9aa169d2160129beb241cb2750ca499b4e90e9"
uuid = "23fbe1c1-3f47-55db-b15f-69d7ec21a316"
version = "0.15.17"

[[LazyArtifacts]]
deps = ["Artifacts", "Pkg"]
uuid = "4af54fe1-eca0-43a8-85a7-787d91b784e3"

[[LibCURL]]
deps = ["LibCURL_jll", "MozillaCACerts_jll"]
uuid = "b27032c2-a3e7-50c8-80cd-2d36dbcbfd21"
version = "0.6.3"

[[LibCURL_jll]]
deps = ["Artifacts", "LibSSH2_jll", "Libdl", "MbedTLS_jll", "Zlib_jll", "nghttp2_jll"]
uuid = "deac9b47-8bc7-5906-a0fe-35ac56dc84c0"
version = "7.84.0+0"

[[LibGit2]]
deps = ["Base64", "NetworkOptions", "Printf", "SHA"]
uuid = "76f85450-5226-5b5a-8eaa-529ad045b433"

[[LibSSH2_jll]]
deps = ["Artifacts", "Libdl", "MbedTLS_jll"]
uuid = "29816b5a-b9ab-546f-933c-edad1886dfa8"
version = "1.10.2+0"

[[Libdl]]
uuid = "8f399da3-3557-5675-b5ff-fb832c97cbdb"

[[Libffi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "0b4a5d71f3e5200a7dff793393e09dfc2d874290"
uuid = "e9f186c6-92d2-5b65-8a66-fee21dc1b490"
version = "3.2.2+1"

[[Libgcrypt_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgpg_error_jll", "Pkg"]
git-tree-sha1 = "64613c82a59c120435c067c2b809fc61cf5166ae"
uuid = "d4300ac3-e22c-5743-9152-c294e39db1e4"
version = "1.8.7+0"

[[Libglvnd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll", "Xorg_libXext_jll"]
git-tree-sha1 = "7739f837d6447403596a75d19ed01fd08d6f56bf"
uuid = "7e76a0d4-f3c7-5321-8279-8d96eeed0f29"
version = "1.3.0+3"

[[Libgpg_error_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "c333716e46366857753e273ce6a69ee0945a6db9"
uuid = "7add5ba3-2f88-524e-9cd5-f83b8a55f7b8"
version = "1.42.0+0"

[[Libiconv_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "42b62845d70a619f063a7da093d995ec8e15e778"
uuid = "94ce4f54-9a6c-5748-9c1c-f9c7231a4531"
version = "1.16.1+1"

[[Libmount_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "9c30530bf0effd46e15e0fdcf2b8636e78cbbd73"
uuid = "4b2f31a3-9ecc-558c-b454-b3730dcb73e9"
version = "2.35.0+0"

[[Libtiff_jll]]
deps = ["Artifacts", "JLLWrappers", "JpegTurbo_jll", "LERC_jll", "Libdl", "Pkg", "Zlib_jll", "Zstd_jll"]
git-tree-sha1 = "3eb79b0ca5764d4799c06699573fd8f533259713"
uuid = "89763e89-9b03-5906-acba-b20f662cd828"
version = "4.4.0+0"

[[Libuuid_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "7f3efec06033682db852f8b3bc3c1d2b0a0ab066"
uuid = "38a345b3-de98-5d2b-a5d3-14cd9215e700"
version = "2.36.0+0"

[[LinearAlgebra]]
deps = ["Libdl", "libblastrampoline_jll"]
uuid = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"

[[LogExpFunctions]]
deps = ["ChainRulesCore", "ChangesOfVariables", "DocStringExtensions", "InverseFunctions", "IrrationalConstants", "LinearAlgebra"]
git-tree-sha1 = "94d9c52ca447e23eac0c0f074effbcd38830deb5"
uuid = "2ab3a3ac-af41-5b50-aa03-7779005ae688"
version = "0.3.18"

[[Logging]]
uuid = "56ddb016-857b-54e1-b83d-db4d58db5568"

[[LoggingExtras]]
deps = ["Dates", "Logging"]
git-tree-sha1 = "5d4d2d9904227b8bd66386c1138cf4d5ffa826bf"
uuid = "e6f89c97-d47a-5376-807f-9c37f3926c36"
version = "0.4.9"

[[MKL_jll]]
deps = ["Artifacts", "IntelOpenMP_jll", "JLLWrappers", "LazyArtifacts", "Libdl", "Pkg"]
git-tree-sha1 = "41d162ae9c868218b1f3fe78cba878aa348c2d26"
uuid = "856f044c-d86e-5d09-b602-aeab76dc8ba7"
version = "2022.1.0+0"

[[MacroTools]]
deps = ["Markdown", "Random"]
git-tree-sha1 = "42324d08725e200c23d4dfb549e0d5d89dede2d2"
uuid = "1914dd2f-81c6-5fcd-8719-6d5c9610ff09"
version = "0.5.10"

[[Markdown]]
deps = ["Base64"]
uuid = "d6f4376e-aef5-505a-96c1-9c027394607a"

[[MbedTLS]]
deps = ["Dates", "MbedTLS_jll", "MozillaCACerts_jll", "Random", "Sockets"]
git-tree-sha1 = "6872f9594ff273da6d13c7c1a1545d5a8c7d0c1c"
uuid = "739be429-bea8-5141-9913-cc70e7f3736d"
version = "1.1.6"

[[MbedTLS_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "c8ffd9c3-330d-5841-b78e-0817d7145fa1"
version = "2.28.0+0"

[[Measures]]
git-tree-sha1 = "e498ddeee6f9fdb4551ce855a46f54dbd900245f"
uuid = "442fdcdd-2543-5da2-b0f3-8c86c306513e"
version = "0.3.1"

[[Missings]]
deps = ["DataAPI"]
git-tree-sha1 = "bf210ce90b6c9eed32d25dbcae1ebc565df2687f"
uuid = "e1d29d7a-bbdc-5cf2-9ac0-f12de2c33e28"
version = "1.0.2"

[[Mmap]]
uuid = "a63ad114-7e13-5084-954f-fe012c677804"

[[MozillaCACerts_jll]]
uuid = "14a3606d-f60d-562e-9121-12d972cd8159"
version = "2022.2.1"

[[NaNMath]]
deps = ["OpenLibm_jll"]
git-tree-sha1 = "a7c3d1da1189a1c2fe843a3bfa04d18d20eb3211"
uuid = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3"
version = "1.0.1"

[[NetworkOptions]]
uuid = "ca575930-c2e3-43a9-ace4-1e988b2c1908"
version = "1.2.0"

[[Ogg_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "887579a3eb005446d514ab7aeac5d1d027658b8f"
uuid = "e7412a2a-1a6e-54c0-be00-318e2571c051"
version = "1.3.5+1"

[[OpenBLAS_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "Libdl"]
uuid = "4536629a-c528-5b80-bd46-f80d51c5b363"
version = "0.3.20+0"

[[OpenLibm_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "05823500-19ac-5b8b-9628-191a04bc5112"
version = "0.8.1+0"

[[OpenSSL]]
deps = ["BitFlags", "Dates", "MozillaCACerts_jll", "OpenSSL_jll", "Sockets"]
git-tree-sha1 = "ebe81469e9d7b471d7ddb611d9e147ea16de0add"
uuid = "4d8831e6-92b7-49fb-bdf8-b643e874388c"
version = "1.2.1"

[[OpenSSL_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e60321e3f2616584ff98f0a4f18d98ae6f89bbb3"
uuid = "458c3c95-2e84-50aa-8efc-19380b2a3a95"
version = "1.1.17+0"

[[OpenSpecFun_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "13652491f6856acfd2db29360e1bbcd4565d04f1"
uuid = "efe28fd5-8261-553b-a9e1-b2916fc3738e"
version = "0.5.5+0"

[[Opus_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "51a08fb14ec28da2ec7a927c4337e4332c2a4720"
uuid = "91d4177d-7536-5919-b921-800302f37372"
version = "1.3.2+0"

[[OrderedCollections]]
git-tree-sha1 = "85f8e6578bf1f9ee0d11e7bb1b1456435479d47c"
uuid = "bac558e1-5e72-5ebc-8fee-abe8a469f55d"
version = "1.4.1"

[[PCRE2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "efcefdf7-47ab-520b-bdef-62a2eaa19f15"
version = "10.40.0+0"

[[Parsers]]
deps = ["Dates"]
git-tree-sha1 = "3d5bf43e3e8b412656404ed9466f1dcbf7c50269"
uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0"
version = "2.4.0"

[[Pipe]]
git-tree-sha1 = "6842804e7867b115ca9de748a0cf6b364523c16d"
uuid = "b98c9c47-44ae-5843-9183-064241ee97a0"
version = "1.3.0"

[[Pixman_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b4f5d02549a10e20780a24fce72bea96b6329e29"
uuid = "30392449-352a-5448-841d-b1acce4e97dc"
version = "0.40.1+0"

[[Pkg]]
deps = ["Artifacts", "Dates", "Downloads", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"]
uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f"
version = "1.8.0"

[[PlotThemes]]
deps = ["PlotUtils", "Statistics"]
git-tree-sha1 = "8162b2f8547bc23876edd0c5181b27702ae58dce"
uuid = "ccf2f8ad-2431-5c83-bf29-c5338b663b6a"
version = "3.0.0"

[[PlotUtils]]
deps = ["ColorSchemes", "Colors", "Dates", "Printf", "Random", "Reexport", "SnoopPrecompile", "Statistics"]
git-tree-sha1 = "21303256d239f6b484977314674aef4bb1fe4420"
uuid = "995b91a9-d308-5afd-9ec6-746e21dbc043"
version = "1.3.1"

[[Plots]]
deps = ["Base64", "Contour", "Dates", "Downloads", "FFMPEG", "FixedPointNumbers", "GR", "JLFzf", "JSON", "LaTeXStrings", "Latexify", "LinearAlgebra", "Measures", "NaNMath", "Pkg", "PlotThemes", "PlotUtils", "Printf", "REPL", "Random", "RecipesBase", "RecipesPipeline", "Reexport", "RelocatableFolders", "Requires", "Scratch", "Showoff", "SnoopPrecompile", "SparseArrays", "Statistics", "StatsBase", "UUIDs", "UnicodeFun", "Unzip"]
git-tree-sha1 = "65451f70d8d71bd9d06821c7a53adbed162454c9"
uuid = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
version = "1.35.2"

[[PlutoUI]]
deps = ["AbstractPlutoDingetjes", "Base64", "ColorTypes", "Dates", "Hyperscript", "HypertextLiteral", "IOCapture", "InteractiveUtils", "JSON", "Logging", "Markdown", "Random", "Reexport", "UUIDs"]
git-tree-sha1 = "2777a5c2c91b3145f5aa75b61bb4c2eb38797136"
uuid = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
version = "0.7.43"

[[Preferences]]
deps = ["TOML"]
git-tree-sha1 = "47e5f437cc0e7ef2ce8406ce1e7e24d44915f88d"
uuid = "21216c6a-2e73-6563-6e65-726566657250"
version = "1.3.0"

[[Printf]]
deps = ["Unicode"]
uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7"

[[Profile]]
deps = ["Printf"]
uuid = "9abbd945-dff8-562f-b5e8-e1ebf5ef1b79"

[[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"]
git-tree-sha1 = "c6c0f690d0cc7caddb74cef7aa847b824a16b256"
uuid = "ea2cea3b-5b76-57ae-a6ef-0a8af62496e1"
version = "5.15.3+1"

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

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

[[RecipesBase]]
deps = ["SnoopPrecompile"]
git-tree-sha1 = "612a4d76ad98e9722c8ba387614539155a59e30c"
uuid = "3cdcf5f2-1ef4-517c-9805-6587b60abb01"
version = "1.3.0"

[[RecipesPipeline]]
deps = ["Dates", "NaNMath", "PlotUtils", "RecipesBase"]
git-tree-sha1 = "017f217e647cf20b0081b9be938b78c3443356a0"
uuid = "01d81517-befc-4cb6-b9ec-a95719d0359c"
version = "0.6.6"

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

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

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

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

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

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

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

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

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

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

[[SortingAlgorithms]]
deps = ["DataStructures"]
git-tree-sha1 = "b3363d7460f7d098ca0912c69b082f75625d7508"
uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c"
version = "1.0.1"

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

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

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

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

[[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"

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

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

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

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

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

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

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

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

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

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

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

[[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"

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

[[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"

[[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"

[[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"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
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 = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.12+3"

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

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

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

[[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"

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

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

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

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

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

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

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

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

[[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"
"""

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