4.linearequation.jl

### A Pluto.jl notebook ###
# v0.19.22

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

# ╔═╡ 37dc4b11-8849-41ad-9afe-5f792ea1fab8
using PlutoUI

# ╔═╡ 8dc256a3-06be-4df9-aa88-1ef1e574055e
using Test, LinearAlgebra

# ╔═╡ c5aaa7ec-9af4-45eb-86f1-e87cef9c8384
using BenchmarkTools

# ╔═╡ d959c85c-0895-475c-b9f8-57368fc5e744
using Plots

# ╔═╡ cdaafdfa-9252-410f-af0c-e47a145fbfe3
TableOfContents(; depth=2)

# ╔═╡ 2464e2fc-1a00-4572-b54c-f87b3239f9ad
html"<button onclick='present();'>present</button>"

# ╔═╡ f6fa6216-0945-4c98-b507-7c91600d8c2f
md"## About assignments"

# ╔═╡ a0fa8273-cf22-4cb4-ad4b-b83aee874556
md"1. You should submit your homework by opening a pull request to [our Github repo](https://github.com/GiggleLiu/ModernScientificComputing). Xuanzhao Gao will comment on your PR, then you should resolve the issues to get your PR merged. You should submit different PRs for different assignments.
2. We will grade based on your merged PRs. You will not be failed if your submition is complete.
4. You may check the answers of other students, but you must credit him in your PR description, e.g.
```
### Reference:
* PR #3
* PR #4
```
"

# ╔═╡ 78b6f5be-0aec-4e40-8512-fb58c253a7e9
md"# System of Linear Equations
Let $A\in \mathbb{R}^{n\times n}$ be a invertible square matrix and $b \in \mathbb{R}^n$ be a vector. Solving a linear equation means finding a vector $x\in\mathbb{R}^n$ such that
```math
A x = b
```
"

# ╔═╡ c1cf7bba-3d47-4c8c-8f1b-64808c300f72
md"Quiz: In a cage, chickens and rabbits add up to 35 heads and 94 feet. Please count the number of chickens and rabbits."

# ╔═╡ 24e421db-547b-416b-8c37-582e9aa78165
md"### Table of contents"

# ╔═╡ e5c4aeef-5818-406c-be55-b6b010c960db
md"* Gaussian elimination algorithm
* Pivoting technique
* Sensitivity analysis and condition number
* Getting matrix inverse with Gauss-Jordan elimination
* Solving linear equations for special matrices (optional)
"

# ╔═╡ a7203d10-cffd-4f08-80c2-f6b1465210bd
md"## Schetch of solving the linear equation"

# ╔═╡ 3a411c5f-264b-4f81-98ba-53eecdad3d28
md"""
One can solve a linear equation by following these steps:

1. Decompose the matrix $A \in \mathbb{R}^{n\times n}$ into $L \in \mathbb{R}^{n\times n}$ and $U \in \mathbb{R}^{n\times n}$ matrices using a method such as [Gaussian elimination](https://en.wikipedia.org/wiki/Gaussian_elimination) or [Crout's method](https://en.wikipedia.org/wiki/Crout_matrix_decomposition).

2. Rewrite the equation $Ax = b$ as $LUx = b$.

3. Solve for y in $Ly = b$ by [forward substitution](https://en.wikipedia.org/wiki/Triangular_matrix). This involves substituting the values of $y$ into the equation one at a time, starting with the first row and working downwards.

4. Solve for $x$ in $Ux = y$ by [back substitution](https://en.wikipedia.org/wiki/Triangular_matrix). This involves substituting the values of $x$ into the equation one at a time, starting with the last row and working upwards.
"""

# ╔═╡ 4015a2f9-a281-44ab-974c-7cd46041d105
md"""
## Solving tridiagonal linear equation
```math
Lx = b
```
"""

# ╔═╡ 53e51d33-9d03-4473-8ed8-aba1ef9ef105
md"""
where $L \in \mathbb{R}^{n\times n}$ is a lower triangular matrix defined as
```math
L = \left(\begin{matrix}
l_{11} & 0 & \ldots & 0\\
l_{21} & l_{22} & \ldots & 0\\
\vdots & \vdots & \ddots & \vdots\\
l_{n1} & l_{n2} & \ldots & l_{nn}
\end{matrix}\right)
```
"""

# ╔═╡ 02605f8e-b457-11ed-19c4-7d6d14b67507
md"""## Algorithm: Forward-Substitution for Lower Triangular System"""

# ╔═╡ c86b3ac8-3ae2-40bb-8988-d9b9250d627d
md"""
Forward substitution is an algorithm used to solve a system of linear equations with a lower triangular matrix. It involves solving for the unknowns in a forward direction, starting from the first equation and moving towards the last. Here's an example of forward substitution algorithm in the matrix form:

Consider the following system of lower triangular linear equations:
```math
L = \left(\begin{matrix}
3 & 0 & 0\\
2 & 5 & 0\\
1 & 4 & 2
\end{matrix}\right)
\left(\begin{matrix}
x_1\\
x_2\\
x_3
\end{matrix}\right) = 
\left(\begin{matrix}
9\\
12\\
13
\end{matrix}\right)
```

To solve for $x_1$, $x_2$, and $x_3$ using forward substitution, we start with the first equation:
```math
3x_1 + 0x_2 + 0x_3 = 9
```
Solving for $x_1$, we get:
```math
x_1 = 3
```
Substituting $x = 3$ into the second equation (row), we get:
```math
2(3) + 5x_2 + 0x_3 = 12
```
Solving for $x_2$, we get:
```math
x_2 = (12 - 6) / 5 = 1.2
```
Substituting $x = 3$ and $x_2 = 1.2$ into the third equation (row), we get:

```math
1(3) + 4(1.2) + 2x_3 = 13
```
Solving for $x_3$, we get:
```math
x_3 = (13 - 3 - 4(1.2)) / 2 = 1.5
```
Therefore, the solution to the system of equations is:
```math
x = \left(\begin{matrix}\
3\\
1.2\\
1.5
\end{matrix}\right)
```
"""

# ╔═╡ fb583209-eb00-41ce-9fc6-906898327ecd
md"""
It can be summarized to the following algorithm
```math
x_1 = b_1/l_{11},~~~ x_i = \left(b_i - \sum_{j=1}^{i-1}l_{ij}x_j\right)/l_{ii},~~ i=2, ..., n
```
"""

# ╔═╡ 35d1d353-c3bd-4e2d-90e0-e3031a2133c2
md"### The implementation"

# ╔═╡ 5c73b504-f8f2-429a-99a2-5f5d98f7e1e6
md"""
We implement the above algorithm in Julia language.
"""

# ╔═╡ 742a79b6-6ced-467e-961b-814da8b65af7
function back_substitution!(l::AbstractMatrix, b::AbstractVector)
	n = length(b)
	@assert size(l) == (n, n) "size mismatch"
	x = zero(b)
	# loop over columns
	for j = 1:n
		# stop if matrix is singular
		if iszero(l[j, j])
			error("The lower triangular matrix is singular!")
		end
		# compute solution component
		x[j] = b[j] / l[j, j]
		for i = j+1:n
			# update right hand side
			b[i] = b[i] - l[i, j] * x[j]
		end
	end
	return x
end

# ╔═╡ e5e54678-4d3a-40ae-8b39-d5c50a0617b2
md"""
We can write a test for this algorithm.
"""

# ╔═╡ d3e1ef10-3f99-4561-87db-47c0fc47e9e9
@testset "back substitution" begin
	# create a random lower triangular matrix
	l = tril(randn(4, 4))
	# target vector
	b = randn(4)
	# solve the linear equation with our algorithm
	x = back_substitution!(l, copy(b))
	@test l * x ≈ b

	# The Julia's standard library `LinearAlgebra` contains a native implementation.
	x_native = LowerTriangular(l) \ b
	@test l * x_native ≈ b
end

# ╔═╡ 7649913c-523f-4fa3-bf1e-ea1566b55114
md"## The LU decomposition with Gaussian elimination
LU decomposition is a method for solving linear equations that involves breaking down a matrix into lower and upper triangular matrices. The $LU$ decomposition of a matrix $A$ is represented as $A = LU$, where $L$ is a lower triangular matrix and $U$ is an upper triangular matrix.
"

# ╔═╡ 24fa07b8-6a9c-47d6-8f72-9346f1325ab4
md"## The elementary elimination matrix"

# ╔═╡ 851510d9-7703-4545-8c00-ec94d8735a85
md"""
An elementary elimination matrix is a matrix that is used in the process of Gaussian elimination to transform a system of linear equations into an equivalent system that is easier to solve. It is a square matrix that is obtained by performing a single elementary row operation on the identity matrix.
"""

# ╔═╡ af1f932e-a6db-41ce-ace3-170228bb183b
# ```math
# (M_k)_{ij} = \begin{cases}
# 	\delta_{ij} & i= j,\\
# 	- a_{ik}/a_{kk} & i > j \land j = k, \\
#	0 & {\rm otherwise}.
#\end{cases}
#```
md"""
Let $A = (a_{ij})$ be a square matrix of size $n \times n$. The $k$th elementary elimination matrix for it is defined as
```math
M_k = \left(\begin{matrix}

1 & \ldots & 0 & 0 & 0 & \ldots & 0\\
\vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots\\
0 & \ldots & 1 & 0 & 0 & \ldots & 0\\
0 & \ldots & 0 & 1 & 0 & \ldots & 0\\
0 & \ldots & 0 & -m_{k+1} & 1 & \ldots & 0\\
\vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots\\
0 & \ldots & 0 & -m_{n} & 0 & \ldots & 1\\

\end{matrix}\right)
```
where $m_i = a_{ik}/a_{kk}$.
"""

# ╔═╡ a890cdb6-34b6-4a3c-8749-e1543e9764ba
md"""
By applying this elimentary elimination matrix $M_1$ on $A$, we can obtain a new matrix with the $a_{i1}' = 0$ for all $i>1$.
```math
M_1 A = \left(\begin{matrix}
a_{11} & a_{12} & a_{13} & \ldots & a_{1n}\\
0 & a_{22}' & a_{23}' & \ldots & a_{2n}'\\
0 & a_{32}' & a_{33}' & \ldots & a_{3n}'\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
0 & a_{n2}' & a_{n3}' & \ldots & a_{nn}'\\
\end{matrix}\right)
```
"""

# ╔═╡ 9640cc8d-f8c0-4d3b-b51a-3ad3865630bc
md"For $k=1,2,\ldots,n$, apply $M_k$ on $A$. We will have an upper triangular matrix.
```math
U = M_{n-1}\ldots M_1 A
```

Since $M_k$ is reversible (as we will show below), we have
```math
\begin{align}
&A = LU\\
&L = M_1^{-1} M_2^{-1} \ldots M_{n-1}^{-1},
\end{align}
```
"

# ╔═╡ 83e0cdc0-70d3-48d2-bf1c-8ed364652aed
md"## Properties of elementary elimination matrices
1. Its inverse can be computed in $O(n)$ time
```math
M_k^{-1} = 2I - M_k
```
2. The multiplication of two elementary matrices can be computed in $O(n)$ time
```math
M_k M_{k' > k} = M_k + M_{k'} - I
```
"

# ╔═╡ 2b3596d7-17a3-46bb-b687-302c1255a4af
md"## Coding: Properties of elimination matrices"

# ╔═╡ b8cf4822-d91b-47f4-86e4-a36d668b5256
A3 = [1 2 2; 4 4 2; 4 6 4]

# ╔═╡ faa03439-b526-4c27-a9dc-593b5a32cb04
function elementary_elimination_matrix(A::AbstractMatrix{T}, k::Int) where T
	n = size(A, 1)
	@assert size(A, 2) == n
	# create Elementary Elimination Matrices
	M = Matrix{Float64}(I, n, n)
	for i=k+1:n
		M[i, k] =  -A[i, k] ./ A[k, k]
	end
	return M
end

# ╔═╡ 416f9a74-2d88-4788-92b1-847eda6539cc
md"The elimination"

# ╔═╡ b8379e6d-768d-4a43-8370-1583324750f3
elementary_elimination_matrix(A3, 1)

# ╔═╡ e478efbc-a75f-4fb4-8ea8-f24ea7f8a9c2
elementary_elimination_matrix(A3, 1) * A3

# ╔═╡ 73fa4829-33dc-46e7-935d-b4c78bdb02d3
md"The inverse"

# ╔═╡ fc286cee-c912-4395-8af6-77702d5fef31
inv(elementary_elimination_matrix(A3, 1))

# ╔═╡ b4282280-a825-456f-90ec-cfc274c68864
md"The multiplication"

# ╔═╡ 32ce956f-a90f-44fb-8473-438dd2a6ba78
elementary_elimination_matrix(A3, 2)

# ╔═╡ 2a8da2fe-f21e-44ac-9253-583c06dd5134
inv(elementary_elimination_matrix(A3, 1)) * inv(elementary_elimination_matrix(A3, 2))

# ╔═╡ 87de48b5-4d4e-4b8a-9f2c-4b6dc328a502
md"## LU Factorization by Gaussian Elimination"

# ╔═╡ b27b9f35-bac1-4d4c-9543-3733ed21e039
md"""
A naive implementation of elimentary elimination matrix is as follows
"""

# ╔═╡ a4b32a01-2d7f-4933-a3cb-2ab2d0c6cb2e
function lufact_naive!(A::AbstractMatrix{T}) where T
	n = size(A, 1)
	@assert size(A, 2) == n
	M = Matrix{T}(I, n, n)
	for k=1:n-1
		m = elementary_elimination_matrix(A, k)
		M = M * inv(m)
		A .= m * A
	end
	return M, A
end

# ╔═╡ 4d8276dc-9356-47f8-afa6-5329dde63468
lufact_naive!(copy(A3))

# ╔═╡ 405450f7-807d-45a8-94ac-54ef9972b234
@testset "naive LU factorization" begin
	A = [1 2 2; 4 4 2; 4 6 4]
	L, U = lufact_naive!(copy(A))
	@test L * U ≈ A
end

# ╔═╡ 1df2f386-2b55-4fb0-a035-64cf454e2698
md"The above implementation has time complexity $O(n^4)$ since we did not use the sparsity of elimentary elimination matrix. A better implementation that gives $O(n^3)$ time complexity is as follows."

# ╔═╡ fadba4ba-efca-48f7-bf31-959f458f64d5
function lufact!(a::AbstractMatrix)
	n = size(a, 1)
	@assert size(a, 2) == n "size mismatch"
	m = zero(a)
	m[1:n+1:end] .+= 1
	# loop over columns
	for k=1:n-1
		# stop if pivot is zero
		if iszero(a[k, k])
			error("Gaussian elimination fails!")
		end
		# compute multipliers for current column
		for i=k+1:n
			m[i, k] = a[i, k] / a[k, k]
		end
		# apply transformation to remaining sub-matrix
		for j=k+1:n
			for i=k+1:n
				a[i,j] -= m[i,k] * a[k, j]
			end
		end
	end
	return m, triu!(a)
end

# ╔═╡ f2bab70d-847f-495b-ba30-409156a9a874
lufact(a::AbstractMatrix) = lufact!(copy(a))

# ╔═╡ 480b6065-f9ca-4770-baa4-f1451fc046b8
@testset "LU factorization" begin
	a = randn(4, 4)
	L, U = lufact(a)
	@test istril(L)
	@test istriu(U)
	@test L * U ≈ a
end

# ╔═╡ b81714d1-9a3c-4800-b478-68cabbc5d10d
A4 = randn(4, 4)

# ╔═╡ 1f1267d0-9ff2-44a4-b418-a42953159a1e
lufact(A4)

# ╔═╡ 5e426a5f-65cf-4adf-8b2b-ea28534e529a
md"Julia language has a much better implementation in the standard library `LinearAlgebra`."

# ╔═╡ 9d30b63e-f437-420b-8c22-037d34e1be2f
julia_lures = lu(A4, NoPivot())  # the version we implemented above has no pivot

# ╔═╡ 4a18900a-9051-4451-9736-724b19c3bd5d
julia_lures.U

# ╔═╡ aacd052d-b6e7-4f40-b08a-872aa9420c46
typeof(julia_lures)

# ╔═╡ c86a4d3c-adf3-429a-a4b4-8a0aaec83b43
fieldnames(julia_lures |> typeof)

# ╔═╡ 2fda47ad-08f2-45c5-8f47-0eec1aa49aac
md"## Issue: how to handle small diagonal entries?"

# ╔═╡ 0d8dc065-4487-41bd-a80d-45f1fc1bf550
md"""
The above Gaussian elimination process is not stable if any diagonal entry in $A$ has a value that close to zero.
"""

# ╔═╡ 77396c1b-f23d-41da-b9e2-d6980a090217
@bind ϵ Slider(-20:0.01:0.0; default=-2, show_value=true)

# ╔═╡ 97e373df-32cc-491e-8066-e3a33b9b2764
small_diagonal_matrix = [10^(ϵ) 1; 1 1]

# ╔═╡ 1201b122-b071-4f13-83e0-6bec0de08e2b
lures = lufact(small_diagonal_matrix)

# ╔═╡ 7a7e8ac0-2af9-42ec-ab17-54d4f86baf70
md"This issue is not always related to the rank deficiency (or ill-condition). A remedy to this issue is to permute the rows of $A$ before factorizing it.
For example:"

# ╔═╡ a4c024e4-83a7-4356-9b8e-168ce9ee1cbf
lufact(small_diagonal_matrix[end:-1:1, :])

# ╔═╡ 99381db6-0735-4bfe-9d20-1e16e08f13c3
md"This technique is called pivoting."

# ╔═╡ 54db9443-a0f9-48f1-97bb-bcab2ae3cfd5
md"""
## Pivoting technique
"""

# ╔═╡ 178c4193-a45d-460f-a22e-b15f6604b5dd
md"""
LU factoriazation (or Gaussian elimination) with pivoting is defined as
```math
P A = L U
```
where $P$ is a permutation matrix.
"""

# ╔═╡ 5a034215-a5da-4fea-8dc3-004310f5e7be
md"Pivoting in Gaussian elimination is the process of selecting a pivot element in a matrix and then using it to eliminate other elements in the same column or row. The pivot element is chosen as the largest absolute value in the column, and its row is swapped with the row containing the current element being eliminated if necessary. This is done to avoid division by zero or numerical instability, and to ensure that the elimination process proceeds smoothly. Pivoting is an important step in Gaussian elimination, as it ensures that the resulting matrix is in reduced row echelon form and that the solution to the system of equations is accurate."

# ╔═╡ 9bd48ccc-549a-4057-aa91-16ad79536583
md"""
## Gaussian Elimination process with Partial Pivoting
Let $A=(a_{ij})$ be a square matrix of size $n\times n$. The Gaussian elimination process with partial pivoting can be represented as
```math
M_{n-1}P_{n-1}\ldots M_2P_2M_1P_1 A = U
```
"""

# ╔═╡ 95c78df3-2283-4230-8d8d-9996e4d6fe36
md"Here we emphsis that $P_{k}$ and $M_{j<k}$ commute."

# ╔═╡ e9ba7dc4-1494-4601-8eb7-72127e92815d
md"## LU Factoriazation by Gaussian Elimination with Partial Pivoting"

# ╔═╡ 5a8e40b0-1cd7-4a71-94da-64cfd9ab9755
md"""
A Julia implementation of the Gaussian elimination with partial pivoting is
"""

# ╔═╡ 9b4fa87d-c40c-4677-82f2-2fff3b87f4c6
function lufact_pivot!(a::AbstractMatrix)
	n = size(a, 1)
	@assert size(a, 2) == n "size mismatch"
	m = zero(a)
	P = collect(1:n)
	# loop over columns
	@inbounds for k=1:n-1
		# search for pivot in current column
		val, p = findmax(x->abs(a[x, k]), k:n)
		p += k-1
		# find index p such that |a_{pk}| ≥ |a_{ik}| for k ≤ i ≤ n
		if p != k
			# swap rows k and p of matrix A
			for col = 1:n
				a[k, col], a[p, col] = a[p, col], a[k, col]
			end
			# swap rows k and p of matrix M
			for col = 1:k-1
				m[k, col], m[p, col] = m[p, col], m[k, col]
			end
			P[k], P[p] = P[p], P[k]
		end
		if iszero(a[k, k])
			# skip current column if it's already zero
			continue
		end
		# compute multipliers for current column
		m[k, k] = 1
		for i=k+1:n
			m[i, k] = a[i, k] / a[k, k]
		end
		# apply transformation to remaining sub-matrix
		for j=k+1:n
			akj = a[k, j]
			for i=k+1:n
				a[i,j] -= m[i,k] * akj
			end
		end
	end
	m[n, n] = 1
	return m, triu!(a), P
end

# ╔═╡ 13493579-9189-4c9a-9361-430868427b59
@testset "lufact with pivot" begin
	n = 5
	A = randn(n, n)
	L, U, P = lufact_pivot!(copy(A))
	pmat = zeros(Int, n, n)
	setindex!.(Ref(pmat), 1, 1:n, P)
	@test L ≈ lu(A).L
	@test U ≈ lu(A).U
	@test pmat * A ≈ L * U
end

# ╔═╡ f3a506b5-669f-4e8f-9424-e04532b9a87a
md"""
If you are interested in knowing the performance of our implementation, please check the following check box.
"""

# ╔═╡ d1b7f00e-2b7e-4c15-8231-a816c59472f9
@bind benchmark_lu CheckBox()

# ╔═╡ 3bcf8d78-3678-4749-99ac-0a7e43830d2d
if benchmark_lu let
	n = 200
	A = randn(n, n)
	@benchmark lufact_pivot!($A)
end end

# ╔═╡ e206e554-64cd-422d-963d-8284eb9625fe
if benchmark_lu let
	n = 200
	A = randn(n, n)
	@benchmark lu($A)
end end

# ╔═╡ ed52cf76-ed66-44af-b98b-b58551bd217f
md"""
## Complete pivoting
The complete pivoting also allows permuting columns. It produces better numerical stability but is also harder to implement. In most practical using cases, partial pivoting is good enough.
```math
P A Q = L U
```
"""

# ╔═╡ e113247a-b2a5-4acc-8714-342ecb191ac4
md"# Sensitivity Analysis"

# ╔═╡ 9231de22-d474-4548-b984-fcc3dda27113
md"""
Sensitivity analysis in linear algebra is the study of how changes in the input data or parameters of a linear system affect the output or solution of the system. It involves analyzing the sensitivity of the solution to changes in the coefficients of the system, the right-hand side vector, or the constraints of the problem. Sensitivity analysis can be used to determine the effect of small changes in the input data on the optimal solution, to identify critical parameters that affect the solution, and to assess the robustness of the solution to uncertainties or variations in the input data.
"""

# ╔═╡ 25f870c1-e70a-48b1-a0ce-4c2b9a523d60
md"## Issue: An Ill Conditioned Matrix"

# ╔═╡ 0c5a1e3f-7383-4705-b63e-e64f4a513f85
md"""
An ill conditioned matrix may produce unreliable result, or the output is very sensitive to the input. The following is an example of a matrix close to singular.
"""

# ╔═╡ b333fc58-0981-4d0f-ba49-0ee15cc78aad
md"""
```math
A = \left(\begin{matrix}
0.913 & 0.659\\
0.457 & 0.330
\end{matrix}
\right)
```
"""

# ╔═╡ 0beab22f-5411-403c-a116-cf552083382e
ill_conditioned_matrix = [0.913 0.659; 0.457 0.330]

# ╔═╡ 63cddfd1-bdef-4601-ab57-48314952fb73
lures2 = lufact(ill_conditioned_matrix)

# ╔═╡ c5b109da-ab9c-4a44-ab2d-168b9711c2cc
lures2

# ╔═╡ 713267d1-08f0-4c41-9523-46128ce40430
cond(ill_conditioned_matrix)

# ╔═╡ 51433b0f-8fce-4241-bb84-bcea9a687abe
md"## The relative error"

# ╔═╡ 26207f81-1f6e-4708-8c04-422a1133c5ac
md"""
The relevant error in floating number system is the relative error.

* Absolute error: $\|x - \hat x\|$
* Relative error: $\frac{\|x - \hat x\|}{\|x\|}$


where $\|\cdot\|$ is a measure of size.
"""

# ╔═╡ 025101f0-a1c4-47da-bff0-c0ae0cd0da6e
md"## Numeric experiment: Floating point numbers have \"constant\" relative error"

# ╔═╡ 424638de-6651-4fe2-bc44-82f9675dc43d
eps(Float64)

# ╔═╡ 26978647-a790-4ba2-be21-60251c47d919
let
	n = 1000
	reltol = zeros(2n+1)
	for i=-n:n
		f = 2.0^i
		reltol[i+n+1] = log10(eps(f)) - log10(f)
	end
	plot(-n:n, reltol; label="relative error")
end

# ╔═╡ c10acde5-e22a-4f79-be4f-f7b3bbb0ece9
eps(1.0)/1.0

# ╔═╡ 8e055682-baef-4c78-b044-d22d51676844
eps(2.0)/2.0

# ╔═╡ ee1aba8e-5992-4ff1-9ae1-09d35a2abf00
eps(sqrt(2))/sqrt(2)

# ╔═╡ d1c29e2e-e67d-41d3-bbff-9f384cf0c84d
md"## (Relative) Condition Number"

# ╔═╡ 2f789e07-69a9-4786-8d82-e363afa4323f
md"
Condition number is a measure of the sensitivity of a mathematical problem to changes or errors in the input data. It is a way to quantify how much the output of a function or algorithm can vary due to small changes in the input. A high condition number indicates that the problem is ill-conditioned, meaning that small errors in the input can lead to large errors in the output. A low condition number indicates that the problem is well-conditioned, meaning that small errors in the input have little effect on the output.

In short, the (relative) condition number of an operation $f$ with input $x$ measures the relative error amplication power of $f$ with input $x$, which is formally defined as
"

# ╔═╡ df17edcb-8b98-4c2e-a628-1a1381b7cf3e
md"""
```math
\lim _{\varepsilon \rightarrow 0^{+}}\,\sup _{\|\delta x\|\,\leq \,\varepsilon }{\frac {\|\delta f(x)\|/\|f(x)\|}{\|\delta x\|/\|x\|}}.
```
"""

# ╔═╡ 46176c75-f23e-4aef-a54b-58aa43941855
md"Quiz 1: What is the condition number of"

# ╔═╡ fe41dd53-f69f-4382-bde1-af86a3a92aef
md"""
```math
y = \exp(x)
```
"""

# ╔═╡ 8651c71c-eb8b-467b-9c9c-f2a173e5b91d
md"Quiz 2: What is the condition number of"

# ╔═╡ d9d88d11-0dd0-4e80-b476-e9b008dfb84c
md"""
```math
a + b
```
"""

# ╔═╡ 88ad6fff-04d8-477d-a664-c1a6522964cc
md"""
With the obtained result, show why we should avoid subtracting two big floating point numbers?
"""

# ╔═╡ bb5e9e72-8cc7-4c0c-96c2-9c1ca9d1b38c
md"## Measuring the size vectors and matrices"

# ╔═╡ e1026249-5d13-418b-a908-b6dc77175434
md"The vector $p$-norm is formally defined as"

# ╔═╡ f256bf0d-e2bb-49f4-882f-12508e68ff07
md"""
```math
\|v\|_p = \left(\sum_i{|v_i|^p}\right)^{1/p}
```
"""

# ╔═╡ f2da27a2-af5f-4f2a-a6d5-fb98b8dcd261
md"""Similarly, the matrix $p$-norm is formally defined as
```math
\|A\|_p = \max_{x\neq 0} \frac{\|Ax\|_p}{\|x\|_p}
```
"""

# ╔═╡ 49d62ec4-bce0-4620-93cc-278f149f848f
md"## Examples: Vector and Matrix Norms"

# ╔═╡ 33a6261a-2b21-4b37-bcd8-b5c0db6b509f
norm([3, 4], 2)

# ╔═╡ 1956af2c-c88f-483d-8e1c-8ed4004e996d
norm([3, 4], 1)

# ╔═╡ 61c1c35c-f7be-495b-90f5-9a3772f254f5
norm([3, 4], Inf)

# ╔═╡ 43731510-dfe7-48c8-8a4d-d8313d1d762b
# l0 norm is not a true norm
norm([3, 4], 0)

# ╔═╡ a0af42b2-1bba-423c-b5f7-0a9d1a2adbb1
norm([3, 0], 0)

# ╔═╡ 79bc3efb-61a1-4241-b787-2726fb2a4a97
mat = randn(2, 2)

# ╔═╡ 9854ad79-d06e-4d60-86d5-54ade042d49a
opnorm(mat, 1)

# ╔═╡ 4628cb4b-649f-4e84-a6d5-e50109f8cc6c
opnorm(mat, Inf)

# ╔═╡ 424f63ce-f5b2-456f-8c4a-63f18877d730
opnorm(mat, 2)

# ╔═╡ 71e6ed15-bb0f-4510-901c-c9f53b3e701c
opnorm(mat, 0)

# ╔═╡ 898a511d-7c18-4eb0-b68b-3c1dd8208cfe
cond(mat)

# ╔═╡ 5653168f-80b5-4435-a711-d8a1d8f9c429
md"## Condition Number of a Linear Operator"

# ╔═╡ c165a6b1-78da-40b5-b925-a657193ade6a
md"""
The condition number of a linear system $Ax = b$ is defined as
"""

# ╔═╡ a1a55972-4baa-4a29-8c3f-ac7e13036136
md"""
```math
{\rm cond}(A) = \|A\| \|A^{-1}\|
```
"""

# ╔═╡ 0b1c4d61-b748-4e9d-9a29-9ee50fa90e79
md"""
Using the defintion of condition number, we have
```math
\begin{align}
{\rm cond(A, x)}&=\lim _{\varepsilon \rightarrow 0^{+}}\,\sup _{\|\delta x\|\,\leq \,\varepsilon }{\frac {\|\delta (Ax)\|/\|A x\|}{\|\delta x\|/\|x\|}}\\
&=\lim _{\varepsilon \rightarrow 0^{+}}\,\sup _{\|\delta x\|\,\leq \,\varepsilon }{\frac {\|A\delta x\|\|x\|}{\|\delta x\|\|Ax\|}}
\end{align}
```
Let $y = Ax$, we have
```math
\begin{align}
{\rm cond(A, x)}&=\lim _{\varepsilon \rightarrow 0^{+}}\,\sup _{\|\delta x\|\,\leq \,\varepsilon }{\frac {\|A\delta x\|\|A^{-1}y\|}{\|\delta x\|\|y\|}}\\
&=\|A\|\frac{\|A^{-1}y\|}{\|y\|}
\end{align}
```
Suppose we want to get an upper bound for any input $x$, then using the definiton of matrix norm, we have
```math
{\rm cond}(A) = \|A\|\sup_y \frac{\|A^{-1}y\|}{\|y\|} = \|A\| \|A^{-1}\|
```
"""

# ╔═╡ 8ee2c9f5-3dbf-4ab0-a638-5a8c2d5d1367
md"""## Numeric experiment: Numeric experiment on condition number"""

# ╔═╡ 180b4de9-9b37-47c2-905a-0156210be5cc
md"""
We randomly generate matrices of size $10\times 10$ and show the condition number approximately upper bounds the numeric error amplification factor of a linear equation solver.
"""

# ╔═╡ b71889e3-85d0-4873-a94a-63a9c4c0c009
let
	n = 1000
	p = 2
	errors = zeros(n)
	conds = zeros(n)
	for k = 1:n
		A = rand(10, 10)
		b = rand(10)
		dx = A \ b
		sx = Float32.(A) \ Float32.(b)
		errors[k] = (norm(sx - dx, p)/norm(dx, p)) / (norm(b-Float32.(b), p)/norm(b, p))
		conds[k] = cond(A, p)
	end
	plt = plot(conds, conds; label="condition number", xlim=(1, 10000), ylim=(1, 10000), xscale=:log10, yscale=:log10)
	scatter!(plt, conds, errors; label="samples")
end


# ╔═╡ 927272dc-d60c-4263-8139-7f43887df52c
md"## Positive definite symmetric matrix"

# ╔═╡ eac7780a-5e0b-4518-907f-9ff75dfb4b2f
md"""
* (Real) Symmetric: $A = A^T$,
* Positive definite: $x^T A x > 0$ for all $x \neq 0$.
"""

# ╔═╡ 8dced7f2-8eab-41d8-95f7-5c23e0035466
md"## Cholesky decomposition

Cholesky decomposition is a method of decomposing a positive-definite matrix into a product of a lower triangular matrix and its transpose. It is named after the mathematician André-Louis Cholesky, who developed the method in the early 1900s. The Cholesky decomposition is useful in many areas of mathematics and science, including linear algebra, numerical analysis, and statistics. It is often used in solving systems of linear equations, computing the inverse of a matrix, and generating random numbers with a given covariance matrix. The Cholesky decomposition is computationally efficient and numerically stable, making it a popular choice in many applications.

Given a positive definite symmetric matrix $A\in \mathbb{R}^{n\times n}$, the Cholesky decomposition is formally defined as
```math
A = LL^T,
```
where $L$ is an upper triangular matrix.
"

# ╔═╡ d0bf8157-9139-418f-9bd8-2791e52aed8e
md"The implementation of Cholesky decomposition is similar to LU decomposition."

# ╔═╡ 79a5bc5d-a2bf-485a-95ee-07d3b8546e75
function chol!(a::AbstractMatrix)
	n = size(a, 1)
	@assert size(a, 2) == n
	for k=1:n
		a[k, k] = sqrt(a[k, k])
		for i=k+1:n
			a[i, k] = a[i, k] / a[k, k]
		end
		for j=k+1:n
			for i=k+1:n
				a[i,j] = a[i,j] - a[i, k] * a[j, k]
			end
		end
	end
	return a
end

# ╔═╡ 7945c565-d226-437c-83ad-5b1e3149c076
@testset "cholesky" begin
	n = 10
	Q, R = qr(randn(10, 10))
	a = Q * Diagonal(rand(10)) * Q'
	L = chol!(copy(a))
	@test tril(L) * tril(L)' ≈ a
	# cholesky(a) in Julia
end

# ╔═╡ 011e1b39-a73d-4eaf-8847-ae5fa5ffd2f0
md"""# Assignments
1. Get the relative condition number of division operation $a/b$.

2. Classify each of the following matrices as well-conditioned or ill-conditioned:
```math
(a). ~~\left(\begin{matrix}10^{10} & 0\\ 0 & 10^{-10}\end{matrix}\right)
```
```math
(b). ~~\left(\begin{matrix}10^{10} & 0\\ 0 & 10^{10}\end{matrix}\right)
```
```math
(c). ~~\left(\begin{matrix}10^{-10} & 0\\ 0 & 10^{-10}\end{matrix}\right)
```
```math
(d). ~~\left(\begin{matrix}1 & 2\\ 2 & 4\end{matrix}\right)
```
3. Implement the Gauss-Jordan elimination algorithm to compute matrix inverse. In the following example, we first create an augmented matrix $(A | I)$. Then we apply the Gauss-Jordan elimination matrices on the left. The final result is stored in the augmented matrix as $(I, A^{-1})$.
![](https://user-images.githubusercontent.com/6257240/222182865-c2a1aa28-946a-4acb-8df8-f5d7da93c3ee.png)

Task: Please implement a function `gauss_jordan` that computes the inverse for a matrix at any size. Please also include the following test in your submission.
```julia
@testset "Gauss Jordan" begin
	n = 10
	A = randn(n, n)
	@test gauss_jordan(A) * A ≈ Matrix{Float64}(I, n, n)
end
```
"""

# ╔═╡ ca87e541-b520-4231-b949-4675456f6c0b
md"""Simular to computing Guassian elimination with elementary elimination matrices, computing the inverse can be done by repreatedly applying the Guass-Jordan elimination matrix and convert the target matrix to identity.
```math
SN_{n}N_{n-1}\ldots N_1 A = I
```
Then
```math
A^{-1} = SN_{n}N_{n-1}\ldots N_1
```
"""

# ╔═╡ c50dbe8c-6c09-47aa-a798-0673875e1e42
md"""
Here, the Gauss-Jordan elimination matrix $N_k$ eliminates the $k$th column except the diagonal element $a_{kk}$.
```math
N_k = \left(\begin{matrix}

1 & \ldots & 0 & -m_1 & 0 & \ldots & 0\\
\vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots\\
0 & \ldots & 1 & -m_{k-1} & 0 & \ldots & 0\\
0 & \ldots & 0 & 1 & 0 & \ldots & 0\\
0 & \ldots & 0 & -m_{k+1} & 1 & \ldots & 0\\
\vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots\\
0 & \ldots & 0 & -m_{n} & 0 & \ldots & 1\\

\end{matrix}\right)
```
where $m_i = a_i/a_k$.
"""

# ╔═╡ b429c34b-122e-4d3d-b8bd-96dab8b18368
md"""
 $S$ is a diagonal matrix.
"""

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
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[compat]
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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 = "261dddd3b862bd2c940cf6ca4d1c8fe593e457c8"
uuid = "3cdcf5f2-1ef4-517c-9805-6587b60abb01"
version = "1.3.3"

[[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 = "30449ee12237627992a99d5e30ae63e4d78cd24a"
uuid = "6c6a2e73-6563-6170-7368-637461726353"
version = "1.2.0"

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

[[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 = "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 = "ef28127915f4229c971eb43f3fc075dd3fe91880"
uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
version = "2.2.0"

[[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.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 = "94f38103c984f89cf77c402f2a68dbd870f8165f"
uuid = "3bb67fe8-82b1-5028-8e26-92a6c54297fa"
version = "0.9.11"

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

[[deps.URIs]]
git-tree-sha1 = "074f993b0ca030848b897beff716d93aca60f06a"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.4.2"

[[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 = "ed8d92d9774b077c53e1da50fd81a36af3744c1c"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.21.0+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 = "93c41695bc1c08c46c5899f4fe06d6ead504bb73"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.10.3+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"]
git-tree-sha1 = "c6edfe154ad7b313c01aceca188c05c835c67360"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.4+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:
# ╠═37dc4b11-8849-41ad-9afe-5f792ea1fab8
# ╟─cdaafdfa-9252-410f-af0c-e47a145fbfe3
# ╟─2464e2fc-1a00-4572-b54c-f87b3239f9ad
# ╟─f6fa6216-0945-4c98-b507-7c91600d8c2f
# ╟─a0fa8273-cf22-4cb4-ad4b-b83aee874556
# ╟─78b6f5be-0aec-4e40-8512-fb58c253a7e9
# ╟─c1cf7bba-3d47-4c8c-8f1b-64808c300f72
# ╟─24e421db-547b-416b-8c37-582e9aa78165
# ╟─e5c4aeef-5818-406c-be55-b6b010c960db
# ╟─a7203d10-cffd-4f08-80c2-f6b1465210bd
# ╟─3a411c5f-264b-4f81-98ba-53eecdad3d28
# ╟─4015a2f9-a281-44ab-974c-7cd46041d105
# ╟─53e51d33-9d03-4473-8ed8-aba1ef9ef105
# ╟─02605f8e-b457-11ed-19c4-7d6d14b67507
# ╟─c86b3ac8-3ae2-40bb-8988-d9b9250d627d
# ╟─fb583209-eb00-41ce-9fc6-906898327ecd
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