0-julia-intro.jl

### A Pluto.jl notebook ###
# v0.19.26

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

# ╔═╡ 42b4beca-a43f-41de-8068-40caf8847b73
begin
	using Plots
	using PlutoUI
	using PlutoTeachingTools
	using PlutoTest
	using ShortCodes
end

# ╔═╡ a81c1305-c953-43fd-8057-c7afbb25d6e6
begin
	using BenchmarkTools
	using LinearAlgebra
end

# ╔═╡ cb2c5fb1-4fd0-4a39-9243-646d7d8096d3
using QuantumEspresso_jll

# ╔═╡ 6719ddf8-ba5d-4694-9d47-c3106039fdc6
begin
	using Libdl
	using Printf
	
	using CondaPkg
	using PythonCall
	CondaPkg.add("numpy")
	const np = pyimport("numpy")
end;

# ╔═╡ 6340a213-17dc-4b8e-8848-8bc50aa620fa
begin
	using IntervalArithmetic: interval
	using Measurements
	using DifferentialEquations
end

# ╔═╡ 01dd3582-349c-4613-9768-869d09168e4f
begin
	using GenericLinearAlgebra
	using SparseArrays
	using CUDA
end

# ╔═╡ ba51d134-6ee7-4921-afaf-09094a6ac3ba
md"""
# A concise Julia introduction
"""

# ╔═╡ b3c5bc68-c92e-4e35-a94f-5b18163b2f8f
md"""
!!! danger "Presentation materials"
    The content of this talk is also available as a static website on
    [https://mfherbst.github.io/julia-for-materials/](https://mfherbst.github.io/julia-for-materials/),
    as interactive Pluto notebooks on
    [github.com/mfherbst/julia-for-materials](https://github.com/mfherbst/julia-for-materials)
    and as a [youtube recording](https://www.youtube.com/watch?v=dujepKxxxkg).
"""

# ╔═╡ 3948dd91-dc26-4531-ba4d-2a8955add0b6
LocalResource("Logos.png")

# ╔═╡ de1fdecc-1752-4baa-b353-ae26519c4506
md"""Pluto notebooks install all packages and run all cells on startup. Since we do some benchmarks in here, this will likely take some time. Best grab a coffee ;)."""

# ╔═╡ 79995a06-14d4-44a8-89c1-5c66286d0e0a
TableOfContents()

# ╔═╡ c6271577-8ea0-487a-9cf4-116dc3c6d962
md"""
## About Julia

- In February 2022 Julia has [turned 10 years](https://julialang.org/blog/2022/02/10years/).
- Project started by Jeff Bezanson, Stefan Karpinski, Viral B. Shah, and Alan Edelman over the **frustration of the state of scientific software**:


> We are power Matlab users. Some of us are Lisp hackers. Some are Pythonistas, others Rubyists, still others Perl hackers. There are those of us who used Mathematica before we could grow facial hair. There are those who still can't grow facial hair. We've generated more R plots than any sane person should. C is our desert island programming language.
>
> We love all of these languages; they are wonderful and powerful. For the work we do — scientific computing, machine learning, data mining, large-scale linear algebra, distributed and parallel computing — each one is perfect for some aspects of the work and terrible for others. Each one is a trade-off.
>
> **We are greedy: we want more.**

[Why we created Julia](https://julialang.org/blog/2012/02/why-we-created-julia/) -- Jeff Bezanson, Stefan Karpinski, Viral B. Shah, and Alan Edelman
"""

# ╔═╡ 8b838733-2ec0-42f0-9dab-48bb24514173
md"""
- What is a **good scientific programming** language:
    - Fast and scalable
    - Interactive
    - Easy to write and read
    - Vanishing division between users and developers



- All existing languages feature various tradeoffs along these points:
    - State of the art: Mix of programming languages (e.g. FORTRAN & Python)



- **Two language problem:**
    - Where do you cut?
    - When do you go from the prototyping language to the production language?
    - Dealbreaker for iterative performance improvements
    - Extra barrier for newcomers (learning one language is hard, now try two or more)


- **People compose if software composes**
"""

# ╔═╡ 6731e40b-4e51-46de-ae45-c9929b476059
md"""## Arrays and linear algebra

One of the striking features about Julia is the built-in support for arbitrary-dimensional arrays, linear algebra and the seamless and efficient integration with user code.
"""

# ╔═╡ 1501bc67-642a-48bc-b355-e6513d379682
α = [1, 2, 3]

# ╔═╡ 049f14d5-18e3-4493-b618-03a42fffe35d
β = randn(3)

# ╔═╡ dd7d5b1c-1658-4fd7-aa83-2723c7043862
α + 2β  # Just add

# ╔═╡ 6866b606-76ed-44e3-87d8-62e4c3bc9a01
α .* β  # vectorised operations

# ╔═╡ dfc8d867-ce37-41a0-b5c2-2bc44adac8e9
md"So far so obvious, but we can also apply custom functions in a vectorised way:"

# ╔═╡ 724e0d7b-1445-4772-ae3e-673f4122f740
myfun(x) = x^2 + 2

# ╔═╡ 9e83f0e2-322d-4500-bf74-88603ef00b33
myfun.(cos.(α))

# ╔═╡ d647f360-c00c-4970-b735-954b76d7f584
md"**Loops are actually fast:**"

# ╔═╡ bdd7caff-3b42-4487-a89f-33638ff4059f
function myadd!(C, A, B)
	for i in eachindex(A, B)
		C[i] = A[i] + B[i]
	end
end

# ╔═╡ 734c3156-a510-4116-b129-67532dfb70c1
let
	n = 100
	A = rand(n, n)
	B = rand(n, n)
	C = zeros(n, n)
	@btime $C .= $A .+ $B
	@btime myadd!($C, $A, $B)
end;

# ╔═╡ 995e081d-d74c-44db-8d02-e55bb69b44cc
md"**Standard linear algebra** is just available, e.g. transposition, diagonalisation:"

# ╔═╡ d9e3a5bc-6665-4b37-99f9-a1c4e5607a9b
begin
	C = randn(3, 3)
	C = C + C'
end

# ╔═╡ 6972cc50-ab7a-4d2a-84b7-aa5231762061
eigen(C)

# ╔═╡ 1d2eae8d-2bb9-4756-894a-5511ed67049d
md"But crucially we have many variants, that (depending on the type) automatically do the right thing:"

# ╔═╡ 5e24f293-b814-40f9-afcd-597230909e28
eigen(Symmetric(C))

# ╔═╡ 9c72c38a-37d7-42a8-8b8a-8d1488c7f332
eigen(Diagonal([1., 2., 3.]))

# ╔═╡ 03b972e2-bf97-49d8-8839-389e335097f6
@which eigen(C)

# ╔═╡ 513beaeb-af44-42a8-827f-a8dafa486b03
@which eigen(Symmetric(C))

# ╔═╡ 88bb95eb-1112-4455-8483-094ba7f1d5be
length(methods(eigen))

# ╔═╡ 23b99435-30df-4c1d-8375-00377f694810
md"""## Reactive notebooks"""

# ╔═╡ c54bb3ca-1ce9-44a2-9fe4-42dac9ad2a01
md"Interactivity is easy ..."

# ╔═╡ 29fa6317-a637-47c4-97cd-b1494ba21517
@bind ω PlutoUI.Slider(1:0.1:10; show_value=true, default=2.3)

# ╔═╡ 12ade949-c5a5-4bff-a1e7-b7a7e191c52a
let
	t = collect(0:0.01:2π)
	Plots.plot(sin.(ω .* t); label=nothing)
end

# ╔═╡ edc6bc46-4bfa-4052-83b8-57e8682c10ae
md"Debug step by step ..."

# ╔═╡ 1d113418-bc56-4bc0-bd28-30b888462cf6
@test 2 + (2^2) in [i ÷ 2 for i in 10:20]

# ╔═╡ f2f2196b-28d1-4f5f-92f1-7e082a2acd72
md"""## Excellent package management"""

# ╔═╡ c4781847-cb21-4ce4-b4fc-305492fd9143
Resource("https://imgs.xkcd.com/comics/python_environment.png")

# ╔═╡ 602b0f1c-bc9a-40c7-88f7-f443af42913a
md"""
- The package manager comes with the language
- Creates reproducible environments
- Handles dependencies and versions properly
- Unit testing and documentation easy to set up
- Includes management of **3rd party binary dependencies**
- E.g. we can just
"""

# ╔═╡ 0e53bfaa-e7d1-4452-8d47-896224eecccf
QuantumEspresso_jll.pwscf() do pwscf
	withenv("ESPRESSO_PSEUDO" => joinpath(@__DIR__, "pseudos")) do 
		@time run(`$pwscf -in Fe_afm.pwi`)
	end
end

# ╔═╡ 1b9e45b3-e2b3-43c2-9795-839c6156d975
md"""
## Seamless combination with existing codes

- Combining **C and Julia** works both ways (see examples above & below)
  * Note: This includes package management
- Same is true for **Python and Julia**
  * Again includes package management, see [**PythonCall**](https://cjdoris.github.io/PythonCall.jl/stable/), [**CondaPkg**](https://github.com/cjdoris/CondaPkg.jl) and [**JuliaCall**](https://cjdoris.github.io/PythonCall.jl/stable/) for details.
- Other supported languages include R and Java.
"""

# ╔═╡ 676aa47c-c36d-46f5-bc44-50a34303b3e7
md"""
## Julia's speed: Sums
"""

# ╔═╡ 3ecca9bd-bc70-460a-a795-28c4a6275ab9
md"""
Now, we would like to answer the question: *is Julia fast?*

But to some extent this is a misleading question, since one can write slow code in any language... so let's try to answer something else instead: *can Julia be fast?*

Our example will be to sum up the entries in a vector. Written in plain Julia just like:
"""

# ╔═╡ 5753e3a3-2499-4b9c-9c69-a6d400ea64db
function mysum(v)
    result = zero(eltype(v))
    for i in 1:length(v)
        result += v[i]
    end
    result
end

# ╔═╡ cba77bab-1d67-4557-8b95-9466f0be59ba
begin
	vlarge = randn(10^7)   # Large vector of numbers
	times = Dict()         # Collection for timings
end;

# ╔═╡ 2d1e09dd-0348-4ac8-a663-0b799ec38162
md"""
First we do it in **plain C**.
"""

# ╔═╡ b0813754-1a15-4c46-993e-8882b9814a7d
begin
	# Compile some C code and call it from julia ...
	code = """
	#include <stddef.h>
	double c_sum(size_t n, double *v) {
   		double accu = 0.0;
    	for (size_t i = 0; i < n; ++i) {
        	accu += v[i];
    	}
    	return accu;
	}
	"""

	# Compile to a shared library (with fast maths and machine-specific)
	const Clib = tempname()
	open(`gcc -fPIC -O3 -march=native -xc -shared -ffast-math -o $(Clib * "."  * Libdl.dlext) -`, "w") do f
    	print(f, code) 
	end

	# define a Julia function that calls the C function:
	c_sum(v::Array{Float64}) = ccall(("c_sum", Clib), Float64, (Csize_t, Ptr{Float64}), length(v), v)
end

# ╔═╡ 0d935958-7f45-49ba-b630-0334b027a0c1
let
	bench = @benchmark c_sum($vlarge)
	times["C (naive)"] = minimum(bench.times) / 1e6
	bench
end

# ╔═╡ 8c6804cf-5b9a-4793-8f2d-fdf35ad720ee
md"""
Now we try an explicitly and **fully vectorised C code**:
"""

# ╔═╡ 86c21556-ff39-4a2d-a5c6-251d74e60c5d
begin
	code_vectorised = """
	#include <stddef.h>
	#include <immintrin.h>
	
	double c_sum_vector(size_t n, double *v)
	{
	    size_t i;
	    double result;
	    double tmp[4] __attribute__ ((aligned(64)));
	
	    __m256d sums1 = _mm256_setzero_pd();
	    __m256d sums2 = _mm256_setzero_pd();
	    for ( i = 0; i + 7 < n; i += 8 )
	    {
	        sums1 = _mm256_add_pd( sums1, _mm256_loadu_pd(v+i  ) );
	        sums2 = _mm256_add_pd( sums2, _mm256_loadu_pd(v+i+4) );
	    }
	    _mm256_store_pd( tmp, _mm256_add_pd(sums1, sums2) );
	
	    return tmp[0] + tmp[1] + tmp[2] + tmp[3]; 
	}
	"""
	
	# Compile to a shared library (with fast maths and machine-specific)
	const Clib_vectorised = tempname()
	open(`gcc -fPIC -O2 -march=native -xc -shared -ffast-math -o $(Clib_vectorised * "." * Libdl.dlext) -`, "w") do f
	    print(f, code_vectorised) 
	end
	
	# define a Julia function that calls the C function:
	c_sum_vectorised(v::Array{Float64}) = ccall(("c_sum_vector", Clib_vectorised), Float64, (Csize_t, Ptr{Float64}), length(v), v)
end

# ╔═╡ acff670e-5011-4212-8f52-95c1b5477d5d
let
	bench = @benchmark c_sum_vectorised($vlarge)
	times["C (vectorised)"] = minimum(bench.times) / 1e6
	bench
end

# ╔═╡ 9b1329aa-4197-4949-bbb9-42a2275c7c2b
md"""
So how does the **Julia version** do?
"""

# ╔═╡ a86eb565-ff94-440f-8ef7-632a9dcd6c82
let
	bench = @benchmark mysum($vlarge)
	times["Julia (naive)"] = minimum(bench.times) / 1e6
	bench
end

# ╔═╡ 51e1c815-bc3b-4373-a236-c3e7df5d46de
md"""
A bit disappointing... but unlike vectorised C we have not yet tried all tricks! Let's try an **optimised Julia version**.
"""

# ╔═╡ 36d82a73-b463-421f-aac3-7e2ce6905563
function fastsum(v)
    result = zero(eltype(v))
    @simd for i in 1:length(v)    # @simd => instruction vectorisation in the loop
        @inbounds result += v[i]  # @inbounds => suppresses bound checks
    end
    result
end;

# ╔═╡ eef56f3f-4a45-4e03-a173-9dcaa605ab75
let
	bench = @benchmark fastsum($vlarge)
	times["Julia (simd)"] = minimum(bench.times) / 1e6
	bench
end

# ╔═╡ 8c347867-f139-4c3a-b454-29d34f0a1be2
md"""
Notice, how this is still nicely readable code!
"""

# ╔═╡ b4245307-e080-4923-b540-b9b3288cfae4
md"""
So ... how does **python** perform in this comparison ?
"""

# ╔═╡ 449130fa-6a73-42a7-93b9-9adbca17e37e
let
	bench = @benchmark ($(np.sum))($vlarge)
	times["Python (numpy)"] = minimum(bench.times) / 1e6
	bench
end

# ╔═╡ f0fdd605-6f29-4f5b-b9bd-d03554273854
md"""
But numpy is not real Python, **that's C in disguise**.
The naive version (like the one we wrote in Julia) would look something like this:
"""

# ╔═╡ 69806ad3-ce38-4d4f-984d-5985cfbb408e
begin
	pycode = """def pysum(A):
					s = 0.0
					for a in A:
						s += a
					return s
	"""
	pyexec(strip(pycode), Main)
	pysum(v) = pyeval("pysum(v)", Main, (; v))
end

# ╔═╡ 2f095034-9e3d-4f27-97f4-96cbeadbe681
let
	bench = @benchmark ($pysum)($vlarge)
	times["Python (naive)"] = minimum(bench.times) / 1e6
	bench
end

# ╔═╡ 0f6c85be-7d32-4e0c-b085-a04227570a46
md"""
In summary:
"""

# ╔═╡ b6a35fc7-c2f9-4d39-baf8-10d8a94996d6
let
	for k in sort(collect(keys(times)))
		@printf "% 15s => %9.2f ms\n" k times[k]
	end
end

# ╔═╡ 74b4e0e0-bae5-48e0-9746-f77a70988015
md"**Conclusion:** Julia is on the same speed level as C."

# ╔═╡ 8c4d9da1-62ca-4d57-9583-59341ab06d52
md"""
## Composable software from generic code

We just produced the following sum function, which was basically en par in speed with C or numpy:

```julia
function fastsum(v)
    result = zero(eltype(v))
    @simd for i in 1:length(v)    # @simd => instruction vectorisation in the loop
        @inbounds result += v[i]  # @inbounds => suppresses bound checks
    end
    result
end
```
But unlike a C implementation, we are not at all restricted to using a particular data type ... and this let's us do crazy things **even though the code is equally fast**:
"""

# ╔═╡ e2d26a9d-be50-49d9-870f-d709cccadcbb
md"""
(a) **Elevated precision** ... let's consider a nasty case:

Some numbers designed to **sum to 1**:
"""

# ╔═╡ af10e42c-bd61-4acd-ba80-0798c3fbbd3b
function generate(N)
    x = randn(N) .* exp.(10 .* randn(N))
    x = [x; -x; 1.0]
    x[sortperm(rand(length(x)))]
end

# ╔═╡ 72ec7c6a-5189-446f-b00e-b15761a205db
numbers = generate(10000);

# ╔═╡ 31a31c46-56a8-4f94-b19e-54d3ba9a319e
fastsum(numbers)

# ╔═╡ e6b5f14c-d254-4515-aed2-5a37e391e4bc
md"""... looks wrong, let's try elevated precision:"""

# ╔═╡ 8db81000-c957-4765-b3af-fa7408964b88
fastsum(big.(numbers))

# ╔═╡ 43ad5619-9b7a-4b61-bb46-b2d484c58ddd
md"""(b) **Tracking numerical error**"""

# ╔═╡ f126ecce-d639-4971-873b-aa0f99fd71c8
begin
	fastsum(interval.(numbers))
end

# ╔═╡ b972089c-0990-4917-89b9-5ad790bcdbb9
md"""... ups clearly something wrong here! But now we know."""

# ╔═╡ 2a3dfa03-f4bd-47bf-b70e-5cd00e8402a5
md"""(c) **Error propagation**"""

# ╔═╡ 41b6c787-32f1-42a6-9dab-58fd23a5df7e
begin
	vmeas = [1 ± 1e-10, 1.1 ± 1e-12, 1.2 ± 1e-10]
	fastsum(vmeas)
end

# ╔═╡ 96e78c8e-7f85-4e16-9865-70fea98cfe6c
md"""
(d) **Unlocking new features**

Let us solve:

$$\frac{du(t)}{dt} = -cu(t)$$
"""

# ╔═╡ 0443c94a-0822-4b5d-9bc3-42371d2c6454
md"Use measurement error: $(@bind use_measurement_error CheckBox())"

# ╔═╡ ccd03c3a-82b9-430b-b818-58794efb4710
if use_measurement_error
	c     = 5.730 ± 2  # Half-life of Carbon-14 is 5730 years.
	u0    = 1.0  ± 0.1
else
	c  = 5.730
	u0 = 1.0
end;

# ╔═╡ 30cbc826-cece-4fcf-871b-eae7bbe8838f
md"""Selected:
  - **`use_measurement_error` = $use_measurement_error**
  - half-life is **c = $c**
  - initial population is **u0 = $u0**.
"""

# ╔═╡ a99231cd-0e8b-48ee-be0e-0613933afe7f
begin
	# Define the problem
	radioactivedecay(u, p, t) = -c*u
	
	# Pass to solver
	tspan = (0.0, 1.0)
	prob  = ODEProblem(radioactivedecay, u0, tspan)
	sol   = solve(prob, Tsit5(), reltol=1e-8, abstol=1e-8);
	
	plot(sol.t, sol.u, ylabel="u(t)", xlabel="t", lw=2, legend=false)
end

# ╔═╡ 9d639f9d-9396-4db3-9c14-45ace90883aa
md"""Now what if we have **measurement errors?** Just tick the box above and see ..."""

# ╔═╡ 3f3566f9-ec2b-4aa2-9e7e-3f61e37af8d6
md"""
Note that, in some sense, **Julia implemented that feature by itself**.

The authors of Measurements.jl and DifferentialEquations.jl never had any collaboration on this.

It **just works**.
"""

# ╔═╡ 90a6efec-b2f0-4220-8927-c5810f80e0f0
md"**Conclusion:** Julia's language design gives enormous flexibility with little effort."

# ╔═╡ 5665c6e6-68de-4811-bcf2-695710362421
md"""## Powerful HPC abstractions"""

# ╔═╡ 896e240a-ad02-42f3-a76e-a9615df1f2e4
LocalResource("julia_hpc.png")

# ╔═╡ 2b4baa3b-403b-4430-91fc-8abc4ece1693
md"""We consider the simple Davidson algorithm (iterative eigensolver), which can be implemented quite concisely:"""

# ╔═╡ 30f46bb0-e248-471d-b413-578d2b3c62d2
qrortho(X::AT) where {AT <: AbstractArray} = AT(qr(X).Q)

# ╔═╡ b849f68d-1d2d-4ddf-bf4a-d5c62697e37b
function rayleigh_ritz(X, AX, N)
    F = eigen(Hermitian(X' * AX))
    F.values[1:N], F.vectors[:, 1:N]
end

# ╔═╡ fabc2507-2ef2-4bc5-974f-fa542cbdac03
function davidson(A, SS::AbstractArray; tol=1e-5, maxsubspace=8size(SS, 2), verbose=true)
    m = size(SS, 2)
    for i in 1:100
        Ass = A * SS
        rvals, rvecs = rayleigh_ritz(SS, Ass, m)
        Ax = Ass * rvecs

        R = Ax - SS * rvecs * Diagonal(rvals)
        if norm(R) < tol
            return rvals, SS * rvecs
        end

        verbose && println(i, "  ", size(SS, 2), "  ", norm(R))

        # Use QR to orthogonalise the subspace.
        if size(SS, 2) + m > maxsubspace
            SS = qrortho([SS*rvecs R])
        else
            SS = qrortho([SS       R])
        end
    end
    error("not converged.")
end

# ╔═╡ 2be08ff3-d1ef-4acd-a6b0-849f83d06c28
begin
	nev = 2
	A = randn(50, 50)
	A = A + A' + 5I
end;

# ╔═╡ fc5adbc2-95bb-4fee-be6b-5e6f94b7a411
begin
	# Generate two random orthogonal guess vectors
	x0 = qrortho(randn(size(A, 2), nev))
	
	# Run the problem
	davidson(A, x0)
end

# ╔═╡ e39eb6f7-87dc-4aad-8d61-48ec57be76cb
begin
	# Mixed precision!
	λ, v = davidson(Matrix{Float32}(A), Float32.(x0), tol=1e-3)
	println()
	λ, v = davidson(Matrix{Float64}(A), v, tol=1e-13)
	println()
	λ, v = davidson(Matrix{BigFloat}(A), v, tol=1e-20)
	λ
end

# ╔═╡ bf9071d7-5e02-4b6d-8a4c-832186a27d4d
begin
	spA = sprandn(100, 100, 0.2)
	spA = spA + spA' + 10I
	
	spA
end

# ╔═╡ 84ee7a04-16fc-420f-b64d-4598254a0dd3
begin
	spx0 = randn(size(spA, 2), nev)
	spx0 = Array(qr(spx0).Q)

	davidson(spA, spx0, tol=1e-4)
end

# ╔═╡ 5d167ac0-0387-4189-828d-5a01b7bf11c9
md"""Let's run with GPUs:"""

# ╔═╡ da419772-c2d1-483f-b31a-a9839f607570
if CUDA.has_cuda_gpu()
	davidson(CuArray(A), CuArray(x0))
end

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
BenchmarkTools = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf"
CUDA = "052768ef-5323-5732-b1bb-66c8b64840ba"
CondaPkg = "992eb4ea-22a4-4c89-a5bb-47a3300528ab"
DifferentialEquations = "0c46a032-eb83-5123-abaf-570d42b7fbaa"
GenericLinearAlgebra = "14197337-ba66-59df-a3e3-ca00e7dcff7a"
IntervalArithmetic = "d1acc4aa-44c8-5952-acd4-ba5d80a2a253"
Libdl = "8f399da3-3557-5675-b5ff-fb832c97cbdb"
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
Measurements = "eff96d63-e80a-5855-80a2-b1b0885c5ab7"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
PlutoTeachingTools = "661c6b06-c737-4d37-b85c-46df65de6f69"
PlutoTest = "cb4044da-4d16-4ffa-a6a3-8cad7f73ebdc"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
Printf = "de0858da-6303-5e67-8744-51eddeeeb8d7"
PythonCall = "6099a3de-0909-46bc-b1f4-468b9a2dfc0d"
QuantumEspresso_jll = "74603b90-2fcf-5710-a7d7-830b31b8b33c"
ShortCodes = "f62ebe17-55c5-4640-972f-b59c0dd11ccf"
SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"

[compat]
BenchmarkTools = "~1.3.2"
CUDA = "~4.2.0"
CondaPkg = "~0.2.18"
DifferentialEquations = "~7.7.0"
GenericLinearAlgebra = "~0.3.10"
IntervalArithmetic = "~0.20.8"
Measurements = "~2.9.0"
Plots = "~1.38.12"
PlutoTeachingTools = "~0.2.11"
PlutoTest = "~0.2.2"
PlutoUI = "~0.7.51"
PythonCall = "~0.9.13"
QuantumEspresso_jll = "~7.0.1"
ShortCodes = "~0.3.5"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised

julia_version = "1.9.0"
manifest_format = "2.0"
project_hash = "e348ded55e38dbebbfc283252ab619fcca659915"

[[deps.ADTypes]]
git-tree-sha1 = "dcfdf328328f2645531c4ddebf841228aef74130"
uuid = "47edcb42-4c32-4615-8424-f2b9edc5f35b"
version = "0.1.3"

[[deps.AbstractFFTs]]
deps = ["LinearAlgebra"]
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uuid = "621f4979-c628-5d54-868e-fcf4e3e8185c"
version = "1.3.1"
weakdeps = ["ChainRulesCore"]

    [deps.AbstractFFTs.extensions]
    AbstractFFTsChainRulesCoreExt = "ChainRulesCore"

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

[[deps.Adapt]]
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weakdeps = ["StaticArrays"]

    [deps.Adapt.extensions]
    AdaptStaticArraysExt = "StaticArrays"

[[deps.ArgTools]]
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version = "1.1.1"

[[deps.ArnoldiMethod]]
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git-tree-sha1 = "62e51b39331de8911e4a7ff6f5aaf38a5f4cc0ae"
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[[deps.ArrayInterface]]
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    ArrayInterfaceBlockBandedMatricesExt = "BlockBandedMatrices"
    ArrayInterfaceCUDAExt = "CUDA"
    ArrayInterfaceGPUArraysCoreExt = "GPUArraysCore"
    ArrayInterfaceStaticArraysCoreExt = "StaticArraysCore"
    ArrayInterfaceTrackerExt = "Tracker"

    [deps.ArrayInterface.weakdeps]
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    BlockBandedMatrices = "ffab5731-97b5-5995-9138-79e8c1846df0"
    CUDA = "052768ef-5323-5732-b1bb-66c8b64840ba"
    GPUArraysCore = "46192b85-c4d5-4398-a991-12ede77f4527"
    StaticArraysCore = "1e83bf80-4336-4d27-bf5d-d5a4f845583c"
    Tracker = "9f7883ad-71c0-57eb-9f7f-b5c9e6d3789c"

[[deps.ArrayInterfaceCore]]
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[[deps.ArrayLayouts]]
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[[deps.Artifacts]]
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[[deps.Atomix]]
deps = ["UnsafeAtomics"]
git-tree-sha1 = "c06a868224ecba914baa6942988e2f2aade419be"
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[[deps.BFloat16s]]
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git-tree-sha1 = "dbf84058d0a8cbbadee18d25cf606934b22d7c66"
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[[deps.BandedMatrices]]
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git-tree-sha1 = "6ef8fc1d77b60f41041d59ce61ef9eb41ed97a83"
uuid = "aae01518-5342-5314-be14-df237901396f"
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[[deps.Base64]]
uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f"

[[deps.BenchmarkTools]]
deps = ["JSON", "Logging", "Printf", "Profile", "Statistics", "UUIDs"]
git-tree-sha1 = "d9a9701b899b30332bbcb3e1679c41cce81fb0e8"
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[[deps.BitFlags]]
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[[deps.BitTwiddlingConvenienceFunctions]]
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git-tree-sha1 = "0c5f81f47bbbcf4aea7b2959135713459170798b"
uuid = "62783981-4cbd-42fc-bca8-16325de8dc4b"
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[[deps.BoundaryValueDiffEq]]
deps = ["BandedMatrices", "DiffEqBase", "FiniteDiff", "ForwardDiff", "LinearAlgebra", "NLsolve", "Reexport", "SciMLBase", "SparseArrays"]
git-tree-sha1 = "ed8e837bfb3d1e3157022c9636ec1c722b637318"
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[[deps.Bzip2_jll]]
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git-tree-sha1 = "19a35467a82e236ff51bc17a3a44b69ef35185a2"
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[[deps.CEnum]]
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[[deps.CPUSummary]]
deps = ["CpuId", "IfElse", "Static"]
git-tree-sha1 = "2c144ddb46b552f72d7eafe7cc2f50746e41ea21"
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[[deps.CRlibm]]
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[[deps.CRlibm_jll]]
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git-tree-sha1 = "e329286945d0cfc04456972ea732551869af1cfc"
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[[deps.CUDA]]
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git-tree-sha1 = "280893f920654ebfaaaa1999fbd975689051f890"
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[[deps.CUDA_Driver_jll]]
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git-tree-sha1 = "498f45593f6ddc0adff64a9310bb6710e851781b"
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    DiffEqBaseMonteCarloMeasurementsExt = "MonteCarloMeasurements"
    DiffEqBaseReverseDiffExt = "ReverseDiff"
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    DiffEqBaseUnitfulExt = "Unitful"
    DiffEqBaseZygoteExt = "Zygote"

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[[deps.DiffEqCallbacks]]
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[[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.ZygoteRules]]
deps = ["ChainRulesCore", "MacroTools"]
git-tree-sha1 = "977aed5d006b840e2e40c0b48984f7463109046d"
uuid = "700de1a5-db45-46bc-99cf-38207098b444"
version = "0.2.3"

[[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"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.7.0+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", "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.micromamba_jll]]
deps = ["Artifacts", "JLLWrappers", "LazyArtifacts", "Libdl"]
git-tree-sha1 = "fc7a2e6bb2a48e08ec739fd1ea855203a3f568dc"
uuid = "f8abcde7-e9b7-5caa-b8af-a437887ae8e4"
version = "1.4.3+0"

[[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"
"""

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